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Obstruction theory Ng, Tze Beng 1973

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f OBSTRUCTION THEORY by TZE-BENG NG B.S c , U n i v e r s i t y of Warwick, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept t h i s t h e s i s as conforming to the required standard. , THE UNIVERSITY OF BRITISH COLUMBIA March, 1973. In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the iquirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columb.^,, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r ext e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . It i s understood that copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Tze-Beng Ng Department of Mathematics The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date 19th'i March 1973 .ABSTRACT The aim of this dissertation at the outset is to give a survey of obstruction theories after Steenrod and to describe the various techniques employed by different researchers, the intricate perhaps subtle relation from one technique to another. Owing to the d i f f i c u l t y in computing higher co-homology operations, one i s led naturally to K-theory and the Eilenberg-Moore spectral sequence. However, these and other recent developments especially those in the study of stable Postnikov systems go beyond the intention of this modest survey. i i i TABLE OF .CONTENTS Chapter'1. Introduction. 1 Chapter 2. F i b e r Spaces. 2.1. D e f i n i t i o n s and Notations. 10 2.2. P r i n c i p a l F i b e r Spaces. 13 2.3. Transgression i n Fiber Spaces. 17 2.4. On Decomposition of F i b r a t i o n and L i f t i n g Problem. 22 2.5. G-spaces. 26 Chapter 3. C l a s s i c a l Obstruction Theory. 3.1. Obstruction to Extension. 30 3.2. Primary Obstruction. 33 3.3. Extension Theorems and C l a s s i f i c a t i o n of Maps. 35 Chapter 4. Global Obstruction. 4.1. D e f i n i t i o n s of Homotopyand Homology Obstructions. 40 4.2. Generalized C e l l Complexes and Obstructions. 45 4.3. Extension Theorems. 51 4.4. n-systems. 53 4.5. Secondary Obstruction and Construction of a Postnikov System. 55 4.6. A Postnikov Decomposition. • 57 4.7. L o c a l i s a t i o n . 61 4.8. Secondary Difference Obstruction. 62 Chapter 5. P r i n c i p a l F i b e r Bundles. 5.1. D e f i n i t i o n s . 66 5.2. P r e l i m i n a r i e s . 67 5.3. Obstructions f o r P a i r o f Fi b e r Spaces 69 5.4. A p p l i c a t i o n s . 72 5.5. Decomposition of P r i n c i p a l F iber Bundles and Moore-Postnikov Invariants. 74 5.6. The Main Theorems. 79 5.7. Obstruction and C h a r a c t e r i s t i c Classes 81 5.8. Conclusion. 91 Chapter 6. Modified Postnikov Towers and Obstructions to L i f t i n g s . 6.1. Modified Postnikov Towers. 94 6.2. Construction of Induced Maps between Postnikov Systems. 98 6.3. The k-i n v a r i a n t s f o r a L i f t i n g . 104 6.4. Obstruction Theory f o r Orientable Fiber Bundles. 112 6.5. An I l l u s t r a t i o n . 116 6.6. Some Examples o f the Use of Postnikov Towers. 121 Bibliography. 125 ACKNOWLEDGMENT I must thank P r o f e s s o r Denis Sjerve f o r h i s guidance, f o r the many e n l i g h t e n i n g conversations and h e l p f u l suggestions, f o r b r i n g i n g t o my a t t e n t i o n [50] and [51], very much so f o r making some o f the d e c i s i o n s i n c u t t i n g down the s i z e o f t h i s t h e s i s and f o r read i n g the f i n a l d r a f t . I must a l s o thank P r o f e s s o r U. Suter f o r reading t h i s t h e s i s . F i n a l l y I am pleased t o express my g r a t i t u d e to the Mathematics Department f o r generously provided a teaching a s s i s t a n t s h i p and a Summer resea r c h grant w h i l e I am a student o f Maths, a t the U n i v e r s i t y o f B r i t i s h Columbia. Tze-Beng Ng March, 1973. CHAPTER 1. INTRODUCTION One of the ol d e s t problems i n (algebraic) topology i s to f i n d l i f t i n g s f o r a map a: K KB to E, the domain of another map £: E — —> B. An even more i n t e r e s t i n g problem i s the number of non-homotopic l i f t i n g s of a, and hence to determine algebraic (problem) i n v a r i a n t s f o r a homotopy c l a s s i f i c a t i o n of such l i f t i n g s . Suppose we cannot l i f t a, assuming that we do not know t h i s i n advance, can we give an " i n v a r i a n t " to confirm the impossible? In many cases we can do so and such an i n v a r i a n t i s branded with the term obstr u c t i o n (to l i f t i n g ) . Suppose we can l i f t a, i . e . we have the following commutative diagram: where 8: K »• E i s a l i f t i n g of a i . e . Cog ~ a . Then we know the following diagram: H n(E) -> H (B) H (K) commutes, we may therefore look at the following algebraic problem: H (E) n n H n(K) ->Hn(B) When can we f i n d {<}> : H (K) > H (E)} such that Y n n n £. od> = f o r a l l n ? * n * (1) I f so, when i s {<)> ) induced by a map 3 : K * E ? (2) Do {<j>n} commute with boundary operator ? (3) Do {<j) } commute with the change of c o e f f i c i e n t operator ? n Consider the following diagram: H (E)®Z sr n p H (K) ® Z n p : y H (B) ® Z a ® 1 n p obtained by tensoring the preceding diagram with the integer mod p, Z (p a p o s i t i v e prime). And we consider l i f t i n g s of t h i s problem. Again the same question can be asked: When i s $ representable as some <j>®l The advantage of looking at t h i s problem i s i t s e f f e c t i v e n e s s . J u s t looking at the homology i s not enough as some simple examples w i l l show. So we increase the structure by looking at the cohomology r i n g : 5 (E) y s y * ^ * H (K) •< - H (B) a * * When can we f i n d homomorphism 6 : H (E) >-.H (K) such that 6o£ = a ? The cohomology r i n g i s more i n t e r e s t i n g because of i t s st r u c t u r e . Thus 9 must be a r i n g homomorphism. We may strengthen the question by r e q u i r i n g 6 to commute with c o e f f i c i e n t operators, with the Poincare d u a l i t y operator i n case E and K are Poincare complexes, with the * boundary operator 6 and with the Bockstein operator. But we know we have cohomology operations and we can regard cohomology as modules over the Steenrod mod p algebra Gl(p), f o r example. Our next requirement on 6 w i l l be that i t should commute with a l l cohomology operations. In addition to r e q u i r i n g 6 be an Ol(p)-module homomorphism, we may require * that i t should also be an OUp)(H (B;Z ))-module homomorphism, where TP * * * the module structures on H (E;Z^) and H (K'"Zp) a r e induced by 5 and * * a r e s p e c t i v e l y , and Gtfp)(H (B;Z )) i s the semi-tensor product algebra XP which i s equal to H (B;Z ) ®Ol(p) as a vector space and with P m u l t i p l i c a t i o n as follows: * For v«>a, u® 0 e H (B,Z ) ® O U P ) / (V®a)«(u®0) = I (-1) 1 ( v a u)«aJB where i|>(a) = £ a , © a ' . and i i>: CH(p) -> Ol(p)®0l(p) i s the diagonal map. Once again we have the question: When i s the module homomorphism induced by a map 0: K •> E ? In another d i r e c t i o n one can consider the homotopy l i f t i n g problem, that i s , when can we f i n d a homomorphism g: TT a (K) • "%(E) making the following diagram: * * ( K ) , * ( B ) # commutative i . e . £„oB = a„ ? When i s 8 induced by a map K > E ? In # tt general we cannot t e l l . But i f £: E -*- B i s a covering space then we have the following theorem: LIFTING THEOREM [24, pp.89]. Suppose K i s connected and l o c a l l y pathwise connected. Then a can be l i f t e d to a map 8: K >• E i f f the following algebraic problem: if. (E) pf 1 f l ( K ) o — " V T r i ( B ) ft i s solvable, i . e . i f f there e x i s t homomorphism (3 s a t i s f y i n g K^o$ = ct^, Consider the following s i t u a t i o n : K B »-a with a "V a', i . e . a. = a' £ [K, B] , where [K, B] denotes the homotopy classes of maps from K to B. We want some conditions to ensure that the l i f t a b i l i t y of a depends only on i t s homotopy cla s s a e [K, B]. In case £ i s a f i b e r space we have the following: COVERING HOMOTOPY THEOREM. Let £=(E, £, B, F) be a f i b e r space , X l o c a l l y compact and paracompact, and A closed i n X. Suppose f : X — s- E i s a f i x e d given map and h: Xxi > B i s a homotopy of a = £of: X • B s a t i s f y i n g h|xx{0} = gof. Suppose furt h e r that there i s a " p a r t i a l l i f t i n g " of h on A, i . e . a map h': Axiuxx{0} *• E s a t i s f y i n g £oh' = h|AxiWxx{0> and h'|x*{0} = f. Then there e x i s t a homotopy of f , h: Xxi >- E s a t i s f y i n g (1) Koh = h , (2) h|xx{0} = f , and (3) h|Ax!WXx{0} = h". Now, since we know that given any map £: E >• B we can convert i t i n t o a map E" >-B , which i s a Hurewicz f i b r a t i o n and where E i s a deformation r e t r a c t of E ' (see f o r example [44, pp.84]), we may assume that £ i s a f i b r a t i o n r i g h t from the beginning. Dual to the l i f t i n g problem we have the extension problem: When can we f i n d a map 8: E > K such that the following diagram, E _ > K i s commutative, i . e . p|x = a ? In general the answer i s not always "yes". There are some obvious counter examples. In case 8 e x i s t s we want some conditions to ensure that a l l o' ^ a be extendable over E. In t h i s d i r e c t i o n we have the well known homotopy extension theorem of Borsuk: -HOMOTOPY EXTENSION THEOREM.' Let g : E > K be a map, X closed i n E and 3|X = a. Suppose one of the following conditions i s s a t i s f i e d (1) E and X are t r i a n g u l a b l e . (2) K i s t r i a n g u l a b l e . (3) K i s an ANR. (4) (E, X) i s an ANR p a i r . Then any homotopy h of a can be extended to a homotopy of 8 . ( see [20] or [24].) We now have the following a l g e b r a i c problem: When can we f i n d 4>: H #(E) • H^(K) s a t i s f y i n g 4>oi# = ? H A(E) H A (X) - H A(K) When i s <J> induced by a map E • K ? Again some easy examples w i l l show that looking at the homology i s not enough so we consider the corresponding problem i n cohomology: * H (E) * v. i N-H (X) «- - * H. (K) When can we f i n d $: H (K) & it it -*• H (E) such that i o$ = 0. ? When i s i t induced by a map E > K ? Since we know H (E) can be considered as a module over the algebra of cohomology operations, we require $ to' commute with cohomology operations, the boundary operator and the Bockstein operator, i n a d d i t i o n to $ being a r i n g homomorphism. We have a broad and so p h i s t i c a t e d spectrum of cohomology theories, which a l l together make t h i s problem i n t e r e s t i n g and at the same time re v e a l i n g the d i f f i c u l t i e s involved i n t r y i n g to obtain even a p a r t i a l s o l u t i o n . We may consider the corresponding problem i n K-theory: K*(E) ,1 v -Kff(X) *• K (K) We require that $: K (K) >• K (E) , i f i t e x i s t s , should corranute with the Adams operations t/j . Suppose we consider i n the category of G-spaces,.where G i s a t o p o l o g i c a l group, a diagram . E X s or -> B B and seek e i t h e r an equivariant l i f t i n g or an equivariant extension. Then take the corresponding KG-theory problem: K*(E) K ! (E ) 0 K M X ) <-or K#(B) K G ( B ) «-seeking i n each case 6 or $ such that they should commute with the ij> I f we consider the KU-theory, we can then e x p l o i t our e x i s t i n g knowledge of the theory o f c h a r a c t e r i s t i c classes i n the d i r e c t i o n of giving a negative answer to the problem. (This i s not the only way i n which the theory of c h a r a c t e r i s t i c classes can be exploited). In some cases t h i s has been done s u c c e s s f u l l y without using K-theory, (see [28] and [40]) . In summary, l e t F: "\7 *• <o be a functor from some s u i t a b l e category of spaces (e.g. * 7 = Top, CW-complexes, etc.) to an abelian category G (e.g. category of graded abelian groups, category of Gl(p)-modules, e t c . ) , then F takes the geometric problem i n t o one which i s algebraic. This gives an obstruction. In p a r t i c u l a r , F can be any cohomology theory. This d i s s e r t a t i o n can be said to have grown out of studying the foundation l a i d down by researchers, i n t r y i n g to understand t h i s very complicated and o l d problem i n algebraic topology. CHAPTER 2. FIBER SPACES This chapter i s a discussion of the p r e l i m i n a r i e s needed f o r the l a t e r chapters. In what follows we s h a l l assume f i b e r spaces i n the s e m i - s i m p l i c i a l category {43] , unless otherwise stated. In t h i s sense, we s h a l l discuss properties of f i b e r spaces as i f we were dis c u s s i n g them i n the se m i - s i m p l i c i a l category without s p e c i f i c a l l y mentioning i t . § 2 . 1 . D e f i n i t i o n s and Notations 2.1.1. By a f i b e r space i n the sense of Serre we mean a quadruple (E, p, B, F) such that p: E •> B s a t i s f i e s the covering homotopy property f o r any f i n i t e complex. E i s c a l l e d the t o t a l space, -1 B i t s base space, and p (x) = E - F i s c a l l e d the f i b e r over x e B. 2.1.2. Given a map f : X >• Y i n the category of based spaces and based maps, we assume the d e f i n i t i o n s o f the mapping c y l i n d e r of f and of the mapping cone C^ of f. We note the following property: The diagram X - > Y M f homotopy commutes and Y i s a strong deformation r e t r a c t of M^. 2.1.3. Given a f i b e r space (E, p, B, F ) , we can replace i t by a homotopy equivalent f i b e r space, s t i l l denoted by (E, p, B, F) by abuse of notation, such that there e x i s t s a subset MC B and a homotopy equivalence X: M >• E with poX = i G , where i 0 i s the i n c l u s i o n of M i n B. Proof. By 2.1.2, we get the following homotopy commutative diagram: E r B ^ • M < p r Let r be the r e t r a c t i o n and form the induced f i b e r space (E , p', M , r p F ) . E < E P B M r p In p a r t i c u l a r , p i s homotopically equivalent to p" and i f we l e t X': E >• E^ be the homotopy equivalence such that p"oX'= i 0 = i n c l u s i o n of E i n M^ , then by the covering homotopy theorem we have the r e s u l t simply replace (E, p, B, F) by (E , p", M , F ) . 2.1.4. Given any based map f : X *- Y, there i s a standard construction by which we can replace f by a f i b r a t i o n f " : X'" >• Y and X may be taken to be a deformation r e t r a c t of X^. (see [44, pp.84]). 2.1.5. We s h a l l assume the knowledge of the standard construction of the Barratt-Puppe sequences with respect to a based map f: X y Y. That i s , there are two sequences: f i a Sf S i X > Y —=-> y SX y SY — y SC^ >- • • • f f and fij ttf a j f -y SIE y QX y Q.Y y E f y x y Y such that given any pointed space, W, we have the following long exact sequences: [X, W] < [Y, W] •< [C , W] -< [SX, W] •< [SY, W] •< ••• and ••• y [w, fix] >• [w, fiY] y [w, E ] y [w, x] y [w, Y] They are exact sequences of groups as f a r as SX (fiY) and are exact sequences of abelian groups as f a r as S 2X (Q2Y) . 2.1.6. We s h a l l assume the following consequence from the d e f i n i t i o n of f i b e r spaces. For a f i b r a t i o n F 1 •> E ——y B, the sequence: i s exact. ijj Pn 3n TT (F) ~5—y TT (E) ~ 5 —>• TT (B) y TT , (F) —> •' n n n n - l 2.1.7. From the Serre s p e c t r a l sequence we obtain the following r e s u l t : . THEOREM (SERRE). Let (E, p, B, F) be a f i b e r space with B simply connected. Suppose that H^(B) = 0 f o r 0 < i < p and that H_. (F) = 0 f o r 0 < j < q. Then there i s an exact sequence: ±* P* T H , (F) >• H , (E) >• H , .(B) > H „ (F) • ••• p+q - lV n p + q - l v P+q-1 p+q-2 h ... y H (E) y 0 where T i s the transgression i n the Serre s p e c t r a l sequence. §2.2. P r i n c i p a l F i b e r Spaces 2.2.1. Consider the standard Hurewicz f i b r a t i o n : Q.Y ——y PY > Y . We can form from i t the induced f i b r a t i o n i f we are given a map f: B >• Y . This induced f i b r a t i o n i s c a l l e d the p r i n c i p a l f i b r a t i o n induced by f. I t i s denoted by (E , p, B, fiY). So we have the f i b e r square: fiY fiY E. Pullback f -> PY B >• Y In p a r t i c u l a r , E = {(b, 6) e BxpY ; f(b) = a(6)}. This construction has the fo l l o w i n g obvious p r o p e r t i e s : (1) f h implies E - E . (2) I f a: B' ->• B and b: Y Y^ are homotopy equivalences, then E, . - E_ . bofoa f (3) The sequence: [W, E ] »• [W, B] y [W, Y] i s exact f o r any space W. 2.2.2. From 2.2.1(3) we have the following immediate consequence: LEMMA. I f g: W >• B i s a map, then g l i f t s to E i f f f^g 2.2.3. There i s an action of ClY on the t o t a l space E^ given by u: fiY><Ef y E , taking (a, (b, $)) to (b, av0), where a*B i s the path obtained by "adding" a and 8. Therefore there e x i s t s a natural action of the group [W, fiY] on the set [W, E ]. The foregoings, 2.1.5, 2.2.1 and 2.2.3, have corresponding statements f o r maps of p a i r s and f i b r a t i o n s of p a i r s . Below we s h a l l discuss n-connectedness and n-connected spaces. This i s important as very often we would l i k e to have n-connected spaces as f i b e r s i n some f i b r a t i o n s . This i s indeed always the case and n-connected spaces occur quite n a t u r a l l y . 2.2.4. Using usual f i b e r space arguments we can obtain the following u s e f u l theorem: THEOREM. Suppose X and B are simply connected and that f: X y B i s a map. Then f ^ : H_^  (X) >• H^ (B) i s an isomorphism f o r i < n and an epicmorphism f o r i < n i f f the same i s true f o r f „: TT. (X) • TT. (B) . # 1 1 As a c o r o l l a r y we note that the c o f i b r e (mapping cone) of f i s n-connected i f f the f i b r e i s (n-1)-connected. I f t h i s i s the case we c a l l f n-connected. Using the r e l a t i v e Serre homology (exact) sequence (see 2.3.1) , we can prove the following easy c o r o l l a r y to 2.2.4. 2.2.5. COROLLARY, Suppose B i s (n-1)-connected,- n > 2, and F i s (m-1)-connected, ni > 2. Then f o r p: (E, F) • (B, * ) , H^(p) i s an isomorphism f o r i < m+n-1, and an epi morphism f o r i < m+n. In other words, p: (E/F) y B i s (m+n)-connected. A dual statement holds f o r cohomology. 2.2.6. THEOREM (SERRE EXACT SEQUENCE FOR HOMOTOPY). Given X f > Y —-—y c„ , with X (n-1)-connected and C. (m-1)-connected f f (n, m >_ 2) . Then the induced map p: X >• E^ i s (n+m-2)-connected. Hence there i s an exact sequence: TT „ ( x ) y ir „(Y) y TT , _(c_) y n+m-3 n+m-3 n+m-3 f Proof. Consider the homotopy exact sequence f o r E. >• Y -^—y C. : l f -* V i ( Y ) ~*" V ^ V ~* W ~~" V Y > W and the homology exact sequence f o r the c o f i b r a t i o n X y Y > C„ : H (Y) p+1 H .. (C.) p+1 f -»• H (X) P + H (Y) P - H p ( C f > X (n-1) -connected implies i _ : H (Y) >• H (C,_) i s an isomorphism f o r * P P f p < n-1 and an epi morphism f o r p < n . Apply 2.2.4 to give the same conclusion f o r i„: TT (Y) > TT (C.) . Looking at the homotopy exact # P P f sequence above implies that E^ i s at l e a s t (n-1)-connected. We then have the homology exact sequence f o r the f i b r a t i o n , E. -> Y C, and hence the following diagram: H p(Y) H p(Y) H . H p ( C f ) -> H (E, ) p-1 1 -v H (Y) P-1, ->H p(C f) — H p_ x(X) - H p _ l ( Y ) with the rows exact f o r p <_ n+m-1 . Apply the f i v e lemma to get an isomorphism f o r p <_ n+m-2 . Apply 2.2.4 to get p an isomorphism f o r Tf k < n+m-3 . Hence there i s an exact sequence: TT „ (X) n+m-3 -> TT „ (Y) n+m-3 ~y it _(C_) n+m-3 f This completes the proof. 2.2.7. THEOREM. Suppose XCY, X i s (n-1)-connected and that Y/X (m-1)-connected (n, m >_ 2) . Then <f>: Tr (Y, X) >- (Y/X) i s an isomorphism f o r i <_ m+n-2 and an epi morphism f o r i <^  m+n-1 . Proof. Apply the following diagram: ( l i f t i n g ) „E. -> Y -*• Y/X with the homotopy exact sequences: TT (X) > TT (Y) • TT (Y, X) y TT , (X) > rr . (Y) P P P P - l P - l IT (E. ) p 1 IT (Y) • TT (Y/X) p p IT , (E.) — y TT . (Y) p - l 1 p - l and the f a c t that p i s (n+m-2)-connected by.2.2.6. An a p p l i c a t i o n of the f i v e lemma gives that <j> i s an isomorphism f o r p <_ n+m-2 and an epi morphism f o r p <_ n+m-1 . 2.2.8. FREUDENTHAL SUSPENSION THEOREM. I f B i s (n-1)-connected (n > 2), then s: TT . (B) • IT. , (SB) i s an isomorphism f o r i < 2n-l — l l + l and an epi morphism f o r i <_ 2n-l . f i Proof. Consider the obvious c o f i b r a t i o n B > * y SB and the p r i n c i p a l f i b r a t i o n induced by i which i s ClSB . Then there i s a l i f t i n g <J>: B •> fiSB given by <f> (x) (t) = t*x the adjoint to I d e n t i t y : SX >- SX . <{>,,: TT. (B) y TT. (QSB) i s the suspension. C l e a r l y <i> i s ft l i (2n-l)-connected by 2.2.6; and the r e s u l t follows. §2.3. Transgression i n F i b e r Spaces. 2.3.1. Leray-Serre Spectral Sequence. Let (E, p, B, F) be a f i b e r space with B and F, ( l o c a l l y f i n i t e ) complexes. Let B CB be a subcomplex and E = p 1 ( B ). ( B may o o o o be empty ). Then there i s a f i l t r a t i o n of H.(E, E ; T), where T i s a * o l o c a l c o e f f i c i e n t system with f i b e r , G, an A-module and A, a PID, and a canonical s p e c t r a l sequence f o r cohomology { ( E r , d r ) , r >_ 2 } such that (1) E _ > Gr H ( E , E ; T), where Gr.H.(E, E ; V) i s the p,q p q 0 * * 0 d i f f e r e n t i a l graded group coming from the f i l t r a t i o n , and ( 2 ) E 2 = H ( B , B ; H (F; G)), where H (F; G) i s a system of P/q P 0 q q l o c a l c o e f f i c i e n t over B [ 7 ] . * There i s also d u a l l y a f i l t r a t i o n of H ( E , E ; T) and a o canonical s p e c t r a l sequence f o r cohomology, { ( E ^ , d^ _) ; r >_ 2 } , with (.1) E P , Q > G r P H Q ( E , E Q ; T) and ( 2 ) E P , Q H P ( B , B ; H q(F; G) ) . 2 0 2 . 3 . 2 . Remark. I f H (F; G) i s a simple system as i s the case when ( B , B q ) i s simply connected or the s t u c t u r a l group i s connected, then we may can o n i c a l l y i d e n t i f y H (F^; G) with H (F; G) and write H q(F; G) i n place of H g(F; G). 2.3.3. Some Exact Sequence. (A) Let (E, p, B, F) be a f i b e r space with F, (m-1)-connected and base B, arcwise connected. Let B C B and E = p ^ ( B ) C E. Then , 0 o ^ o there i s an exact sequence: m+1 * ••• *• H n(B, B ; D - P—*- H n(E, E ; D > H n~ m(B, B ; H m(F; G) • 0 0 0 * ... y o > H M ( B , B Q; D H m(E, E q ; D —-»• H ° ( B , B q ; H m(F; G) ) j m + l ,-. * . -i i „ ,m+l ^—y H ^ B , B ; D P—y (E, E ; T) y H^B, B ; ^ ( F ; G) ) 0 0 0 H m + 2 ( B , B o; D f o r n <_ m , where T i s a c o e f f i c i e n t system over B, G i s i t s f i b e r and p*: H 3 (B, B ; T) y H D (E, E ; T) (j < m+1) i s a c t u a l l y the map o o — induced by p: ( E , E ) > ( B , B Q ) . ( B ) Suppose f u r t h e r that the p a i r ( B , B Q ) s a t i s f i e s rr. ( B , B ) = 0 , j < k-1 , k > 2 . Then, using s p e c t r a l homology argument, we have a s i m i l a r s p e c t r a l sequence to (A). * 0 y H N ( B , B q ; D -2—y H N ( E , E Q ; D y 0 y y 0 y H M + K ( B , B q ; D - J U H M + K ( E , EQ ; D — H ° ( B , B q ; ^ ( F ; G)) y H M + K + 1 ( B , B q ; D H M + K + 1 ( E , E Q , T) »• H ^ B , B Q ; ^ ( F ; G) ) — m+k+1 m.. , 0 ^ — > H M + K + 2 ( B , B O ; r>. In p a r t i c u l a r , i f V i s a simple system of c o e f f i c i e n t , we have that p : H N ( B , B ) >• H N ( E , E ) i s an isomorphism f o r n < m+k-1 and o o " — * an monomorphism f o r n <_m+k . We note that p i n the sequence i s a c t u a l l y the map induced by p: ( E , E Q ) >• ( B , B Q ) . We remark that dual statements and sequences f o r homology hold f o r (A) and ( B ) . 2.3.4. Transgression. Let T be a l o c a l c o e f f i c i e n t system with f i b e r , G, an A-module and A, a PID . Let { ( E ^ , d^); r >_ 2} be the cohomology Serre s p e c t r a l sequence coming from 2.3.1 f o r a f i b e r space ( E , p, B , F ) . The 0 r ir+1 0 d i f f e r e n t i a l d , : E ' >• E ' i s c a l l e d the transgression. Let r+1 r+1 r+1 2  r r T (F, G) denote the submodule of H (F; G) which corresponds under the O r 0 ^ r r f isomorphism, E ' = H ( B ; H (F; G)) ~ H (F; G) (= the f i x e d r O r submodule under the a c t i o n of TT^  ( B ) i n H (F; G) ) , to the term E ^ . So, we have the following commutative diagram: «0/r r+1 r+1, 0 Er + 1 Er + i T r ( F , G) ^ H r + 1 ( B ; G)/M r + 1 „r+l,0 v „r+k+l,l-k . _ , . „ , Since we know d, : E, • E, i s zero f o r k > 2 so that k k k — r+1,0 „r+l,0 . . , . -r+1,0^ „r+l , n ^ r+1,0 E k >• E ' i s onto, implying E 2 — H (B; G) > r+1 0 r+1 r+1 r+1 i s onto, i . e . , E r + 1 ' = H (B<" G ) / M r where M i s some submodule of H (B; G). In.fact we can even c a l c u l a t e what M i s . C a l l T also the transgression. Elements of T (P, G) are sa i d to be transgressive r [48]. In f a c t i t can be shown that i f we define T (F, G) = -1—* r+1 <5 p H (B, *; G), where 6 i s the coboundary homomorphism f o r the r — * - l r p a i r (E, F) ,and S (B, G) = p <$H (F; G) , .then T i s the same as the r r+1 —* map also denoted by x, T (F, G) »- S (B, G)/Ker p , given by — * — T(U) = [u'] where 6 (u) = p (u") and p: (E, F) >• (B, *) . T has the following n o n - t r i v i a l p r o p e r t i e s : * (1) Ker T = Im 0 , where 0 i s the i n c l u s i o n F C E . * it (2) S (B, G) = Ker p . * * * (3) I f we define a: S (B, G) • T (F, G)/Im 0 by —* —* a(v) = [v'] , where p (v) = 6(v'), then Ker a = Ker p . a i s c a l l e d the suspension. (4) R e l a t i v e Transgression. Let F >• E >• B be a f i b e r space and * £ B C B. Let E = p 1 (B ) E and p: (E, E ) >-(B, B ) . o o o o o Take the pullback of (<$, p ), (11 , H ) . That i s we have the diagram: II U >• H ( E ; G) 0 If. P . B . * * - * H ( B , B q ; G) • H ( E , E q ; G) where IT and are p r o j e c t i o n s . Define S ( B , B q ; G) = ff^U and A A A A T ( E ; G) = 1IJJ , and maps, a : S ( B , B. ; G) • T ( E ; G)/Im 6 and o 2 o o o o T q : T ( E ; G) >- S ( B , B q ; G)/Ker p to be the ones induced by - 1 - 1 ^2^2. ^ 1 ^ 2 r e s p e c t i v e l y . 2 . 3 . 5 . T q has the following noteworthy p r o p e r t i e s : ( 1 ) I f we denote the map B >• ( B , B ) by j and the i n c l u s i o n * * F c E ( B i s non-empty.) by k , then T k = j T o o J o * ( 2 ) T q i s an H (B)-morphism. ( 3 ) <(>T = T ij> f o r any primary cohomology operation, <j>, and t|> i s i t s suspension. ( 4 ) Suppose we assume that TT^ ( B ) operates t r i v i a l l y on H (F> Z ) and that H . ( B , B ; Z ) = 0 f o r i < (a-1) and H . ( F ; Z ) = 0 f o r l . o — • 3 0 < j < b; i t i s not hard to show that the following sequence i s exact. * -T i ... • H 1 ( E ) — H ( B , B ) >• H ( E ) — ^ H ( E ) • 0 0 o a+b-1 ,„ . • • • • H ( E Q ) , where i s the composition, H I + 1 ( B , B Q ) • - • -»• H I + 1 ( E , E ) >• H 1 + 1 ( E ) / * and i i s the homomorphism induced by i n c l u s i o n i : E C E . (We have dropped the c o e f f i c i e n t i n the case of i n t e g r a l cohomology). * (5) L e t t i n g T = xk , i f k : B C B and 0 < t < a+b-1 (a, b, 1 o o — * * * as i n 2.3.5(4) ) and i f Ker p D Ker k i n dimension t and k i s onto i n dimension t , then the sequence, 1 T Hfc(E) H t(E ) — > H t + 1 ( B ) , 0 i s exact. (6) N a t u r a l i t y of 2.3.5(4). Given a f i b e r space map: F = = = = = F Suppose H_. (F; Z ) = 0 f o r 0 < j < b and f A : B M B ^ Z ) >• ( B ; Z ) i s an isomorphism f o r o < r < a-1 and an epicmorphism f o r r = a-1. Then * —* the sequence i n 2.3.5(4) i s s t i l l defined and exact with i = f and ( B , B q ) thought of as (M f, B q ) where M i s the mapping c y l i n d e r [58]. Following Thomas [58] , we describe i n the next section a means of decomposing a f i b r a t i o n . §2.4. On Decomposition of a F i b r a t i o n and the L i f t i n g Problem 2.4.1. Let (E, p, B, F) be as before a f i x e d f i b r a t i o n . Take the standard Hurewicz f i b r a t i o n induced by a map to: B > C. We assume that p UJ - * so that we get a l i f t i n g q: (E, F) y (E , Qc) with p^oq = p where p^ i s the induced f i b r a t i o n suggested by the following diagram: E y* 0) / p E y B y C P l Obviously the map induced on the f i b e r , v , i s the map q|F: F QC We say v i s geometrically r e a l i z e d by the p a i r (w, q). Using the Barratt-Puppe sequence: [E, QC] >• t(E, F ) , (E , BC)] — ^ t(E, F) , (B, *) ] — [ E , C] * i i [F, fiC] ======= [p, nc] * -1 — — define Eu> = i p^ *. [ p ] C [F, fiC] where [ p] i s the homotopy c l a s s of p: (E, F) •*- (B, *) . I t i s e a s i l y seen that Eto = a l l homotopy classes t h a t can be geometrically r e a l i z e d by (w, q) f o r some l i f t i n g q of p. The following gives a c h a r a c t e r i s a t i o n of Eu. THEOREM, -ato = Ea) , where a i s the suspension defined i n 2.3.4(3). 2.4.2. We would l i k e to pursue the method o u t l i n e d i n 2.4.1 to decompose q f u r t h e r . We add the assumption that TT^(B) acts t r i v i a l l y * on H (F; G), as i s the case when B i s simply connected. Throughout t h i s s e c t i o n t h i s assumption i s made. 24 Suppose F has non-zero homotopy groups ( F) i - n dimensions n ( l ) , n ( 2 ) , ... , w i t h o < n ( l ) < n(2) < ••• ; and i f n ( l ) = 1, we s h a l l assume t h a t rr^ (F) i s a b e l i a n . Let i e H.n ^  (F; TT ,„ , (F) ) be the fundamental c l a s s o f F. 1 n ( l ) From the exact sequence f o r the f i b r a t i o n , we see th a t i i s t r a n s g r e s s i v e . L e t (u = - T ( I ^ ) , then we have the f o l l o w i n g commutative diagram: -> F •*• QC1 = K ( T T N < 1 ) (F) , n ( l ) ) E, B E K(TT ( F ) , n ( l ) + l ) n ( l ) = C. w i t h = K t ^ ^ j » n ( l ) + l ) and F^ = the f i b e r o f the f i b r a t i o n , •i : F K ^ 7 r n ( l ) ^ F ^ ' n ^ 1 ^ • F o r bY 2.4.1, we have a l i f t i n g , q^, of p t o E^ w i t h q ^ F = . Then by the exact homotopy sequence f o r the f i b r a t i o n , F^ >• F t h a t -> ^C^, and the c o n n e c t i v i t y o f ttC^ we have f 0 , r < n ( l ) v r r r ( F ) , r > n ( l ) Thus, we have s u c c e s s f u l l y k i l l e d one homotopy group of the f i b e r , n (2) Now, l e t i 2 E H (F^; 7 r n ( 2 ) ^ F ^ ^ e t* i e c h a r a c t e r i s t i c c l a s s o f F^ Again, i t i s t r a n s g r e s s i v e f o r the f i b r a t i o n , F^ •> E >• E.^  Let w = - T ( I ) c H n (E , TT (F) ) . Repeating the procedure, we obtain a sequence of cohomology classes and spaces: E , to , E^, u^, ••• ••• . The co.'s are r e f e r r e d to as the k-invariants of E. x To study the k-invari a n t s , we have to look i n t o the v a r i a t i o n under two l i f t i n g s , f ^ , f ^ . When can we use the k-invariants to decide the l i f t a b i l i t y of maps from X to B ? Can they be used at a l l ? In otherwords,' we r e a l l y hope that the k-invariants can determine the l i f t a b i l i t y of a map f: X y B . Questions n a t u r a l l y a r i s e as to how one can c a l c u l a t e with these k-inv a r i a n t s , once i t i s decided that they indeed can be used. In the next chapter, we s h a l l describe some c l a s s i c a l obstruction t h e o r i e s . To complete t h i s chapter, i t i s thought s u i t a b l e to say something about G-spaces since there i s an obstruction theory f o r G-spaces, and because equivariant cohomology can be used to obtain the c l a s s i c a l o b s t r u c t i o n theory. Before we close t h i s s e c t i o n , we state the following r e s u l t s f o r future reference. 2.4.3. R e c a l l that fiC i s an H-space. So fiC acts on the induced f i b r a t i o n . We have the following r e s u l t where y: fiC*E^ »• E^ i s the group action defined i n 2.2.3. LEMMA. The fol l o w i n g diagram i s commutative, where p^ i s the induced p r i n c i p a l f i b r a t i o n and q^ i s a l i f t i n g of p to E^. Also TT: fiCxE y E i s the obvious p r o j e c t i o n . ftCXE E, -> B Proof. Just checking the d e f i n i t i o n s and e x p l i c i t maps, u and COROLLARY. Suppose F > E y B i s a f i b r a t i o n with F (m-1)-connected, C and E^ as before. Then, by. 2.'3.5 (6) , 2.1. 3, the above lemma together with 2.3.5(4), we have the following exact sequence: * x v . . . > H N n c x E ) ^ H 1 + 1 ( B , E) > h 1 + 1 ( E 1 ) * H 1 + 1(fiCXE) > H 2 m(fiCxE) where v = uo(lxq^) . §2 . 5 . G-spaces 2.5.1. Re c a l l the d e f i n i t i o n o f a G-space: I f G i s a t o p o l o g i c a l group, f o r example a compact L i e group, then by a G-space X, we mean a t o p o l o g i c a l space X together with a G-action on X defined i n terms o f a map u: G XX > X which i s continuos and which s a t i s f i e s the following condition ( a s s o c i a t i v i t y ) g 1'(g 2*x) = (g g )*x , f o r a l l x e X , where • denotes the a c t i o n . By a G-map between G-spaces we mean a map, f: X > Y, which commutes with the a c t i o n of G, i . e . , the following diagram: GXX l x f GXY i s commut at i v e . Suppose we are given another t o p o l o g i c a l group, H, and a homomorphism, <j>: H y G. Suppose X i s a H-space and Y i s a G-space. A map, f: X y Y, i s sai d to be ^-equivariant i f the following diagram: ,u . H><X GXY i s commutative, where, with abuse of notation, we denote the group action by the same l e t t e r . We can define G-complexes, i n s i m i l a r fashion as G-spaces and G-subspaces, and t a l k about equivariant (generalized) cohomology theory. 2.5.2. Remarks. > (1) One can define a l o c a l c o e f f i c i e n t system on G-complex, K, as a covariant functor -> Abel , where K i s the category o f G-subcomplexes of K and Morphism(A, B) = {g e G; gA C B}, and deduce analogous equivariant cohomology theory with l o c a l c o e f f i c i e n t system ; and hence one can have equivariant cohomology sequences f o r p a i r , t r i p l e , etc. . By Grothendieck*s r e s u l t (see [47, 4.7, pp.181]) , the l o c a l c o e f f i c i e n t systems on K form an abelian category = Funct(K, Abel). K (2) One can s i m i l a r l y , i n an algebraic way, define equivariant homotopy, and deduce the corresponding exact homotpy sequence f o r pa i r s and a Hurewicz type theorem. 2.5.3. S i m i l a r l y , given p: X >• Y , an equivariant map between two G-spaces, we say that p i s a G-fiber map i f and only i f i t has the equivariant l i f t i n g property with respect to G-complexes. In t h i s category of G-spaces and G-maps,.we have the following important r e s u l t . THEOREM. An equivariant map, p: X >- Y, i s a G-fiber map i f f H H H H p|X : X y Y i s a (Serre) f i b r a t i o n f o r every H c G, where X denotes the stationary space of X under H. Proof. Omitted.. (See Bredon [9]). Remarks. (1) One can define Eilenberg-MacLane G-complexes or G-spaces and obtain the corresponding c l a s s i f i c a t i o n theorem f o r them. (2) The suspension functor and loop functor can be defined i n a similar' way. Obviously we also have Hopf G-spaces with the • • corresponding Q (Hopf)-space structure. (3) The r e s u l t s i n 2§1 and 2§2 s t i l l hold true f o r G-spaces except f o r a s l i g h t m odification f o r 2.2.5. (4) We now know we can define a G-spectra so that a theory of equivariant s p e c t r a l homology and cohomology can be used. 2.5.4. THEOREM. There i s a s p e c t r a l sequence { ( E P , q , d )} such that (1) E P ' q ^ E x t P ( H (K, L; Z) , V)) and 2 • -q (2) E P ' q = > H * + q ( K , L; D , G where T i s a (generic) l o c a l c o e f f i c i e n t system f o r K, and (K, L) i s p a i r of G-spaces. The proof of t h i s theorem i s omitted and can be found i n [9] Remarks. (1) The r e s u l t s i n section 3 of chapter 2 s t i l l hold true. (2) Obviously we obtain analogous r e s u l t s as i n chapter 2 section 4 f o r equivariant l i f t i n g . C H A P T E R . 3 . . . C L A S S I C A L : O B S T R U C T I O N T H E O R Y In t h i s chapter, we s h a l l give a discussion of the c l a s s i c a l obstruction theories, l i s t i n g the p r o p e r t i e s and the main c l a s s i f i c a t i o n theorems. This discussion r e l i e s h e a v i l y on Steenrod's book 154]. §3.1. Obstruction to Extension Let L be a subcomplex of K, and suppose E i s (n-1)-connected ( n >_ 1 ) ( therefore arcwise connected ) and (n-1)-simple. Suppose xve are given a map, f: L >• E, we ask i f there i s an extension of f to K. We s h a l l assume f a m i l i a r i t y with the d e f i n i t i o n of o b s t r u c t i o n cochain. I f f i s already given on the q-skeleton of K, i . e . , on L uK q, then the o b s t r u c t i o n to extending f to L W K ^ i s a cochain c q + " ^ ( f ) e q+1 o> , C (K, L; TT (E) ) where TT (E) denotes the l o c a l c o e f f i c i e n t system q q over E, defined by the homotopy groups {TT (E, e); e e E}. I t has the q following p r o p e r t i e s : (1) c q + 1 ( f ) i s a cocycle. (2) f Q - f± => c q + 1 ( f 0 ) = c q + 1 (£.,_>. . (3) c q + 1 ( f ) = 0 i f f f can be extended to L u K q + 1 . (4) Suppose we have two extensions, f^, : L ^ K q > E , with f Q | L = f - j L ; and suppose that f 0 | L v K q _ 1 ~ . f ^ L ^ i c 3 - 1 r e l L. Then the homotopy k induces a map, K: ( K x i ) q w L x l ->'"E , defined i n the obvious way by K| (x, 0) = f Q ( x ) , K(x, 1) = f ^ x ) . Define d q ( f ^ , k, f^) to be equal to the obstruction to extending K to ( K x i ) q + 1 w L X I . Now d q ( f , k, f ) l i v e s i n c q + 1 ( K X L , L x l ; y ( E ) x l ) O X . q q ^ which i s isomorphic to C (K, L; TT ( E ) ) . We- also denote the image of q. d q ( f Q , k, f 1 ) i n C q(K, L; T? ( E ) ) by d q ( f Q , k, f.^) and i n future we cr cr ^ only t a l k of d ^ ( f Q , k, f^) as i n C^(K, L; T T ^ ( E ) ) . T O be p r e c i s e , d q ( f Q , k, ^ j x i = ( - l ) q + 1 { c q + 1 ( K ) - c q + 1 ( f 0 ) x 0 - c ^ f ^ x l } where I 'Vq+l q+1 — i s regarded as-a 1 - c e l l and c (K) = c (K)xi. Hence, 6 d q(f Q, k, f ^ j x i = 6 ( d q ( f Q , k, f^xl) = c q + 1 ( f 0 ) x l - c q + 1 ( f l ) x l , which implies that « d q ( f 0 , k, f l ) = c q + 1 ( f 0 ) - c ^ 1 ^ ) . (5) d q ( f Q , k, enjoys the following property: I f f Q , f ^ , f 2 : L u K q >• E with f = f = ±2 on L, and suppose , k l K2 _ f Q | L w K q = f |L«-»Kq « f 2 | L ^ K q ( r e l L ) where the homotopies are r e l a t i v e to L , then d q ( f 0 , k 2 # V f 2 ) - d q ( f Q / k x, f x> + d q c f ; L , k 2, f 2 ) . (6) d q ( f 0 , k, f 1 ) = 0 i f f k: f 0 | L ^ K q 1 * f 11L ^K*3 1 can be extended to a homotopy f ^ L ^ R " 1 - f ^ | L ^ K q . (7) I f f Q : L u K q K E i s a map, l e t d E C q(K, L; TT (E) ) . Then f ^ l L w ^ 1 may be extended to f ^ on L ^ K q such that d q ( f 0 , 1, f x ) = d . (8) Let f: L u K ^ y E be as before and suppose c q + " ^ ( f ) = 0. cr+2 cr+1' Then { c ( f ; f an extension of f to L v K } i s a s i n g l e cohomology c l a s s . I t ' s vanishing i s a necessary and s u f f i c i e n t condition f o r f to extend to L u K q + 2 . Denote t h i s c l a s s by c q + 2 ( f ) . I t i s sometimes c a l l e d a secondary obstruction. (9) Let f: L u K q - — y E be given. Then f | L u K q _ 1 i s extendable to L u K q + 1 i f f c q + ^ ( f ) i s a coboundary. (10) Analogous to ( 8 ) , we have the following: I f f ^ , f 1 : L w K q y E are two maps, and k: f_jL'~'K q 2 -f^|LOK q 2 ( r e l L ) i s extendable to a homotopy k': f ^ | L v j K q 1 -f j L u K * 1 " 1 ( r e l L ) , then d q _ 1 (f | L u K q _ 1 , k, f j L u K 3 " 1 ) = 0 and the secondary d i f f e r e n c e obstructions { d q ( f ^ , k'', f ) ; a l l k'} form a si n g l e cohomology c l a s s , d ^(£Q/ f^)•-The homotopy k: f | L K q 2 -f ^ L u K * 3 " 2 ( r e l L ) i s extendable to ( L ^ K q ) x i i f f d " - q ( f 0 , f±) = 0 . (11) Using the homotopy extension theorem we hwve the following: I Cf-1 Let f ^ , f ^ : K y E be two maps such that fQ|L>->K = f ^ | L v j K q X . Then, there e x i s t s a homotopy H: ^ f * r e l L«^K q 2,. where f.jLuj . c ' 1 = f ^ | L ^ K q , i f f the d i f f e r e n c e cocycle ^ ( Z Q I IF f-^ E C^(K, L; TI ( E ) ) i s a coboundary i n K — L . q (12) I f f , f.^ : L ^ K q •> E are two maps such that f j L v K q _ 1 k f l L W K q _ 1 then c q + 1 ( f n ) * c q + 1 (f.) and 6d q(f , k, f ) 0 1 1' 0 1 o 1 = c q + 1 ( f 0 ) - c q + 1 ( f 1 ) . ( This follows from 3.1(4) ). (13) N a t u r a l i t y . Let h: K > K' be a map. I t induces a map h: (K, L) > (K', L ' ) . Then, c ( f o h ) = c(f) = h # c ( f ) e H*(K, Lj TT (E)), whenever c ( f ) and c ( f ^ ) are defined. S i m i l a r l y i f f * r f ^ : L'V_J K' q — y E are two maps such that £Q|l' = ^ l L ' a n ^ suppose that k'': f ' | L ' ^ K < q 1 - f ' l L ' ^ K ^ 1 r e l l/, then h induces maps f A , f , o " 1 U 1 and homotopy k such that d ( f Q , k, f ) = d(f^oh, k, f^oh) = h * d ( f j , k', f£) , whenever they are defined. We next define primary obstructions and secondary obstructions, primary and secondary d i f f e r e n c e obstructions. §3.2. Primary Obstruction We s h a l l assume E to be (n-1)-simple, arcwise connected f o r the remainder of t h i s chapter. We l i s t the following conditions which we would l i k e E to s a t i s f y . (A) H- , + 1(K, L; v (E)) = 0, f o r j £ n-1. (B) H2 (K, L; TT_. (E)) = 0, f o r j <_n-l. (C) H D~ 1(K, L; TT (E)) - 0, f o r j <_ n-1. 3.2.1. Suppose conditions (A) and (B) are s a t i s f i e d . Given f; L y E, by3.1(3) f can be extended to L u K n . Then 3.1(8) applies to give that the set { c n + " ^ ( f ) ; f any extensions of f to Li~/Kn } i s a single cohomology c l a s s , c ( f ) . We c a l l t h i s c l a s s the primary obstruction o f f. Properties 3.1(2), 3.1(3), 3.1(5), 3.1(6), 3.1(7) n+l and 3.1(13) hold analogously f o r c ( f ) . 3.2.2. Suppose conditions (B) and (C) are s a t i s f i e d . I f f Q , f^-: K y E are maps with f Q | L = f - j L , then by 3.1(10) we say that d n ( f _ , f,) e H n(K, L; TT (E)) i s the primary d i f f e r e n c e of f 0 1 n U and f^. Suppose conditions (A), (B) and (C) are s a t i s f i e d . We s h a l l further assume conditions (B) and (C) f o r L as well as f o r (K, L ) . Then we have that analogous r e s u l t s hold f o r 3.1(5) and 3.1(13). And d n ( f y / f-^) s a t i s f i e s the following property: f | L K " = f j l K N i f f d n ( f A , f,) = 0. o 1 1 1 0 1 In place of 3.1(4) we have the following lemma. LEMMA. Lf f g , f ^ : L *• E are two maps, then the primary d i f f e r e n c e , d n (f , f ) , l i v e s i n H n(L; TT^ (E) ) by conditions (B) and (C) for L and 3.1(10). Moreover, 6d n ( f n , f.) = c n + 1 ( f ) - o n+1lt.) , 0 1 o 1 where 6: H n (L; v (E)) • H n + 1 (K, L; W (E)) a n d c ^ f f j , c " n + 1 ( f 1 ) n n u x are given by 3.2.1. §3.3. Extension Theorems and C l a s s i f i c a t i o n of Maps 3.3.1. THEOREM. With assumptions (A) and (B) of §3.2 or i+1 ^ IT_, (E) = 0, f o r j £ n-1. I f furthermore H J (K, L; n\ (E)) = 0 f o r n < j < dim (K — L ) , then a map £: L • E has an extension to K i f f c n + 1 ( f ) = 0. Proof. Apply the analogue of 3.1.3 twice. 3.3.2. THEOREM. With assunptions (B) and (C) of §3.2. Further-more assume H J (K, L; IT (E) ) = 0 f o r n < j <_ dim (K — L) . Suppose f0' f l ! K *" E a r S m a p s s u c h t n a t £Q | L - then f Q - f ^ r e l L i f f d n ( f Q , fx) = 0. Proof. Just apply 3.2.2 twice. 3.3.3. Hence pursuing i n t h i s d i r e c t i o n we have the following c l a s s i f i c a t i o n theorem. THEOREM. (With the usual assumption on E, i . e . , E i s (n-1)-simple and arcwise connected). Suppose conditions (B) and (C) of §3.2 i s s a t i s f i e d . That i s , H^(K, L; TT ( E ) ) = 0 = H-'~1(K, L; TT_. (E)) f o r j £ n-1 and H"' (K, L} 7 T \ ( E ) ) = 0 = H- , + 1(K, L; TT_. (E) ) fo r j > n. Let f: K >- E be a map. Then the set of (r e l a t i v e ) homotopy classes of maps f':K • E ( r e l a t i v e to L) with £'|L = f | L are i n one to one n ^ correspondence with H (K, L; TT^ (E)) under the assignment [fl ydn(f% f) . Proof. The f a c t that t h i s correspondence i s one to one and well defined i s 3.3.2 and the addition formula. To show.it i s onto, we need the fol l o w i n g theorem. 3.3.4. THEOREM. Same hypothesis as f o r theorem 3.3.3. Let — n ^ f: K y E be a map. Then f o r each cl a s s d e H (K, L; IT^E)) , there e x i s t s extension f * of f|L to K such that d n ( f , f ) = d. Proof. Take a reoresentative e e d. Then 3.1(7) says that f | L u K n 1 can be extended to g: L U K " > E such that e = d n ( f | L u K n , 1, g). But 0 = 6 d n ( f | L u K n , 1, g) = C n + 1 ( f | L o K n ) - C n + 1 (g) n+1 = -c (g) so that by 3.1(3), g can be extended to K; l e t such an extension be f ' . Then 3.1(10) applies to give d n ( f , 1") = d. 3.3.5. Suppose E i s (n-1)-connected, n > l . I f n = 1, we assume that TT^(E) i s abelian. Let f, f ^ : L >- E be maps. Suppose f ^ i s extendable to L u K n + 1 ) then c n + ^ ( f ) = 6d n ( f , f^) . 3.3.6. Using arguments analogous to that i n N. Steenrod [54], we can prove the following: n+1 ^ THEOREM. Suppose H J (K, L; TT_. (E)) = 0 f o r j > n and the conditions (A), (B), and (C) are s a t i s f i e d f o r (K, L) as well as f o r L (e.g. TT.(E) = 0 f o r j < n; i f n = 1, assume TT (E) i s abe l i a n ) . Then a map f: L *• E i s extendable to K i f f d n ( f , f Q | L ) i s i n the image of i * : H N(K; T? (E)) > H" (L; TT (E)) , where f : K >• E i s a f i x e d n n o — n ^ » . map. I f f does extend to K, then f o r each such d' e H (K; Tr^(E)) wxth i (d') = d n ( f , f ^ L ) , there e x i s t s extension f of f with d n ( f , f Q ) = d ' We s h a l l now proceed to define the primary obstruction to contracting E to a point. 3.3.7. Let E be a (n-1)-connected CW complex (n >_ 1; i f n = 1, assume that TT^ (E) i s abe l i a n ) , and e^ a poin t i n E. Let g^: E ——>• E be the constant map given by 9Q( S ) = 0Q f ° r a l l e e E. Define the primary obstruction'to contracting E to a point e^ to be k e = ^"^O' 1 E ) e H n ( E ; V E ) ) ' k n enjoys a l l the properties of a di f f e r e n c e obstruction. I f n > 1, eo then E i s arcwise connected and so any two constant maps are homotopic. This implies that k n i s independent of the choice of e . Hence we eo n 0 denote t h i s c l a s s by k . In p a r t i c u l a r , we have the following r e s u l t s . (1) Given f: L > E and L C K. Then, by 3.3.5 f i s extendable to L u K n + 1 i f f c n + 1 ( f ) = 6fV\ f o r c n + 1 ( f ) = 6d" n(g 0of, l o f ) = 6f*d n ( g Q , 1) = 6f*k n. (2) Corresponding to 3.3.6 we have the following theorem. THEOREM. I f H' ) + 1(K, L; TT (E))= 0 f o r j > n and i f TT_, (E) = 0 f o r j < n ( i f n = 1, assume T T^(E) i s abelian) , then a map f: L > E — n * n n ^ i s extendable to K i f f d ( l o f , g Qof) = f k e H (L; T T ^ E ) ) i s i n the * n 'V' n ^ image of i : H (K; ^ ( E ) ) »• H (L; TT^ (E)) which i s induced by _ *_ * n i n c l u s i o n . I f f does extend to K, then f o r each d' with i d = f k there e x i s t s an extension f " of f: L *• E to K such that ( d " ( l o f , g Q o f ' ) = ) f ' V = d " . (3) A s i m i l a r statement to 3.3(3) holds, where we replace — n * n d ( f , f) by f ' k and so the correspondence there now becomes * n [ f ] , < • f k . r e l L 3.3.8. G-spaces. We come back to questions r e l a t i n g G-spaces and extensions. The technique i n obtaining the previous r e s u l t s ( ' c f . N. Steenrod [54] ) can be c a r r i e d over to equivariant extensions i n G-spaces with s l i g h t modification, and " a d d i t i o n a l " assumptions. A l l the usual assumptions i n the previous sections are retained H and we require also that E to be j-simple f o r j < n and f o r a l l HCG. Here E 1 1 denotes the st a t i o n a r y space of E under H. Analogous r e s u l t s to §3.1, §3.2 and the preceding sections of §3.3 hold word f o r word except that everything i s now i n the category of G-spaces and equivariant maps and so cohomology becomes equivariant cohomology and homotopy, equivariant homotopy. (For d e t a i l s see G. E. Bredon [9]). 3.3.9. I t i s sa i d that one could have developed the theory of equivariant extension and then everything could be c a r r i e d to the non-equivariant case, simply by considering G = {e}. Remarks. (1) We have now come to the end of t h i s chapter; and i f we consider the map f as a cross-section to the bundle (E, p, B, F ) , then we obtain the c l a s s i c a l obstruction theories to extending cross-sections and o f course also the equivariant c l a s s i c a l obstruction theory to extending equivariant cross-sections of G-fiber bundle. (2) We note that i n the case of bundles our c o e f f i c i e n t system i s chosen to be TT (F) , where F i s (n-1)-connected, since we want to n extend a cro s s - s e c t i o n to a cross-section, not merely an extension. Hence, i n place of E i n the previous sections we i n s e r t F and we often assume F to be n-simple. That i s to say a l l the conditions on E i n the previous sections become conditions on F . In the next chapter we s h a l l discuss a new development of obstru c t i o n theory f o r f i b e r spaces, u t i l i s i n g a decomposition of f i b e r spaces due to Moore-Postnikov. CHAPTER 4. GLOBAL OBSTRUCTION In t h i s chapter we s h a l l stay i n the category of based spaces and based maps. (E, p, B, F) w i l l denote a f i b e r space i n the sense of Serre. B w i l l always be assumed to be simply connected. The exposition i s based on R. Hermann [18]. §4.1. D e f i n i t i o n s of Homotopy Obstruction and Homology Obstruction. Let (E, p, B, F) be a f i b e r space. Suppose i t has a cross-section, f: B >• E, then from the homotopy exact sequence f o r the f i b e r space, we get the following s p l i t t i n g : ir.(E) = i „ ( T r . ( F ) ) © f „ (ir. (B)) f o r a l l j , 3 ff 3 # 3 where i : F »- E i s the f i b e r i n c l u s i o n . This determines an onto-homomorphism, f: TT_. (E) >-»• TT_, (F) , which i s the composition, { i / / P'J p r o j . (4.1.1.) TT , (E) == i„TT. (P)e-f*Tr. (B) ~ ^ TT . (F) 9 TT . (B) »• TT. (F) . 3 # 3 " *? D s 3 3 3 We need the following t e c h n i c a l lemma to obtain a s i m i l a r type of map, H (E) H.. (F) . 4.1.2. LEMMA. Suppose (E, p, B, F) admits a cro s s - s e c t i o n , f, and suppose B i s (n-1)-connected (n >_ 2), and F i s (m-1)-connected. Then i • H.(F) »• H.(E) i s one-to-one f o r j < m+n-2, and * 3 3 ... H.(E). = f r tCH^(B))ei*(H'^(F)) f o r 1 £ j <_ m+n-2. Proof. By 2.3.3 p^: EL (E, F) y H (B, *) i s an isomorphism f o r j < m+n ; and f : H, (B) + H.(E) i s one-to-one f o r f i s a cross-3 3 section. Consider the following diagram: Hj + 1 ( E ) P* H <B> H j + 1 ( E , F) P* = -+ H. (F) 3 - 1 * H. (E) where p^ i s onto since f i s a cross-section. Thus j i s onto i n dimensions greater than 1 and l e s s than or equal to (m+n-1), implying that i . : H .(F) y H . (E) i s a monomorphism f o r 0 < j < m+n-2 by 3 3 — — exactness. Therefore we have the following s p l i t t i n g : H . ( E ) = i (H.(F))© f. (H . (B)) f o r 0 < j < m+n-2 . 3 * 3 * 3 - — 4.1.3. Thus by 4.1.2 we can define f: H.(E) y H.(F) f o r 3 3 1 < j < m+n-2 to be the composition, H . ( E ) = i:H. (F) © f . H . (B) {I* ' P*> pro] , -> H. (F)©H. (B) y H. (F) 3 3 3 We observe that the n a t u r a l i t y of the Hurewicz homomorphism, $L,implies the commutativity of the following diagram: f TT. (E ) D it H . (E ) 3 -y TT . (F) 3 -y H . ( F ) 3 f o r 1 < j < m+n-2 4 . 1 . 4 . Let B C B with B f d> . Consider the induced f i b r a t i o n o o E CZ o P E, B C B where E can be taken to be p V ). Suppose f: B y E i s a cross-o o o o section of the f i b e r space (E , pIE , B , F ) , we s h a l l consider the o 1 o o problem of extending f to a cro s s - s e c t i o n over B. With 4 . 1 . 1 i n mind we define the homotopy obstruction to extending f to a cro s s - s e c t i o n over B to be the following compositions: P # 8 # F ( 4 . 1 . 5 ) to ( £ ) : TT . (B, B ) ^ TT. (E, E ) —-*•»• rr (E ) y T r._ 1 (P) f o r j = 2, 3, i.e. , to. (f) = f ^ 1 Suppose we have that H. ( B , B Q ) = 0 = H_. ( B q ) f o r 1 <_ j <_ m and £ F i s (n-1)-connected. Then p : H.(E, E ) y H . ( B , B ) i s an D O 3 o isomorphism f o r j <_ m+n-1 (see 2 . 3 . 3 ( B ) ) ; and so we can define the homology ob s t r u c t i o n to extending f to a cross-section over B to be the following compositions: p~* - 1 .9^ f (4.1.6) V. (f) : H. ( B , B ) • H. (E, E ) y H. (E ) ^ H. (F) D 3 o £34 D . 0 3 _ 1 0 D - 1 f o r 2 <_ j <_ m+n-2 ; i . e . , v.. (f) = f ^ p * " " 1 . From d e f i n i t i o n 4.1.4 we see that the following diagram i s commutative. '"u. (f) (4.1.7) TT.(B, B Q ) - 1 — > -rr , (F) ft. v.(f) H.(B, B ) J • H. .(F) D O D-l j = 2, ••• , m+n-2. 4.1.8. At t h i s point i t i s c l e a r how one can obtain the cohomology obstruction to extending f to a cross-section over B. However, we are more i n t e r e s t e d at the moment i n the homology obstructions, { v_. (f) }, and the homotopy obstructions, { to_. (f) }. The l a t t e r have the following p r o p e r t i e s . (1) I f f i s extendable to B, then to_. (f) = 0 f o r a l l j , (2) Consider the f i b r a t i o n , fiB, with B c B and f: B • E as before. Then f i s a cross-section of o o o the loop space f i b e r i n g . The following diagram i s commutative, w. (£) TT_, ( B , B Q ) Tr;._1(F) 7 T . , (fiB, fiB ) 3 " • TT. „(flF) 3-1 o 3 - 2 where a i s the loop isomorphism. That i s , we have invariance under the loop functor. (3) N a t u r a l i t y . Given two f i b e r spaces, (E, p, B, P) and (E', p", B^, F ' ) , and a f i b e r map, {g, h, k}: . g F >• F' Suppose B C B and B ' c B ' are such that k(B .) c B ' , and that f : B and f : B" »- E' are cross-sections making the diagram, o o . k B -> B E -> E' o o commutative. Then {g, h, k} induces a commutative diagram, OK (f) TT.(B , BO) TT. ( B ' , B") 3 o + TT (F) J - • w j ( f ' } Tr. (F") f o r j = 2, 3, • • • . (4) I f f ^ , f ^ : B y E are cross-sections such that f x | B Q ^2^ Bo ' ^ e n o n e c a n define d i f f e r e n c e obstructions to deforming f onto f over B as follows: We know loop spaces have an H-space structure, V: RXxfiX y SIX . We use t h i s s t r u c t u r e to define a map, g: (SIB, QBq) * (SIS.,. *) , by g(x) = Slf (x) v Slf^(x "S , where x i s the homotopy inverse of x. With t h i s we define the differ e n c e obstruction to be the compositions: - .-1 d. (f , f ) : TT. (B, B ) > TT. , (SB, SIB ) — ^ IT. . (StE) —^> 3 1 2 3 0 3-I 0 3-I - -1 a TT. , (SIF) y TT. (F) . D-l 3 d.(f , f ) f i t s i n the following diagram: TT_. ( E ) — ^ — — y TT ( F ) (4.1.9) w (E, E Q ) -> TT . (B, B ) 3 0 We would l i k e to consider some of these co.'s as cocycles. This 3 leads us to the following notion of a GCW decomposition due to W. Barcus and R. Hermann. The obje c t i v e i s to obtain a gl o b a l d e f i n i t i o n of obstruction. §4.2. Generalized C e l l Complexes and Obstructions 4.2.1. Definitions. A generalized'cellular decomposition of a space, B, is a sequence of subspaces B ( 0 ) ~ { b ) C B ( 1 ) c C B ( n ) c r ... C B 0 such that (4.2.2) H . ( B ( n ) , B ( n _ 1 ) ) = 0 for j f n, n = 1, 2, ••• . 4 6 A subspace, B q, i s s a i d to be adapted to such a decomposition C abbreviated GCW decomposition ) i f the following i s s a t i s f i e d . ( 4 . 2 . 3 ) H. (B n B ( n ) , B A B ' " " 1 ' ) = 0 f o r j ^ n, and j o o H.(B w B ( n ) , B W B ( n - 1 ) ) = 0 f o r j ^  n . j o o For example, the skeleton decomposition of a CW complex, X, i s a GCW decomposition and any CW subcomplex, X , i s adapted to a s k e l e t a l decomposition. We s h a l l now proceed to describe an obstruction theory with respect to such GCW complexes. 4 . 2 . 4 . Let B ( 0 ) = {b } C B ( 1 ) c C B W C - d B o be a GCW decomposition f o r B, and B O C B be a subspace of B adapted to the given GCW decomposition f o r B. Define an algebraic chain complex, &> =' { C (B, B ) , 9 } as follows: n o Denote by B ^ the union B U B ' K ' and define o ( 4 . 2 . 5 ) C (B, B ) = H (B , B u ' ) , and n o n 9: C (B, B ) > C AB, B ) , n o n - 1 o Mn) ^ ( n - 1 ) M n - 2 ) . to be the boundary operator, 9*, of the t r i p l e , (B , B , B ). I t i s c l e a r that 9 2 = 0 . By usual homology argument we can show that H ( S ) C^. H (B, B ) f o r a l l n. n n o 4 . 2 . 6 . We can now proceed to see how our previous concept comes i n t o p l a y i n f i b e r spaces. The main d i f f e r e n c e from the c l a s s i c a l obstruction theory of f i b e r bundles i s that the d e f i n i t i o n of obstru c t i o n cochain i s defined i n terms of the global p r o p e r t i e s of the homotopy groups of the f i b e r . Let (E, p, B, F) be as before. Suppose we are now given a cross-section f of (E , p E , B , F) over B , which i s adapted to a o 1 o o o . GCW decomposition of B. Assume B q ^ <j>. of course. Analogous to the c l a s s i c a l theory we define the obstruction cochain on the complex, defined i n 4.2.5, by y n ( f ) : C^ ^ T r n_- L (F) as the homotopy obstruction to extending f over B ^ ; i . e . , w n(f) i s the composition, M n ) -(n-1) " n ( f )  C n = V B ' B > - n - l ( P ) ' where the isomorphism i s given by the r e l a t i v e Hurewicz theorem. We s h a l l proceed to show that u n ( f ) i s indeed a cocycle and l i s t i t s properties to j u s t i f y i t s name. 4.2.8. (1) 6co(f) = 0. /^x ^ * -(n-1) -(n-1) (2) Suppose £ , f 2 : B >• E are cross-J -(n-1) , ^ _ i-(n-2) _ i-(n-2) , sections already given on B such that f^|B = f 2 | B . ( -(n-1) = p - l - ( n - l ) K ) W e can then define the di f f e r e n c e o b s t r u c t i o n cochain by , n - l . _ _ . ... ... _. ,-(n-l) -(n-2). d (f, , f„) = the composition C , TT . (B , B ) -1 2 n-1 n-1 d (f , f ) -3 * TT (F) . n-1 Then S d " " 1 ^ , = Af^ - (Af,,) • Proof. We s h a l l only prove (1)> (2) follows almost t r i v i a l l y by checking the d e f i n i t i o n of d n 1 (£^r a n ,3 diagram 4.1.9. Consider the following diagram: n £5 n ' s n n-1 n Mn) ^ Mn) ^2% Mn) H & n > ,*)< TT ( B W , *) rr ( E 1 " ' , F) n, n •—f n •i# p. L2# "3# 3# Hi.-1# i _ c «_2 /^n+1) Mn) ^3# Mn+1) M n ) ^ l ^ ( n ) * TT .(F) n-1 f o r n _> 2 . Observe that i t i s commutative. F i r s t 9 „ . u i . , , = 0 by exactness. 3# 4# 6.0(f) = tt(f>3 - f a ^ i i ^ a - « 2 # ( V i # > " 1 3 * f i3# 83 # 3 4 # P 3#*2 1 = ? i3# ( 33# i4# > 35# P3#* t2 1 " °' 4.2.9. Now, we s h a l l pass on to describe the corresponding Mn-1) homology obstruction to extending f : B • E to a cros s - s e c t i o n Mn) Mn-1) over B . We assume that F i s (m-1)-connected and B i s (q-1)-connected, and n <_ m+q-2, so that we can apply 4.1.2 and therefore 4.1.6 makes sense. Using the commutativity of diagram 4.1.7, we define v n ( f ) : C (B, B ) tZ-M'rr' (F) * H _ (F) n o n-1 n-1 to be the map shown above; i . e . , i t i s the map making the diagram, 49 w (f) C (B, B ) n o C (B, B ) n o -*• TT .(F) n-1 v"(f) **" H (F) n-1 commutative, 4.2.10. v n ( f ) i s c l e a r l y a cocycle by 4.2.8; and by the commutativity of diagram 4.1.7 we observe that v n ( f ) i s also the composition, 1 p * -(n) -(n-1) (4.2.11) C (B, B ) -rr-> H ( E W , E( ' n o = n ) »• H . ( E( n 1 ] ) • H , (F). n-1 n-1 4.2.12. THEOREM. In addition to the hypothesis of 4.2.9, suppose B i s (p-l)-connected and n < m+q-2, m+p-2. Then under the o — canonical homomorphism, K: H n(B, B ; H n (F)) y Horn(H (B, B ); H (F)), o n-1 n o n - i K ( V n ( f ) ) i s the following composition, (4.2.13) H (B, B ) y H (E, E ) »• H . ( E ) y H . (F) n o ^ n o n-1 o n-1 Proof. The canonical homomorphism, K, i s given by K(Y) = Y — n — f o r any y e H ( B , B ; H , ( F ) ) , where y i s constructed by p i c k i n g a o n-1 representative y of y such that the following diagram i s commutative, fc <B, B q ) — ^ ^ ( B , B O ) ( 0 N ( B , B Q ) -*• 2,(B, B ) n o -y H .(F) n-1 / Y H (B, B ) n o 50 Y i s unique upto isomorphism. Nov;, we only need to show that i f we — — n n ^ ^n take y = v (f) and y = v (f) , then y = \> (f) w i l l be equal to the composition , ' H (B, B ) = H (E, E ) > H . ( E ) • H (F) n o n o n-1 o n-1 Consider the following diagram: (4.2.14) v (f) PT* C (B, B ) = H (S ( n ), S ^ ^ H ( E ( n > , g ^ ^ J l ^ H 1 ( ^ ( n - 1 > ) - ^ H y , (P) n o n = n ' n-1 . n-1 > n Mn-2h 2* ,Mn Mn-2) 4 2* ,Mn-2L Z (B, B ) == H • (B , B V -^T— H (E , E ) > H (E ) n o n = n , n-1. Mn) MO) 3* Mn) MO) 3* MO) H ( B V n ; , B l U ; ) -fr-^ H ( E W , E l ') —^-»- H (E ) = H . ( E ) n n n-1 n-1 o MO) P 4 * MD) 4* MO) H (B, B ) = H (B, B ) •< - H (E, E ) • H , ( E ^ ; ) = H . ( E ) n o n s n n-1 n-1 o A l l maps are defined here since n < m+p-2, m+q-2. p i s an isomorphism since n <_ n+m-1. p^ A i s an isomorphism since n <_ (n-1) + (m-1) f o r Mn) MO) m > 2. (B , B ) i s Min(n-1, p-1, q-1)-connected; and so p i s an — 3* isomorphism. S i m i l a r l y p i s an isomorphism. We remark that we may take fe> ,(B, B ) = C (B, B ) , (B, B ) = B (B, B ), and n+X o n+x o n o n o "SJ^fB, B Q) = z n ( B / B Q) /* an<3 by assuming the commutativity of the diagram we see that K (v (f)) = fS.J?... . We observe that the diagram 4.2.14 .4 4 . i s commutative except f o r the following p o r t i o n . H 1 ( ^ ( n - 1 ) ) " ~ h H ,(F) n-1 n-1 H _ (E ) n-1 o -(n-1) We know i s (p-l)-connected and B i s (q-1)-connected. Then we have the following s p l i t t i n g s and maps. ( See 4.1.2. ) { i * 1 ' P*} p r o j . H . (E ) > H . (F) © H, (B ) >• H , (F) 3 0 eg 3 3 0 3 1 r . - l , ft(n-l), . 1*__'_- p*f „ , „ s ^ „ ,-(n-l) H ( E v " ') pro: • • Hj (F) e Hj (B N" """ ) > Hj (F) j <_ n + Min (p, q) - 2. The above diagram i s obviously commutative. This completes the proof of 4.2.12. 4.2.15. Remark. 4.2.12 shows that <(v(f)) i s independent of the GCW decomposition. I t i s not easy to see or show how the higher obstructions can be shown to be independent of the GCW decomposition. §4.3. Extension Theorems In t h i s section we c o l l e c t together the main r e s u l t s concerning onstructins to extending cross-sections to a f i b e r space, (E, p, B, F ) , and give a c h a r a c t e r i s a t i o n , analogous to that of the c l a s s i c a l obstruction defined on a f i b e r bundle using i t s l o c a l homotopy proper t i e s . 4.3.1. Homogeneity Property. Let (E, p, B , F) be a f i b e r space. Assume B to be a CW complex and take f or i r s GCW decomposition the CW skeletons, { B N }. Let B 0 C B be a subcomplex. Then B q i s adapted to such a GCW decomposition. Suppos f: B N ^ U B q »- E (n >_ 2) i s a cross-section over B N Consider w n(f) e c N ( B , B ; TT , (F)) as defined i n 4.2.7. Then o n-1 (1) oj n(f) = 0 = £ • f can be extended to a cross-section over a. n B . (2) I f d e c N ( B , B Q ; TT^ (F)) i s another cocycle such that d n i—n—2 i s cohomologous to w (f) , then f | B •> E can be extended to- a cros s - s e c t i o n , f ^ : B N 1 >• E, such that w n ( f ) = d. 4.3.2. THEOREM. Let (E, p, B , F) be a f i b e r space, B , a CW complex with a GCW decomposition, B ' ^ C B ' ^ C '** C B ^ ' c " C B , where B ^ are subcomplexes of B. Assume F i s j-simple f o r j <_ n-1 . (n—1) Then a c r o s s - s e c t i o n f: B >• E can be extended to a cross-section over B ^ N ^ when i t s homotopy obstruction, w n(f)> i s zero. Proof. Take the s k e l e t a l decomposition of B ^ . Then we know (n—1) B i s adapted to such a GCW decomposition. Apply 4.3.1 to get the existence of an extension, f., to f, ,: B ^ ' 3 ^ U B ' " > E. Since 3 3-1 tip' (f ^) E H^CB^, B ^ N ^ ) = 0 f o r j ^ n, the inductive a p p l i c a t i o n of 4.3.1 gives an extension, %n_^r °f f t o B ^ ' n X . Now, io n(f) = 0 implies w n ( f ) = 0 ( see R. Hermann [18] ); and another a p p l i c a t i o n n—1 of 4.3.1 completes the proof. §4.4. n-systems We s h a l l proceed to show that we can define secondary and higher obstructions by u t i l i s i n g a notion due to Moore-Postnikov of decomposition of a f i b r a t i o n , and to show that the obstructions are somehow r e l a t e d to the k-invariants of the f i b r a t i o n under s u i t a b l e conditions. In t h i s section (E, p, B, F) i s a f i x e d f i b e r space with a connected f i b e r and with B being a simply connected CW complex. 4.4.1. D e f i n i t i o n s . We s h a l l denote by B, the s k e l e t a l k GCW decomposition of B. We have seen the idea of k i l l i n g homotopy groups of the f i b e r i n §2.4. We now gen e r a l i z e " t h i s idea. ( c f . [43]). For a given space, F, the p a i r , (F g ) , where F i s a space and g a. f i b e r map, n n n n F" > F > F , w i l l be c a l l e d a n-Postnikov system f o r F i f n n (4.4.2) g n # : ^ (F) • ^ ( F j i s an isomorphism f o r k <_ n and ir^ (F ) = 0 f o r k > n. I t then follows that r o , i f k < n u, (F') k n (F) , i f k > n We s h a l l denote such a Postnikov system f o r F by (F, g , F^, F^) when there i s no ambiguity, ( c f . § 2 . 4 ) . ' 4.4.3. The t r i p l e s , (E , F , h ), which denote a family indexed c n n n by n, where F E -> B and F' •+ E n -+ E n . n are f i b e r spaces such that the following diagram: 9*. n -> F n F -y E -> E i s commutative, and where (F, h F, F , F') i s a n-Postnikov system n 1 n n for F, are c a l l e d a n-Postnikov system f o r E. This i s the essence of Moore-Postnikov decomposition of f i b r a t i o n . 4.4.4. A p p l i c a t i o n of the homotopy exact sequence to the f i b r a t i o n , F' y E >• E , together with 4.4.2 gives that n n h : TT (E) > TT (E ) i s an isomorphism f o r k < n. n k k n — C a l l the c h a r a c t e r i s t i c c l a s s of the n-Postnikov system, g n F' y F y F , the n-Postnikov i n v a r i a n t of F, sometimes also n n c a l l e d the k-in v a r i a n t of F. Denote i t by k n + 2 e H n +^ (F , TT (F)) . . jr v n n+1 That i s , k n + 2 = T (I ,) , where a ,, e H n + 1(F''; TT ,. (F)) i s the n+1 n+1 n n+1 fundamental c l a s s of F" and T i s the transgression of the f i b r a t i o n , g n n F' -y F • • •> F . S i m i l a r l y , c a l l the c h a r a c t e r i s t i c c l a s s of the n n h n-Postnikov system, F" >• E — E , i c ^ 2 e H n + 2 (E , TT . (F)) , n n n n+1 the n-Postnikov i n v a r i a n t f o r E. 4.4.5. Since h F = g we see that k , r e s t r i c t e d to F, i s n 1 • n , n+2 . . _ *-n+2. . , -n+2 . k . - i n v a r i a n t f o r F; h^(k ) i s zero, because k i s i n the image of the transgression and t h i s follows e i t h e r by Serre exact sequence or by argument i n § 2.4. 4.4.6. Thus, i f we are given a cr o s s - s e c t i o n , f: B y E Oi i a* n+2 n+2 of ( E n + 1 , P | E n + 1 ' B n + 1 > E) , l e t to (f) e H (B; TT^ +^ (F)) be the obstruction to extending f over B _ . h of: B , >• E w i l l be a • n+2 n n+l n cross-section to the f i b r a t i o n , E ——> B ; and since rr, (F ) = 0 f o r n k n k > n , h f can be extended to a section f : B > E by 4.3.1(1). n n We s h a l l show i n the next section that ^*,-n+2. to(f) = f (k ) . §4.5. Secondary Obstruction and Construction of a Postnikov System In t h i s section, F ——>• E -P-+- B, i s a f i b r a t i o n with F s a t i s f y i n g n, (F) = 0 f o r m < k < n o r k < m . Then, using the Serre exact sequence,.we can deduce the following t e c h n i c a l lemma. 4.5.1. LEMMA ( c f . 128, §10.7] ) . Let f : B , * E be a n+J. cross-section ( assuming i t e x i s t s ) and l e t G be an abelian group. I f a e H m(F; G), then there i s an unique c l a s s , a, i n H m(E; G) such that * * i (a) = a and f (a) = 0. This c l a s s i s the same f o r any two cross-sections that agree on B^ . 4.5.2. Construction of a Postnikov System. Let a(f) be the unique c l a s s given by 4.5.1 such that * * rn m m f (a(f)) = 0 and i (a(f)) = i e H ( F ; rr ( F ) ) . ( x = fundamental m cl a s s of F ) . Let v"1 e H m(K(Tr (F) , m) ,* TT (F ) ) be the fundamental cl a s m m • * ^ Consider a map, g: E > K t i r ^ C F ) , m) , s u c h t h a t g (i ) = a ( f ) . g ex i s t s by the well known c l a s s i f i c a t i o n : H M ( E , Tr ( F ) ) < y [ E , K(TT (F ) , m) ] m m * M i l given by the correspondence, g (i ) •* >• [g] . Then g, r e s t r i c t e d to F , i s an ri-Postnikov map f o r F . (If i t i s not a f i b r a t i o n , we can always convert i t in t o one; however, i t i s always a f i b r a t i o n i n t h i s case.) Thus we have the following s i t u a t i o n : (4.5.3) m E gxp = h -y F = K (TT (F) , m) m m > K(TT (F ) , m) xB m . pro 3. •y K(TT ( F ) , m) m = E n This i s p r e c i s e l y a n-Postnikov system f o r E by 4.4.3. Mi+2 n+2 Let k e H (E ; TT ( F ) ) be the corresponding Postnikov n n+1 in v a r i a n t f o r E. Let X: B '-y E be the i n c l u s i o n map and l e t n -> E be an extension of h f : B , n n n+1 -y E. Denote by pro j . : E y K(TT ( F ) , m) the obvious p r o j e c t i o n on the f i r s t f a c t o r . n m Then (proj.of") (i ) = 0 by 4.5.3, i . e . , f - A . Thus we have the lemma. * o.h+2 * 'Vn+2 4.5.4. LEMMA. A (k ) ( - f Ck ) ) = 0)(f) . §4.6. A Postnikov Decomposition Let (E, p, B, F) be a f i b e r space. Assume 7!n^^ ^ 0 a n <3 TT (F) = 0 f o r k fi { n(j) }, where n(l) < n(2) < n(3) < I f n(l ) = 1 , we assume TT^ (F) to be abelian. Consider the following decomposition, o u t l i n e d i n chapter two : , n ( i ) ) (F), n ( i - l ) ) , n ( l ) ) where h., r e s t r i c t e d to F , i s g f o r i = 1, 2, 3, ••• ; and i i ( F , g., F . = K(T T . . . ( F ) , n ( i ) ) , F T ) i s a Postnikov system f o r F . Let i i n (l) i p^: E^ >• B be the f i b r a t i o n , determined by the decomposition, which has F. as f i b e r , l 4.6.1. Consider a CW decomposition, B^ C B^ C C B . Suppose f: B ... > E i s a cross-section. Then the obstruction n(}) cohomology c l a s s , u)(f), l i e s i n H n ^ + 1 ( B ; 7 r n ( j ) ^ F ^ * I f J = 1/ t n e f i r s t o b struction c l a s s , a/1 ^ + X ( f ) , i s simply the c h a r a c t e r i s t i c c l a s s of the f i b e r space, p: E »- B. Consider the diagram, h j-1 E J > E. y 1 I I I B ... r- B n ( 3 ) <• We have an extension, f t B >- E. , , of h. J : B ... >• E. , , D-l 3-1 n ( 3 ) 3-I since h^_^f has no obstruction. We wish to r e l a t e the secondary and higher obstructions to the f i r s t obstructions of some n-Postnikov maps of type F ' *• E , »• E . * n n+l n 4.6.2. Conversely given f : B > E j _ - j _ ' a cross-section of (E. , p. , B, F. • ) , l e t £.. denote f"|B Then f., can be l i f t e d 3-1 3-1 3-1 1 n ( 3 ) 1 to E, since i t has no obstruction to l i f t i n g . 4 . 6 . 3 . Denote by k j _ 1 ( E ) e H n ( j ) + 1 ( E . . ; TT ...(F)) the j - l n(]) c h a r a c t e r i s t i c c l a s s of the Postnikov system, K(rr ...(F), n(j)) >• E. »• E. n(;j) J j j-l . We have the following p r o p o s i t i o n r e l a t i n g the obstruction with the k-invariant of E. 4 . 6 . 4 . PROPOSITION, u) (f) = f "* ( k 3 " 1 (E) ) , where f." i s a cross-section of (E , p. ,, B, F._ ) and also an j x j A j i extension of h. , f : B ... >• E. , given as i n 4 . 6 . 1 . 3 - 1 n ( 3 ) 3 - 1 Proof. See R. Hermann [ 1 9 ] . ^ 1 4 . 6 . 5 . We s h a l l proceed to give a c h a r a c t e r i s a t i o n of k (E) e H n (BXK(TT . ( F ) , n ( l ) ) , TT . . ( F ) ) . With the previous assumptions, n i A; Yi \ Z) suppose there i s a cros s - s e c t i o n , f : B y E, over B ... By n( l ) n ( l ) lemma 4 . 5 . 1 , n (1) ( 4 . 6 . 6 ) there e x i s t s unique a(f) e H (E; ^(u^)) such that ( 1 ) f * ( a ( f ) ) = 0, and (2) i * ( a ( f ) ) = i n ( 1 ) e H n ( 1 ) ( F ; rr (F)) . n( l ) * r\j ^ Take g_^: E y K ^ 7 ^ ^ ( F ) > n ( l ) ) such that ^ ( i - ^ ) = a ^ f ^ ' where i s the fundamental c l a s s of K(TT (F) , n ( l ) ) . Define E, to be the product n ( 1 ) 1 B X K ( T T (F) , n ( l ) ) as we have done before i n 4 . 5 . 3 and obtain the n i l ; f i b r a t i o n given by h^ = 9^XP= E *" E^, and the corresponding k-'vl n(2)+l i n v a r i a n t k (E) e H (BXK(TT ,,, (F) , n ( l ) ) , TT (F)) of the , n ( l ) n (2) h l f i b r a t i o n , F^ • E y E^. -1 We have the following useful c h a r a c t e r i s a t i o n of k (E), i f 7 r / 0\ W ===' Z or Z £ a p o s i t i v e prime. 4.6.7. THEOREM. With the above hypothesis of 4.6.5 and 4.6.6 <\,1 v/e have that k (E) i s uniquely determined by (1) h^ i c^E)) = 0 , and —1 (2) k (E) , r e s t r i c t e d to K ( T T . , . ( F ) , n ( l ) ) , i s the Eilenberg-n (1) MacLane i n v a r i a n t of F , provided i t does not vanish. 4.6.8. Let X: B > E^ be the i n j e c t i o n . Then, by 4.5.3, we * ' V l see that X (k (E)) = u ( f ) . R e c a l l the following f a c t that the extension, f , of h of s a t i s f i e s f ' * ( k 1 ( E ) ) = io(£). (4.6.4) We look at the following example f o r a s p h e r i c a l f i b r a t i o n , n (1) F = S >• E • B, n ( l ) > 2 . S. D. Liao [28] gave a formula f o r a s p h e r i c a l f i b e r bundle with s t r u c t u r a l group, the orthogonal group, as follows: p*(u(f)) = S q 2 ( a ( f ) ) + p * ( y ( S q 2 a ( f ) ) ) ^ a ( f ) , where ¥: H n ^ + 2 ( E ; Z^) >• H 2(B; Z^) i s the " i n t e g r a t i o n over the f i b e r " homomorphism. Apply 4.6.7 to x = S q 2 ( a ( f ) ) + p ¥ ( S q 2 a ( f ) ) a ( f ) . * n(l)+2 Using the Gysin sequence we get x = p (b) f o r some b e H (B; Z^), because 6 : H n ^ + 2 ( E ; Z^) > H n ^ + 3 ( E , E n ^ 1 j / ' z 2 ^ takes x to zero. Thus, h*{ p r o j . * S q 2 (i ) + ¥ (Sq 2a (f j) + pro j . *b } = 0 i n By 4.6.7 we get p (u(f)) = p (X k X ( E ) ) = p (b) = x . 61 Example 4.6.8 i s taken from R. Hermann [18]; the reason i s that we can compare the way, S.D. Liao [28] obtains t h i s formula, which i s e s s e n t i a l l y being used i n theorem 4.6.7, with the ease that 4.6.8 obtains the same formula. However, Liao gave several c h a r a c t e r i s a t i o n s , i n p a r t i c u l a r , an e x p l i c i t d e s c r i p t i o n of ¥(Sq 2a(f)) [28, §19.3] . §4.7. L o c a l i s a t i o n The above example leads us to the method of using reduction mod SL (SL a prime) . There has been a l o t of work done i n t h i s d i r e c t i o n , e s p e c i a l l y that of Serre, and studies made, i n the r e l a t i o n of the Steenrod algebra to reduction mod £ homotopy with the use of the "natural homomorphism of the second kind" from homotopy groups to homology groups, and c h a r a c t e r i s a t i o n of c e r t a i n s p h e r i c a l homology clas s e s . ( Chou [13], Mahowald [30, 31], Thomas [59], Yo [62], Yu [63] ) The l a t t e r i s important to a l a t e r - chapter and also to a p p l i c a t i o n s to question o f the r e a l i z a b i l i t y of a s p h e r i c a l c l a s s as an embedding. We s h a l l only sketch the idea of reduction mod SL i n obstruction theory. 4.7.1. Notation as before. (E, p, B, F) i s a f i x e d f i b e r space. Suppose f: B ^ + ^ > E i s a c r o s s - s e c t i o n and i a prime number. Let w ( f ) ^ denote the o b s t r u c t i o n c l a s s reduced mod SL corresponding to w(f) i n H n + 2 ( B ; TT (F))<S>Z„. F i r s t a t e c h n i c a l lemma. 4.7.2. LEMMA. Suppose TT . (F) Z = 0 f o r j <_ n, i . e . , rr. (F) has no £-torsion f o r j <_ n. Then to(f)^ i s transgressive. In f a c t , w ( f ) ^ = T ( . I ^ ) where \^ i s the reduction modulo I of the fundamental c l a s s , \, of H n + 1 ( F ; TT . CF) ) . n+I Hence i represents a homotopy class of map F > TT (F) © Z Thus we can employ the method of §4.6 to the secondary obstruction mod £ to the case when TT , (F) ® Z = 0 f o r j < m o r m < j < n . I n the m P case of a s p h e r i c a l f i b r a t i o n , F = S y E y B, m >_ 2, we can show,.similarly as i n 4.6.8, that -if f: B , y E i s a cross-section n+1 then we have Liao's formula: (4.7.3) p * ( 0 3 ( f ) £ ) =<pj(a(f)) - p%«pj(a(f)))^a(f) . §4.8. Secondary Difference Obstruction We want to show how our previous discussions can help to describe d i f f e r e n c e obstructions and t h e i r p r o p e r t i e s . 4.8.1. Notations and assumptions as i n §4.6. Let (E, p, B , F) be a f i x e d f i b e r space. Suppose f^, f ^ : B n + 1 y E# a r e t w o cross-sections. Assume F i s (m-1)-connected and TT . (F) = 0 f o r m< j < n . D — Then, by 4.6.6, we have the following f i b r a t i o n s : h = p x g i : E y E = BXK(TT (F), m) , and n 1 n m = P x g „ : E y E = B*K(Tr (F , m) , n 2 n m — . ^n+2 Mi+2 n+2 and corresponding k - i n v a r i a n t s , k, (E) , k„ (E) e H (E ; TT , (F)) . 1 2 n n+1 * rn Tn Thus, by 4.6.6, i ( a ( f 2 ) - a ( f )) = i - i = 0 . Hence a ( f ) - a(f±) p (b) f o r some b e TT (F)) by exactness. Let i E m m H M ( K ( i T (F) , m) , TT (F)) be the fundamental c l a s s and l e t a: E m m n • y K(TT (F) , m) be the p r o j e c t i o n ; then (aoh') \ = a(f„). Thus m n m 2. b = f*p*(b) = f * ( a ( f 2 ) - a C ^ ) ) = f * a ( f 2 ) * * -ki\j = 'f.H' a T I n m Let Y, : B *• K(TT (F) , m) corresponds to b under the isomorphism, b m H m(B; TT (F)) < > [B, K(ir (F) , m) ] , given by X 7 < > X ; that i s , m m m Y , i = b . Then y, \ = (ah'f_) i implies that b m b m n l m (4.8.2) Y K I B ^ . - oh'f b 1 n+1 n 1 Consider the graph of / G(Yj ; )) : B *" , given by G(Y B)(b') = (b', Y b(b')) f o r a l l b' s B. Then, by 4.8.2, 4.6.2 and 4.5.4, we have the following lemma. * ^n+2 4.8.3. LEMMA. G (Y, ) (k' (E)) = to(f,) . b 2 1 4.8.4. From here we put a d d i t i o n a l conditions on TT N (F ) . We n+1 s t i p u l a t e that TT (F) has a p a i r i n g , TT , (F) ® TT (F) * TT (F) , n+l n+l n+1 n+l so that we can apply the Kunneth theorem. Let w denotes i t s cup-product with respect to t h i s p a i r i n g . We assume also that N "vr\+2 ' ' n * * n+2 (4.8.5) k 2 (E) = I p n ( b ) w v (6 ) £ H n + ^ ( E n ; ir (F)) , j=0 2 2 * where V: E >• K(TT (F) , m) i s the p r o j e c t i o n , b. E H (B; TT . (F)) , n m j n+l 9. e H*(K(f (F) , m) , IT ,, (F)) , dim(b A) = 0 and 6, = n+2 . 3 m n+1 0 N N * —n+2 r * 4.8.6. PROPOSITION. G(y.) (k (E)) = I b.v/y, (6.) , and ^ j=0 3 3 N * • o)(f ) - w ( f ) = I b v y . i e ) . j = i ° J Proof. The f i r s t statement follows from the d e f i n i t i o n of Gty. ). The second comes from the f i r s t and the f a c t that b * 'Vn + O * V n + ? <o(f2) = f^ (k^ (E)) = X (k 2 (E)) (4.5.4) and * rhn+"? Oi(f±) = G(Y b) (k 2 (E)) (4.8.3). . * "Vn+? * * * * * * For X (k^ (E)) = X p ( b Q ) ^ X V (0 Q) = b ^ X V (6 Q) = V f 2 V ( V = b 0 ^ Y ^ ( 9 0 ) bY 4.8.2 ; i t i s then e a s i l y seen that the second statement i s true. This completes the proof. We are now ready to state the main r e s u l t of t h i s s e c t i o n . ' * 4.8.7. Every cl a s s 0 e H (K(TT (F) , m) , TT ...(F)) determines a m n+l primary cohomology operation. ( See f o r example [44].) Denote i t by the same l e t t e r , 0: H m(X; TT (F)) >• Em+3 (X; TT (F)) . L e t t i n g X = B , m n+l * we see that YMOJ) corresponds to 0.. (b) . Thus, with previous notations and assumptions, we have the following theorem. —n+2 THEOREM. Suppose k (E) s a t i s f i e s 4.8.5. For a f i x e d cross-s e c t i o n , f : B , y E. there i s a cross-section over B _ i f f there n+l n+2 i s a cohomology c l a s s , b e H m(B; ^ ( E ) ) , such that (4.8.8) a)(f) + I (b) = 0 j = l 3 D Proof. ( = > ) By 4.8.6. * % (<= ) Let y corresponds to b = y ( l m) • Then Y: B y KCTT IF), m) gives a graph, G(y): B y E . Since m n h^: E y E^ induces isomorphisms f o r homotopy groups i n dimension les s than or equal to n, G(y) can be l i f t e d to a cross-section, f l : Bn+1 E' with G^ Y^l Bn+l = h n f l ' T ] i e : n 4 - 8 ' 6 and 4.8.8 apply to give w(f.) = 0 ; and so i t has a cross-section over B ... 1 n+2 We have so f a r not consider any obstruction f o r p r i n c i p a l f i b e r bundle, nor d i d we use any theory of c l a s s i f y i n g spaces and c l a s s i f y i n g bundles to a large extent. We s h a l l proceed to the next chapter describing obstructions to extending cross-sections of p r i n c i p a l f i b e r bundles. The technique follows more or l e s s the Moore-Postnikov type decomposition. CHAPTER 5. PRINCIPAL.FIBER.BUNDLES In t h i s chapter we s h a l l discuss an obstruction theory u t i l i s i n g the idea of a Postnikov r e s o l u t i o n which has now become the most powerful t o o l i n algebraic topology. Among the p r i n c i p a l f i b e r bundles discussed are U(n), SO(n)-and SP(n) bundles. We s h a l l more or less state the c l a s s i f i c a t i o n theorem f o r p r i n c i p a l G-bundle; and much i s assumed about the construction of u n i v e r s a l bundles and c l a s s i f y i n g spaces [41]. §5.1. D e f i n i t i o n s A p r i n c i p a l f i b e r bundle, £ = (E, p, B, G), i n the sense of Steenrod, i s a f i b e r bundle with the f i b e r , G, a t o p o l o g i c a l group, acting continuously on E and compatibly with the co-ordinate neighbourhood system of E; that i s , the l o c a l t r i v i a l i s a t i o n s , { ( J i ^ : U x G •> p 1 (U) }^  , s a t i s f y the f o l l o i n g conditions: I f b E D, and g,, g e G, then 1 2 CD 3 2**u t b' g l ) = *u ( b' g 2 * g l ) " (2) g«x = x f o r a l l x e E <i—*> g = e . C3) g 2«(g 1*x) = ( g 2 ' g 1 ) x f o r a l l g , g 2 e G and any x e E We s h a l l always make the following assumptions i n t h i s chapter unless otherwise stated. I f £ = (E, p, B, F) i s a f i b e r bundle, the base space, B, i s always a l o c a l l y f i n i t e connected s i m p l i c i a l polyhedron. B n w i l l denote the n-skeleton of B. We s h a l l also assume 67 that the f i b e r , F, i s connected and i s n-simple i n a l l dimensions. We s h a l l be dealing with f i b e r bundles i n the sense of Steenrod and so we need some f a c t s about p a i r of f i b e r bundles over B. §5.2. P r e l i m i n a r i e s 5.2.1. COVERING HOMOTOPY THEOREM. Let (£, where £ = (E, p, B, F) and = (E', p % B, F') , be a p a i r of f i b e r bundles over the base space, B. Let X be l o c a l l y compact and paracompact, and suppose X i s closed i n X. I f f : (X, X ) y (E, E') i s a l i f t i n g o f o o h|xx{0} : X > B, then there e x i s t s a map, H: (Xxi, X Q X I ) > (E, E') , such that H|XX{0} = f and poH = h . 5.2.2. LEMMA. Let (£, %,') be a p a i r of f i b e r bundles over B; and suppose B^ i s a subspace of B. Let k:B^C B denote the i n c l u s i o n map. Then, f o r m >_ 2 , the following induced map: Proof. Apply the usual covering homotopy theorem and the covering homotopy theorem, given as i n chapter one. •*• TT (E, E') i s an isomorphism and we have the s p l i t t i n g : TT (E, E'') © TT (B, B ) . m m o Proof. Consider the diagram: o m B o rr ( E , E ' ) m B o TT (E, E") m k_ TT (E , E" ) m B B o o j . i s an isomorphism f o r m > 1 , since rr ( E . E ) ^  TT (B. B ) ~ * — m B '— m o — o Tr m(E, Eg ). This implies i s onto and j # i s a monomorphism. So f o r o m >_ 2 t h i s gives the two short exact sequences shown. k A i s onto by the b u t t e r f l y lemma and the f a c t that k^ i s onto, k^ i s also a mono-morphism. Because i f k (x) = 0, then k'k (x) = 0. So by exactness there.exists y e TT (E', E ' ) such that k. (x) = j A ( y ) . 0 = j'k (x) m j * j * ( y ) = 0 implies y = 0, since j i s an isomorphism. Thus k^(x) = 0, i . e . , x = 0. Hence Ker( k #) = 0 and k^ i s a monomorphism. Thus we can construct a s p l i t exact sequence, 0 »- TT (E', E" ) m B o -> TT (E, E ' ) m B o -> TT (E, E') m and so we have the s p l i t t i n g , -> 0 TT (E, E" ) ^ T T (E, E') 9 u (B, B ) m B m m o o 5 . 2 . 3 . COROLLARY. Hypothesis as i n 5 . 2 . 2 . Taking B q = *, we see that f o r m > 2 we have the following isomorphism: k. : TT (F, F') 5- rr (E, E") . * m m With. 5 . 2 . 2 and 5 . 2 . 3 , one can e a s i l y prove the following c h a r a c t e r i s a t i o n of the s i m p l i c i t y of the p a i r , (E, E'). 5 . 2 . 4 . THEOREM. (1) Suppose (F, F") i s m-simple. Then TT^ ( B ) acts t r i v i a l l y on TT (F. F') i f f (E, E') i s m-simole. m C2) I f CF/F') i s (m-1)-connected and (F, F") i s m-simple, then the induced map, k.: H (F, F") > H (E, E') i s an • * m m isomorphism i f and only i f TT^ ( B ) operates t r i v i a l l y on 1 7 ( F* F')» The proof o f 5 . 2 . 4 r e l i e s on the following p r o p o s i t i o n , which we state here f o r future use. 5 . 2 . 5 . PROPOSITION. Suppose (F, F') i s m-simple. Let to e TT CE', x) , P(XQ) = b Q , o E T T M ( F / F') and k: (F, F') >- (E, E') be the i n c l u s i o n . Then k* (p^ ( to ) • ct ) = to © k # ( a ) , where " © " denotes the usual operation of TT. (E' 7 X ) on TT (E, E', x ) 1 o m o C see Steenrod 154 ] ) . Proof. See Lundell [ 2 9 ] . § 5 . 3 . Obstructions f o r P a i r of Fiber Spaces In chapter 3 we have discussed an obstruction theory to the existence of cross-sections f o r f i b e r spaces. In an analogous way ( N. Steenrod [54] ), i f given a p a i r of f i b e r spaces over a s i m p l i c i a l a. complex, one can define a l o c a l c o e f f i c i e n t system, t tF, F -*) , over B and develop an obstruction theory f o r p a i r of f i b e r spaces. Let (.£, £') be a p a i r of f i b e r bundles over a s i m p l i c i a l complex, B, such that the p a i r , ( F , F " ) i s (n+l)-simple. Then, i f f: (B, B n ^ B o ) *• (E, E ' ) i s a cr o s s - s e c t i o n , there i s a deformation cochain (. i n f a c t a cocycle ) d n + 1 ( . f ) e c n + 1 ( B , B ; TT (F , F ' ) ) o n+l [29, 54] having the following p r o p e r t i e s . 5.3.1. (1) d n +^" (f) = 0 i f f there i s a cross-section f B »- E s a t i s f y i n g f ' ( B n + 1 U E ) C E ' and f - f r e l (B n *J B ). o o (2) f =- g r e l B q = > d n + 1 ( f ) = d n + 1 (g) . (3) 6 d n + 1 ( f ) =0. — n+l (4) d (f) i s a t o p o l o g i c a l i n v a r i a n t . (5) I f d (f) = 0, then f can be deformed r e l a t i v e to B n \, B q to a cross-section, f : B > E , such that f ( B n + 1 ( j B ) d. E " . ( See chapter 3, 3.2.1 or [54].) We next r e l a t e the connection between the deformation cochain n+l and the fundamental c l a s s , i , of ( F , F') i n case ( F / F ' ) i s n-connected. Consider the boundary operator, 9„: TT (F, F ' ) >• TT (F ' ) . # n+l n In a n a t u r a l way, i t induces the following map of l o c a l systems: rr , (F, F") > TT (F') and hence a map, n+1 n n+1 'v n+1 ^ 9 #: C N + ± ( B , B Q ; i r ^ t F , F") ) » c" ^ ( B , B Q ; I T ^ F ' ) ) , also denoted by 5 . tr We have the following p r o p o s i t i o n connecting the deformation cochain with the obstruction cochain. 5.3.2. PROPOSITION. Suppose f : ( B , B N O B ) >- (E, E') i s a o ~, ^ ,n+l . „. n+1 , _ I _n „ . cross-section. Then 9,.d (f) = c (f B <->B ) . # ' o Proof. Straightforward. Just check the d e f i n i t i o n , o f c ( f | B u B ) . From now on, we assume (F/F') i s n-connected and (F, F') i s (n+1)-simple. Then 5.2.4 t e l l s us that TT^(B) acts t r i v i a l l y on TT , (F, F ' ) i f and only i f (E, E') i s (n+1)-simple. So we assume TT. (B) n+l l operates t r i v i a l l y on TT (F, F") , so that 5.2.4(1) and 5.2.4(2) hold; n+l i . e . , (1) TT . (E, E") i s simple; and (2) H.(F, F') >-H.(E, E') i s n+1 j j an isomorohism f o r j £ n+1. n+1 5.3.3. LEMMA. Let i be the fundamental cl a s s of (F, F"). — n+1 * *—*—1 n+1 n+1 Then d (f) = f k i (i ) e H (E, E ' ; TT (F, F )) and B n+1 o — n+1 * *—*-1 n+1 n+1 c (f) = f (3„(k i - " ( i " X ) ) ) e H ( B , B ; TT ( F ' ) ) , where # o n i * " 1 : H n + 1 ( F , F ' ; * N + 1 & , ) — • H n + 1 ( E , E'; ) / k*: H n + 1 ( E , E'.- TT ( F , F \ ) ) + H ^ C E , E' ; TT ^ (F, F ' ) ) and n+1 • — B n+l * n+1 n+1 0 f. : H n + ± ( E , E ' ; TT (F, F ' ) > »• H" X ( B , B ; TT (F, F ' ) > . B n+1 o n+1 o Proof. See [29] n+l *—*—1 n+l I f we define A ( £ , C ) = k i ( l ), then 5.3.3 gives — n+l * n+l — n+l * n+l (5.3.4) tf n X ( f ) = f X LK, V)i c n (f) = f 9 #A n + J-(£, r ) . §5.4. Applications In t h i s s e c t i o n , we s h a l l t a l k about p r i n c i p a l f i b e r bundles, state the well-known c l a s s i f i c a t i o n theorem and describe the a p p l i c a t i o n o f the l a s t sections to p r i n c i p a l G-bundles. 5.4.1. THEOREM. (1) A p r i n c i p a l bundle, (E, p, B, G, G), i s equivalent to the product bundle i f f i t admits a cross-section. (2) ( C l a s s i f i c a t i o n ) There i s a one-one correspondence between cross-sections of a p r i n c i p a l f i b e r bundle, (E, p, B, G) , and maps,' { h: E > G; h(g.x) = g.h(x) f o r . a l l x e E and for a l l g e G }• We assume the d e f i n i t i o n of a p r i n c i p a l map between p r i n c i p a l bundles due to Steenrod; i . e . , i t i s a t r i p l e (<f>, k, I) such that the diagrams: <J> . u G - G G x E - > E B -> E' <j>xk G' x E' -> E' -y B are commutative. Then the map, h^: E. >• G, corresponding to a cross-section as i n (2) of theorem 5.4.1, i s given by h^(x)'fp(x) = x f o r a l l x e E [7, 54]. With, t h i s c l a s s i f i c a t i o n , we have the following immediate consequence. 5.4.2. C O R O L L A R Y . Suppose B = B " , £ = 1 and notations are as B i n 5.4.1. £) i s a p a i r of p r i n c i p a l f i b e r bundles over B . I f £ has a cr o s s - s e c t i o n , f: B > E, then . f = k f i s a cross-section and <j>hf = h Jc. 5.4.3. With, t h i s c o r o l l a r y , one can define a deformation cochain f o r p r i n c i p a l bundles as a f i r s t step towards a c l a s s i f i c a t i o n of cross-sections, {f: B >• E } , which agree on a subspace or sub-complex, B Q C B . Suppose f ^ , f ^ : B • E are cross-sections of £ = (E, p, B , G, G) such that f Q | B = f xl B* L e t h f , h f : E >• G be given by 5.4.1. o 1 Then define h^ f : B *• G to be the composition, h^ of^. The o 1 o following i s a c l a s s i f i c a t i o n theorem. THEOREM, f - f, r e l B h, ^ - * r e l B . o 1 o f f, o o 1 Proof. See [29, pp.170]. Thus, with t h i s theorem, one can define h : ( B n u B , B n "*"UB 0 1 ° ° >• (G, e) i f one i s given cross-sections, f , f, : B u B > E o 1 o n—1 which agree on B u B 0 * With t h i s map, one can define a cochain, n n — d (f , f ) e C (B, B ; TT (G, e)) unambiguously by o x o n n — i • by d (f , f )(a) = h f |(a, a) f o r a l l a. This d e f i n i t i o n coincides ° 0 1 with that of the d i f f e r e n c e obstruction due to Steenrod; i t has the properties given i n chapter 3, namely 3.1(4), 3.1(5), 3.1(6), 3.1(7) and 3.1 (10). The following p r o p o s i t i o n r e l a t e s the fundamental cl a s s of G with the d i f f e r e n c e cocycle. The proof i s analogous to that of lemma 5.3.3. 5.4.4. PROPOSITION. Let (E, p, B, G, G) be a p r i n c i p a l f i b e r n+l bundle with the f i b e r , G, (n-1)-connected. Suppose f , f, : B >• o 1 are cross-sections such that f IB" = f, IB" 1 . Then h r r : ( B n + 1 , B n o 1 1 1 f f, * _ 0 1 • (G, e) i s defined and h f f ( i n ( G ) ) = d n ( f Q , f ) e H n(B; TT (G)) . o 1 §5.5. Decomposition of P r i n c i p a l F i b e r Bundles and Moore-Postnikov Invariants In t h i s s e c t i o n , we s h a l l begin with the following r e s u l t s about u n i v e r s a l bundles. Throughout £ = (E, p, B, G, G) denotes a p r i n c i p a l bundle with f i b e r and s t r u c t u r a l group, G. G i s assumed to be a connected countable CW-group i n the sense of Milnor [41]. 5.5.1. THEOREM. Any countable CW-group, G, has an associated u n i v e r s a l bundle, ^ = (EG, p, BG, G, G), where BG i s a countable CW-complex and EG i s c o n t r a c t i b l e . ( Milnor 141, II§5] ) 75 We remark that there i s also an u n i v e r s a l s p e c t r a l sequence by which one can r e l a t e the group, G, and i t s c l a s s i f y i n g space, BG ( See [7] or 141]. ) Our method w i l l depend h e a v i l y on Postnikov decompositions. The following p r o p o s i t i o n guarantees the existence of a Postnikov system for G; thus we can then use t h i s to obtain the desired decomposition and hence to describe an obstruction theory f o r p r i n c i p a l bundles with f i b e r and s t r u c t u r a l group, G. 5.5.2. PROPOSITION. Let G be a connected countable CW-group. For any integer N ^_ 1 there i s a f i n i t e CW-group sequence, ^ YN „ YN-1 „ „ Y l G = G . y G y G , , >• »•• G„ y G, , N+l N N-1 2 1 which i s a Postnikov system for G, where each Y n l s a n embedding and G i s homotopically equivalent to G = G ^* Proof. See Lundell [29]. 5.5.3. In p a r t i c u l a r , i f we are given a p r i n c i p a l bundle, £, over a CW-complex, B, with f i b e r and s t r u c t u r a l group, G, then we have a Postnikov decomposition f o r £, which we may assume to have been induced from the Postnikov system f o r G, G C GN+1 ° GN C V l C C G 1 * By 5.5.2 we may even assume that { (£ . , £ ) } are p a i r s of p r i n c i p a l n+1 n f i b e r bundles over B, where £ = (E , p , B, G , G ) i n the Postnikov n n n n n decomposition [ Chapter 4 ]. In the terminology of chapter 4, (E , G , g ) i s a Postnikov system f o r E; i . e . , the diagram, n n n > G n-1 > E n-1 i s commutative, i n p a r t i c u l a r , TT . (E . , E ) = 0 f o r i j4 n+l ; 1 n - l n TT . (E , E ) = TT (G) n+l n-1 n n This f i n i t e decomposition i s abbreviated (£, N, {£ }, {(y , g , 1)}), m m m a decomposition of length N. We would l i k e some kind of c o m p a t i b i l i t y condition on t h i s decomposition. Kahn [25] has shown that given two Postnikov decompositions on two f i b r a t i o n s , then i f there i s a map from one to the other, i t induces a map between the Postnikov systems. Although t h i s construction i s not f u n c t o r i a l , we are s t i l l contented since we only need to know that i f we have a map from one system to the other, we are guaranteed a r e l a t i o n between the k-i n v a r i a n t s . 5.5.4. LEMMA [25; 29]. Let OS, N, {£ }, {(y , g , 1)}) be a n n n decomposition f o r £ = (E, p, B, G, G) . Suppose f : B' >• B i s a map. Let f (?) be the induced bundle. Then f." ) are the bundles i n a n * decomposition of f (.£) . This lemma t e l l s us that IE, N, {E }, { ( Y , g , 1)}) i s induced n n n by a c l a s s i f y i n g map, <j)^ : B »• BG i f a decomposition f o r E i s given. So from now on we assume we already have a f i x e d decomposition fo r the u n i v e r s a l bundle over BG and any decomposition over £ = (E, p, B, G, G) i s induced by i t s c l a s s i f y i n g map, ti : B * BG. 5.5.5. PROPOSITION. Let E = (E, p, B, G, G) be a p r i n c i p a l bundle over a CW-complex, B. Then there e x i s t s a cross-section, f: B n >• E i f f there i s a cross-section, f : B ——> E , , of £ , = n-1 n-1 (E » , p n , B , G » , G _) which i s a term i n the decomposition of E n - l n - l n-1 n—1 of length N >_ n-1. Proof. See Hermann [18] or 4.6.1. The footpath has been set; the r e s t of the work i s analogous to that used by R. Hermann to c a l c u l a t e the obstruction. 5.5.6. Since G _/G i s n-connected, we see that (G ., G) i s n-1 n-1 (n+1)-simple and TT (B) operates t r i v i a l l y on TT (G , , G) . Hence l n+1 n-1 —* n+1 n+1 i : H (E -,.E; * (G , G) ) > B. [& , G; TT , (G , G)) i s n-1 n+1 n-1 n-1 n+1 n-1 an isomorphism. Suppose B q i s a subcomplex of B. Define as i n 5.3.4, .n+l,,. ... * - r * - l , n+l v n+1, . . % K (5 . , K) = k i (x ) e H (E _ , E ; TT CG . , G)) , where B n-1 n-1 o n+1 n-1 *° n+1 n+1 k : H (E , E; ir . CG . , G)) v H (E , E ; TT . (G , G)) i s n-1 n+1 n-1 n-1 o n+1 n-^ 1 induced by i n c l u s i o n , and i i s the fundamental cl a s s of (G , # . G ) . n - l Since 3„: TT I N (G , , G) >• TT CG) i s an isomorphism ( 5.5.2 or 4.4.2 ), # n+1 n-1 n define X * + 1 ( £ ) = 3„x" + 1(E . , E) z Hn+^" (E E : TT (G)) . B # B n-1 n - l o n o o 78 The A's enjoy the following important p r o p e r t i e s : (1) Let h , : . (E, E ) -> (E , , E ) be the map given i n the n-1 o n ~ l o * n+l decomposition. Then h ,(A„ (£)) = 0 . n-1 B o (2) I f £ has a cross-section, f : B \j B q >• E, l e t f ' be an extension of h . f : B N u B • E . , then n-1 o n-1 — n+l,... ,^*,n+l.., , n+l • • ,„\ vc (f) = f A (?) e H ( B , B ; rr (G)). B o n o Proof. See 5.3.4. 5.5.7. In a s i m i l a r way, we can deduce the following connection between the k- i n v a r i a n t s of G and the A's. Since (G . , G) i s (n+1)-n-1 simple, the k - i n v a r i a n t of G, kn+"^ (G) , i s defined, and equals T ( T . n + 1 ( G , G)) e H n + 1 (G TT (G)) , where T i s the transgression i n n-1 n—1 n the f i b e r space, G' »- G >• G . kn+"^ (G) i s also the image of n n i n + ^ ( G ,, G) under the composition, n-1 H n + 1 ( G n - l ' G ; V i ( G n - l ' G ) )7— H n + 1 ( G n - l ; V l , G n - l ' G ) ) ~ ' .-n+1 H n + ± (G ; rc (G)) . n-1 n Let A n + 1 ( £ ) denote the corresponding element to A n + 1 ( 5 ) when B =0. B 0 0 Then we have the fo l l o w i n g : PROPOSITION. (1) i * _ ( A n + 1 ( ? ) ) = k n + 1 ( G ) , where I : G C E n-1 n-1 n-1 n-1 i s the i n c l u s i o n of the f i b e r . (2) N a t u r a l i t y . I f <j>: B' >• B i s a map and * (<j> ,,<(>): <J> K , • ? , i s the induced map from the induced n-1 n-1 n-1 * * n+l decomposotion on <j> £ by a decomposition on £, then ^. jtx (?)) = An+1<**<0> e H n + 1 ( E ' .; TT (G)). n-1 n Thus t h i s p r o p o s i t i o n together with 5.5.6 gives the following r e s u l t s about the decomposition on the u n i v e r s a l G-bundle. PROPOSITION. Let C = (EG, p, BG, G, G) be the u n i v e r s a l bundle 'Xj • ^'Xj • 'Xj 'XJ -f o r G. Suppose i t i s given a decomposition, (.£, N, {£ }, { (y , g , 1)/) m m m of length N. Then (1) T T i ( E n ) =* { f o r i < n+2, rr. , (G) f o r i > n+2; i - l — (2) H. (E ) = { l n f o r 0 < i < n+l, rr , (G) f o r i = n+2; n+l i — n+2 "\J (3) H (E ) = 0 f o r 0 < i < n+l, and H (E ; TT ,(G)) i s a n — n n+l n+2 — c y c l i c Horn (TT (G) , rr (G)) -module generated by X (5(G)) n+l n+l The proof of t h i s p r o p o s i t i o n i s straightforward, by checking on.(1) and applying the Hurewicz theorem and u n i v e r s a l c o e f f i c i e n t theorem to get (2) and (3). ( See [29].) C o l l e c t e d i n the next section are the main theorems f o r p r i n c i p a l G-bundles, which we s h a l l use to obtain information about U(n), SO(n) and SP(n) bundles. §5.6. The Main Theorems In t h i s section, we s h a l l always assume that G i s a connected countable CW-group. Most of the following can be proved by e x p l i c i t l y looking at the Postnikov decomposition as i n chapter 4. We s h a l l assume * n+1 n+1 that p : H (B;'ir (.G)) »- H (E; T T ^ (G)) i s a monomorphism. Then the following theorem determines the obstruction completely. 5.6.1. THEOREM. Suppose (E, N, {E }, { (y , g , D>) i s a m m m decomposition o f length N >_ n. Cl) Let f: B N E be a cross-section of E over the n--skeleton. Suppose f ' : B > E , extends h , f: B N > E , so that, n-1 n-1 n-1 by our c l a s s i f i c a t i o n theorem, h„.-:E , G , i s defined. Then f n-1 n-1 * — n+1 * * n+1 p (c n + 1 ( f ) ) + h .h_.(k n + J-(G)) = 0. n - l t (2) Suppose f ^ , f ^ : B" ^ E are cross-sections and f ^ , f ^ : B N E , are the extensions of h . f and h f to B r e s p e c t i v e l y n-1 n-1 0 n-1 1 so that h_^.^ : B *• G , i s defined. Then f o f i n - x o N + 1 C F L ) - c n + 1 ( f n ) = h* ( k N + 1 ( G ) ) . 0 1 C See Lundell 129,.theorems 6.1, 6 . 2 ] . ) Notice that (2) says that i f k (G) = 0 and the (n+1)-dimensional o b s t r u c t i o n i s defined, then i t i s a sing l e cohomology cl a s s ( c f . 3.1.8 ). Let the primary cohomology operation, determined by kq+"^ (G) e HQ + 1( G , ; TT C G ) ) be denoted by i ^ * " 1 " . Here, of course, we assume that q-1 q TT^CG) = 0 f o r i <_ n-1 or n < i < q. Thus, suppose f: B Q > E i s a cross-section to E - ( E , p, B , G , G ) over the q-skeleton and a n ( f ) E H " C E ; TT (G)) i s the unique element determined by lemma 4.5.1, then, n using theorem 5.6.1, we can e a s i l y prove the following: 5.6.2. THEOREM. Hypothesis as above. (1) P*C c q + 1 ( f ) ) - ^ q + 1 ( a n ( f ) ) ; (2) I f f Q f f l S B q — y E are two cross-sections, then c q + 1 ( f x ) - o q + 1 cf0) = ^ q + 1 c a n C f 0 , f ± ) ) . An immediate cosequence i s the following c o r o l l a r y . 5.6.3. COROLLARY. With the same hypothesis as 5.6.2, suppose f: B q -> E i s a cross-section. Then f : B q + 1 >• E i s a cross- • section i f f c q + 1 ( f ) + ^ q + 1 ( d n ( f , f )) = 0. For d e t a i l s of the proof of theorem 5.6.2, we r e f e r to [29]. §5.7. Obstructions and C h a r a c t e r i s t i c Classes In t h i s s e c t i o n , we give an a p p l i c a t i o n of §5.6 to U(n), SO(n) and SP(n) bundles. In p a r t i c u l a r , we i n d i c a t e how one can consider some c h a r a c t e r i s t i c classes as obstructions. For the homotopy groups of the c l a s s i c a l groups, at l e a s t the stable homotopy groups, see R. Bott [8]; and f o r the cohomology rings of the c l a s s i c a l groups, see Borel [7], Milnor [40] or Epstein and Steenrod [16]. For the cohomology of c l a s s i f y i n g spaces of the c l a s s i c a l groups, see [7] or [40], We s h a l l make use of the un i v e r s a l Chem classe s to define the Chern classes o f a p r i n c i p a l bundle v i a the c l a s s i f y i n g map, s i m i l a r l y f o r Stiefel-Whitney c l a s s e s , Pontrjagin c l a s s e s and Euler-Poincare classes ( Hirzebruch [23], Milnor 140] ). 5.7.1. Let E = (E, p, B, U(n), U(n)) be a p r i n c i p a l U(n)-bundle * a, and E = (EU(n) , p, BU(n) , U(n), U(n)) be the uni v e r s a l bundle. Assume (tr N/ xE )/ {(Y / 9 ' 1)}) i s a f i x e d decomposition of E of length m m m N >_ 2n. Then we have the following : THEOREM. I f X e H ^ 2 k - 2 ' U 2 k - 1 ( " ( n ) } } = H (E2k-2'" Z ) f o r n > k i s as defined previously, then a.* <v 2k 'v P 2 k _ 2 ( c k ) = ± (k-1)IX K ( £ ) , ^ . th where c, i s the k -Chern c l a s s of the un i v e r s a l bundle 123]. k Note that we have the r e s u l t that n\ (U(n))== Z f o r i B 1 (mod 2) and i < 2n-l; TT. (U(n)) = 0 f o r i = 0 (mod 2) and i < 2n-l; TT„ (U(n)) £=; — l — 2n n! For the proof of the theorem we r e f e r to [29]. With notations as before we obtain the follow i n g : 2k-l 5.7.2. COROLLARY. Suppose f : B • E i s a cross - s e c t i o n f o r E = (E, p, B, U(n), U(n)) over the (2k-l)-skeleton of B. I f n > k, the n a t u r a l i t y of Chern classes gives c kU) = ± (k - l ) ! c 2 k ( f ) . Proof. Let $£.: B BU(n) be a c l a s s i f y i n g map f o r £. Then * % * 2k-2*^* 2k-2 C k = ^£^ ck^ = f ^ ^ E P2k-2^°k^' w h e r e f ' i s a n extension of h f , * 2k-2* 2k 'Xt • = ± ( k - l ) l f ' ^ T (5) by theorem 5.7.1, ± '{k-1) If'V k ( * * (£ ) ) by 5.5.7, = ± ( k - l ) ! c (f) by 5.5.6(2). This completes the proof o f 5.7.2. 5.7.2 has the f o l l o w i n g easy immediate a p p l i c a t i o n . 5.7.3. LEMMA. Suppose M 2 n i s an almost complex ma n i f o l d such t h a t the cohomology t o r s i o n c o e f f i c i e n t s i n dimension 2k are r e l a t i v e l y prime to (k-1)! f o r k = 1, 2, ••• , n. Then M 2 n i s p a r a l l e l i z a b l e i f f a l l the Chern c l a s s e s o f i t s tangent bundle v a n i s h . Proof. By 5.7.2, 5.4.1 and the d e f i n i t i o n o f p a r a l l e l i s a b i l i t y [40]. C For the r e l a t i o n between the P o n t r j a g i n c l a s s e s o f M and the Chern c l a s s e s o f M see Hirzebruch [23, Theorem 4.6.1].) The f o l l o w i n g theorem summarizes the c h a r a c t e r i s t i c c l a s s e s as o b s t r u c t i o n s . 5.7.4. THEOREM. L e t S = (E, p, B, U(n), U(n)) be a p r i n c i p a l U(n)-bundle w i t h c l a s s i f y i n g map, <j>^ : B »• BU(n). — 2 (1) The primary o b s t r u c t i o n to a c r o s s - s e c t i o n i s c (f) = c^ ( C ) . This i s the on l y o b s t r u c t i o n i f n = 1, s i n c e U ( l ) = S 1 (2) Assume n > 1. Then, i f c^(£) = 0, the secondary o b s t r u c t i o n 4 i s c 2 (?) e H (B; Z) . (3) Suppose c^(S) = c 2 ^ = ^' then there i s a c r o s s - s e c t i o n , 4 5 4 f Q : B ~—•> E ( f •: B »- E, i f n > 2 ) and i f f ^ : B >• E ( 5 r e s p e c t i v e l y f. : B ——>• E ) i s another c r o s s - s e c t i o n , l e t f f . " : B — 1 o 1 -> E 3 be the extensions o f h ^ f ^ and h^f^ ( r e s p e c t i v e l y h^f^ and h ^ f x Then, (i) i f n = 2, c 5 ( f x ) - c 5 ( f Q ) = Sq 2y 3 ( f Q , f±) e H 5(B; IT (D(n))) = H 5(B; Z 2) , where 7 3 ( f Q r f ± ) = h * ^ ( u 3 ) , u 3 i s a 3 * 3 °31 generator of H (U(n)j Z) and h , H (G ; Z) y H (B; Z); 0 1 ( i i ) i f n > 2, c 6 ( f ' 1 ) - c 6 ( f Q ) « A 2Sq 2 Y 3 (fQ, , where A 2 i s the Bockstein operator associated with the sequence, •2 0 ——>• Z > Z ——y Z 2 y 0 . Hence by the homogeneity of Y 3 ( f Q ' f-^) » w ® have 5 (4) There i s a cross-section, f i B > E when n = 2 ( re s p e c t i v e l y f : B •——*- E when n > 2 ) i f f c (f) = Sq 2(Y 3) f o r some 3 . ^ ^ Y 3 e H (B; Z) ( r e s p e c t i v e l y c (f) = A 2Sq 2(Y 3) f o r some Y 3 e H (B; Z) ) In f a c t , i f n = 3 and c^(£) ' = c 2 (£) = = ®> ^ a e t h i r < 3 obstruction i given by (A 2Sq 2) (c"^ e H 6 ( B ; Z)/A 2Sq 2H 3(B; Z) , where (A Sq ) i s the f u n c t i o n a l cohomology operation, determined by 4>£ and A^Sq 2, and A 2 i s the Bockstein operator associated with the • 2 sequence, 0 > Z >• Z »• Z;> • 0 [49]. 5.7.5. D e f i n i t i o n s and Notations. Let SO(n) denote the s p e c i a l % i orthogonal group of GL(n, R) . Let to_^  e H (BSO(n); Z 2) f o r i = 2, S,'" , t h — 4 i n denote the i u n i v e r s a l Stiefel-Whitney c l a s s , e H (BSO(n)) f o r th i = 1, 2, ••• , [n/2] denote the i un i v e r s a l Pontrjagin c l a s s . — n X n e H (BSO(n)), n even denotes the u n i v e r s a l Euler-Poincare c l a s s . In p a r t i c u l a r , and x a r © of i n f i n i t e order and H (BSO(n); Z 2) = Z 0 $ „ , , u j '( Borel [7] ) . Let E = (E, p, B, SO(n), SO(n)) be a SO (n)-bundle with c l a s s i f y i n g map, B ——> BSO(n). Define the Stifel-Whitney c l a s s , Pontrjagin c l a s s and Euler-Poincare" c l a s s as follows: u. (E), = '•?$.), 1 S -1 oAE) = ^ ( P ^ / xn(o = ^.cxn). By 5.5.2 i t i s leg i t i m a t e to make the following assumption: 'v. 'Xi (5.7.6) E = (ESO(n), p, BSO(n), SO(n), SO(n)) has a'decomposition of length N >_ 4 [n/2] . . . Below we l i s t the homotopy groups of SO(n) [8], which we need, and state the corresponding theorem l i n k i n g the A's with the Pontrjagin classes of the u n i v e r s a l bundle and hence l i n k i n g the Pontrjagin classes with the obstructions. 5.7.7. 'TT (SO(n)) £=: z 2 ; T 7 4(SO(5))fi^ Z 2 ; T T 7 ( S 0 ( 8 ) ) ^ Z + Z ; IT . ' (SO(n)) ^ Z i f n > 4k+l. 4k-l — 4k ^ 4k ^  4k ^ THEOREM. Let A (?) e H ( E 4 k _ 2 ; i r 4 (SO (n))) = H ( E 4 k _ 2 ) i f n >, 4k+l, be as defined i n 5.5.7. Then e 4k ^ r ± 2(2k-l)!A (E) i f k i s odd; P 4 k - 2 ( p k ) = ± ( 2 k - l ) l \ A k ( t ) i f k i s even. 4k-l Hence, i f f: B > E i s a cross-section to £ = (E, p, B, SO(n), SO(n)) over the (4k-l)-skeleton, then r — 4k ' . ± 2(2k-l)!c (f) i f k i s odd; — 4k ± C2k-l)!c (f) i f k i s even. The proof of t h i s theorem i s s i m i l a r to that of 5.7.1 and 5.7.2. Thus with t h i s theorem one can deduce with ease the following r e s u l t . 5.7.8. COROLLARY. Suppose CD H 8 k + 2 ( B ; 7.^ = H 8 k + 1 ( B ; Z) = 0, 8k (2) the t o r s i o n of H (B) i s r e l a t i v e l y prime to (4k-l)! and the t o r s i o n of H 8 k + 4 ( B ) i s r e l a t i v e l y prime to 2(4k-l)! , f o r n >_4k+l. Then £ = (E, p, B, SO(n), SO(n)) has a cross-section, f : B n 1 > E i f f P l ( 5 ) = P 2(5) = V - P I ( n _ 1 ) / 4 1 « ) = 0 • 5.7.9. In t h i s subsection we s h a l l assume n >_ 5, and the . , following f a c t . (A) Suppose n >_ 5, <f>^: B > BSO(n) i s a c l a s s i f y i n g map f o r * <\j £ = (E, p, B, SO(n), SOCn)) and that ®2(V = " J * ^ 0 ^ = °* T h e n t h e 3 generator u^ e H (SO(n)) = Z i s transgressive f o r £. In p a r t i c u l a r , p^CO = ± 2TCU 3) [29] . (1) The following i s a table of some cohomology groups of SO(n) and BSO(n) f o r n > 5. Dimension * H (SO(n) ; Z) H*(BSO(n); Z) 1 0 . 0 2 Z2 0 3 2 Z^ (generator w^ ) 4 Z2 Z (generator p^) (2) The following i s a table of some k-invariants of SO(n) f o r n >_ 5. (a) k 4(SO ( n ) ) =' 0; (b) k 5 ( S O ( 5 ) ) = S q 2 6 3 e H 5(G 3; TT 4(SO(5))) = H 5(G 3; Z 2) ; (c) k 6 ( S O ( 6 ) ) = A 2 S q 2 9 3 £ H 6(G 4; ir 5(SO(6))) = H 6(G 4? Z) <d) k 8 ( S O ( 7 ) ) = + V l E H X ; Z ) ; . (e) k 8(S0(8)) = A/303 + A 3C^e 3 + A 2e x 7 + A ^ 7 £ H 8(G 6; Z+Z) ; (f) k 8 ( S O (n)) = A B^e e H 8(G 6; Z) f o r n >_ 9 ; here 6^ 3 i s a generator of H (GJ ^  Z for i = 3, 4, 6; 8^ i s a generator of H^G L; Z 2) for g i = 3, 4 and 6 r e s p e c t i v e l y ; H (G &; Z+Z) H (G,.) + H (G^) and denotes decomposition with respect to the above 6 6 d i r e c t summand; A i s the Bockstein operator associated with the exact n •n 1 sequence, 0 »• Z »• Z > Z^ »• 0, i s the Steenrod reduced 7 cube, and of course 6^ denotes the seven f o l d cup product of 6^ with i t s e l f . For the c a l c u l a t i o n s , we r e f e r to Lundell [29]. (3) PROPOSITION. Suppose £ = (E, p, B, SO(n), S0(n)) has a 4 4 3 cr o s s - s e c t i o n , f Q : B • E. Then, i f d E H (B; Z 2) and y e H (B; Z), 4 there i s a c r o s s - s e c t i o n , f : B • E, such that a ( f 0 # f ) = a, and y 3 C f Q 7 f±) = y, — 3 * where y ( f Q / f^) i s defined 1 to be hf^f..(.u3) and i s a generator of 3 0 1 H (SO(n)) ~ Z. The proof o f t h i s p r o p o s i t i o n i s analogous to that of the Un-bundle case. Thus with t h i s p r o p o s i t i o n we get the following main theorem f o r SO (n) -bundles, n >_ 5 . (4) THEOREM, (i) The primary obstruction to a cross-section i s u2tO e H 2(B; Z 2) . ( i i ) I f ^ 2(£) = 0, the secondary obst r u c t i o n i s T (u 3) where T i s the transgression i n £ and u 3 i s a generator o f H 3(S0(n)). In p a r t i c u l a r , i f c 4 denotes the obstruction, then P i ( £ ) = ± 2c 4 by 5.7.9(A) . 4 ( i i i ) Let f : B *" E be a cros s - s e c t i o n to K = — 4 (E, p, B, SO(n) , S0(n)) over the 4-skeleton; i . e . , w 2 ( £ ) = c = °* Then the a p p l i c a t i o n of p r o p o s i t i o n (3) above and theorem 5.6.1 gives the following c r i t e r i o n : 5 — 5 o I f n = 5, £ admits a cross-section over B i f f c (f) = Sq Y 3 f o r some y e H (B; Z ) . I f n = 6, £ admits a cross-section over B^ i f f c ^ ( f ) = A 2Sq 2y f o r some y e H 3(B; Z) . 89 I f n = 7, £ admits a cross-section over B i f and only i f 1 * 7 3 1 c 8Cf) = A F y + A d , f o r some Y e H (B; Z) and d e H (B; Z ). 3 3 2 8 - 8 c I f n = 8, £ admits a cross-section over B i f and only i f 1 1 7 7 3 (f) = A. P Y + A (? y" + A d + i d ' , f o r some y , y ' e H (B; Z) and J o J O ^ some d, d' E H (B; Z^). g g T_ I f n > 9, C admits a c r o s s - s e c t i o n over B i f f c (f) = A^ff^Y 3 f o r some Y e H (B; Z). In an analogous way, one deduces the main theorems f o r SP(n)-bundles. 5.7.10. (A) We l i s t some homotopy groups of SP(n) Dimension i < 4n+l T T # ( S P (n)) i = 3 ( mod 4 ) Z i H 4, 5 ( mod 8 ) Z2 i = 0, 1, 2, 6 ( mod 8 ) 0 i < 3 0 i = 3 z i = 4 Z2 <v. 4i (B) Notations. Let e^ E H (BSP(n); Z) denote the symplectic Pontrjagin classes of the u n i v e r s a l bundle, £(SP(n)). Suppose £ = (E, p, B, SP(n), SP(n)) i s a SP(n)-bundle with c l a s s i f y i n g map, * % 4 j_ <(>,_: B >• BSP(n). Then l e t e. C£) = <jv(e.) E H (B; Z) be the symplectic Pontrjagin c l a s s of £ . We assume that £ has a decomposition of length N > 4n. As i n 5.7.9 we have the following theorem. 4k — 4k — (C) THEOREM. Let X . (?) e H ( E 4 k _ 2 ; ir (SP (n))) 4k — C = H ( E4] c_2' Z^ ^ o r k — n ^ be as defined p r e v i o u s l y . Then P 4 k - 2 ( \ ) = 4k 'v ± (2k-l) ! A (£) f o r k odd; 4k 'Xi ± 2C2k-l)!A (£) f o r k even. 4k-l Thus, i f £ = (E, p, B, SP(n), SP(n)) has a cross-section, f : B f o r k < n over i t s (4k-l)-skeleton, then eK(0 — 4k ± (2k-l)!c (f) f o r k odd; — 4k ± 2(2k-l)!c (f) f o r k even. (D) Analogous to 5.7.3, i n the case of an almost 4n quaternionic (4n)-manifold, M , we have the following immediate r e s u l t by ( C ) . 4n i 4n THEOREM. Suppose M i s as above, H (M ; Z^) = 0 f o r i = 5, 6 t mod 8 ) and that the cohomology t o r s i o n c o e f f i c i e n t s are r e l a t i v e l y prime to 2 ( 4 i - l ) ! i n dimension 8 i and are r e l a t i v e l y prime to ( 4 i - l ) ! 4n i n dimension (8i + 4). Then M i s p a r a l l e l i z a b l e i f f e^C?) = e 2(£) = 4n e3(.£) = ••• = e n(-£) = 0 / where £ i s the tangent bundle of M (E) The Main Theorem. Suppose £ = (E, p, B, SP(n), SP(n)) i s a SP (n)-bundle. (1) The primary obstruction to a cross-section i s e^(£) . 91 4 (2) Suppose e^(5) = 0 and f : B ——*• E i s a cross-section. 5 Then there i s a cross-section, f " : B —:—* E i f f there i s an element 3 — 5 o d e H (B; Z) such that c (f) = Sq^d. Proof. (1) follows from theorem (C). (2) follows from the f a c t that k (SP(n)) = Sq z0 , where 0^ i s the fundamental cl a s s of = K(Z, 3), and theorem 5.6.1. §5.8. Conclusion We have seen how one can use the idea of Moore-Postnikov decomposition i n computing the higher obstructions. The main d i f f i c u l t i e s l i e i n computing the k-i n v a r i a n t s . In the more general case, one considers p r i n c i p a l f i b r a t i o n s and thus one needs to know some higher cohomology operations u s u a l l y with twisted c o e f f i c i e n t s , determined by some k- i n v a r i a n t s . Of course, t h i s means one i s reduced to looking at cohomology with twisted c o e f f i c i e n t s . In t h i s d i r e c t i o n , one has some deep connections with the o l d problem of determining the s o l u t i o n to the l i f t i n g problem: Y H £ x > B One may assume H to be a f i b r a t i o n . Let to be a cohomology c l a s s i n * KerCH ). Then, taking o> as a c l a s s i f y i n g map, one obtains a p r i n c i p a l fibration,i{»: E > B, over B and H automatically l i f t s to ty with to 92 * a l i f t i n g , q: Y >• E . Let 3c be any f i x e d c l a s s i n Ker (q ) . Then one can define k(E) = LJf k , where the union i.s taken over a l l f: X —:—»• E to * such that ifiof = E . For k(E) to be non-empty' one must have £ to = 0 . I f E to = 0, i . e . k(.£) 0, then one can show that k(£) i s a coset i n the image of some (twisted) cohomology operation, determined by some r e l a t i o n 159; 38]. The r e s u l t appears a b i t cumbersome (even i n the o r i e n t a b l e case), even more so i f we generalize to the case o f two * c l a s s e s , k , k e Ker(q )• McClendon [39] considers the tower with K - p r i n c i p a l f i b r a t i o n s : f I k X >- B — > C where the C. i s a product of L,(G, n)'s and L,(G, n) i s the generalized i <J> <J> Eilenberg-MacLane complex, a f i b e r space, K(G, n) >- L (G, n) >• K = K t i r ^ B ) , 1) and G i s considered as a ir^ (B)-module v i a a homomorphism7 a): TT, (B) > Aut (G) 117]. Thus the k.'s are twisted. Let k. (f) be as 1 i i defined before. He i s able to prove the following theorem. THEOREM. Suppose there i s a positive integer, N, such that a l l the C.'s are products of L. (.G, n)'s for N+l < n < 2N. Then there i s a stable B-operation, $ = $" ; I X , ftC ] • •>• I X , C .J C an additive .x, f I K i K relation ) such that k.(f) i s a coset of the subgroup Im($) C[x, C.] 94 . CHAPTER.6. MODIFIED POSTNIKOV TOWERS AND . .OBSTRUCTIONS TO LIFTINGS In t h i s chapter we give a modest account of a modified Postnikov tower which, has p r i n c i p a l f i b r a t i o n s f o r each stage and we use higher order cohomology operations without t w i s t i n g to describe the k-invariants f o r a l i f t i n g . For t h i s reason we s h a l l assume the f i b e r bundles we s t a r t with, are o r i e n t a b l e . This presents no r e s t r i c t i o n to the technique . used; i n f a c t , one can generalize t h i s to a r b i t r a r y f i b e r bundles and t h i s would have involved higher order twisted cohomology operations [ McClendon, 37,38; E. Thomas, 59]. We s h a l l give only the d e s c r i p t i o n as f a r as orie n t a b l e f i b e r bundles, leaving the not-so-obvious g e n e r a l i -s a t i o n to the ex c e l l e n t accounts by McClendon and Thomas. § 6 . 1 . Modified Postnikov Towers The main idea i s to replace the usual Postnikov tower by one i i - l where the f i b e r space, p^: E E , has as i t s f i b e r a product of Eilenberg-MacLane spaces and the f i b e r of h.: E > E 1 s a t i s f i e s some nice p r o p e r t i e s . In ad d i t i o n we require some algebraic conditions on h^ which are a g e n e r a l i s a t i o n of the construction i n chapter 4, v i a the idea of s p h e r i c a l c l a s s e s . 6.1.1. D e f i n i t i o n . Suppose X i s a complex and J = Z or Z^ ( q some p o s i t i v e prime ). Let a E H (X; J ) . a i s s a i d to be ( J ) s p h e r i c a l 95 k * i f there e x i s t s a map, f: S —:—> X, such that f (a) i s a generator of k k H (S ; J ) . We want to work slowly towards the conditions required of the f i b e r of h. : E > E 1 . i 6.1.2. D e f i n i t i o n . Let X be an (n-1)-connected space f o r n >_ 2 such that TT (X) i s f i n i t e l y generated. For k < 2n - 1 define ^ ( x; Z ) K • — q to be the subset of H (X; Z ) s a t i s f y i n g the following conditions: (1) Each a e ij (X; Z ) i s (Z ) s p h e r i c a l . q q k * (2) I f f : S >• X i s such that f (a) i s a generator of a a k k H (S ; Z), then the c o l l e c t i o n of homotopy classes of such f ' s , {[f ]} «k,„ „. i s a l i n e a r l y independent set i n TT, (X) considered a a e y ( x? Z) . k as a Z-module, and { [ f j } & e tf * ( x . z > " { [ f j >a e . ^ ( X ; z ) i s l i n e a r l y independent over Z , where [ f ] i s the image of [f ] under q a a the n atural homomorphism, TT, (X) * TT, (X) 0 Z k k q (3) I t i s maximal with respect to (1) and (2). C a l l ^ k ( X ; Z ) a regular s p h e r i c a l set. 6.1.3. Our aim i s to define a modified Posinikov tower ( abbreviated MPT ) where the f i b e r of each h^: E —:—>• E 1 i s t-r e g u l a r ( see d e f i n i t i o n l a t e r ). To have t h i s at a l l we need the existence of a regular s p h e r i c a l set. The following theorem guarantees t h i s . THEOREM. Any (n-1)-connected space, X, (n >_ 2) with Tr k(X) f i n i t e l y generated i n the stable range ( k <_ 2n-.l ) has f o r each prime k q and f o r q = 0, a regular s p h e r i c a l set, ^ (X; Z ) with the convention that Z = Z , i f q = 0 . q 96 Proof. See M. Mahowald 130]. 6.1.4. Let (E, p, B, F) be a f i b e r bundle with F (n-1)-connected (n >_ 2) . Then, by 2.1.3, we see that we have the following commutative diagram: E P Wo - ' M C B where X i s a homotopy equivalence. Thus consider the sequence f o r the induced bundle, (E , p E" , M, F ) , over M: 0 . H o * H k ( E u ) > H k(F) -S-> H k + 1 ( E u , F) . ° k ° * k We say F i s t-regular i f (F; Z ) ^ 0 and <S t5 (F; Z ) = 0 f o r a l l <2 q primes q, q = 0, and f o r a l l k <_ t . With l i t t l e d i f f i c u l t y one can see that F i s t - r e g u l a r implies F i s also t"-regular f o r a l l t" < t. 6.1.5. Suppose (E, p, BG, F) i s an u n i v e r s a l bundle with BG a c l a s s i f y i n g space f o r an abelian group G and with F being a G-space. Then i t can be e a s i l y seen that (E , P|E , M, F) ( notation as above ) i s an u n i v e r s a l object f o r a l l F-bundles with cross-section. We s h a l l often assume t h i s without mentioning i t . 6.1.6. Define t5, (F) = U W ^ t F ; z ) . A representation t q k=l q f o r t5 (F) i s a map, h: F >• II II II K (Z , k) = Y fc q k=l a £ t 5 k ( F ; Z ) * q q * such that h (oc ) = a where a i s the image of the c h a r a c t e r i s t i c c l a s s a a of K (Z , k) i n Y and K (Z , k) i s an Eilenberg-MacLane space of type a q a q 97 (Z , k ) . q Thus a modified Postnikov tower of dimension t f o r (E, p, B, F) i s a tower s i - l X i- 1 h. . i - i E -> E B such that (1) p^: E 1 *• E 1 1 i s a f i b e r space with f i b e r x1 = n K (J , k ) a a a aeA where A i s a f i n i t e index set, J i s e i t h e r Z or Z (some prime q) ; a q (2) h. , : E *• E 1 1 i s a f i b e r space with i t s f i b e r X 1 t.-l - l l regular f o r some t. < t , where t. > n. = dimension of the f i r s t non-* l — i — i zero homotopy group of X 1 ; and •> X 1 i s a representation of t5, (X 1) . (3) h. Ix 1 : X 1 l 1 t. l 6.1.7. Remarks. (1) I f t ^ = n ( i ) f o r a l l i , then the MPT reduces to the tower described i n chapter 4. (2) There i s a theorem guaranteeing the e x i s t -ence of a MPT o f dimension t f o r a G-universal bundle, £, where G i s a arcwise connected t o p o l o g i c a l group, and the f i b e r , F, i s (n-1)-connected 98 (n > 2) with TT_, (F) f i n i t e l y generated f o r j <_ .t < 2n . Hence i t also guarantees the existence of a MPT f o r any (principal) G-bundle with F as f i b e r 130]. The next section describes some prop e r t i e s of Postnikov systems and how the k-invariants are r e l a t e d n a t u r a l l y . We s h a l l then come back to the question of computing k-invariants f o r l i f t i n g s . This has been an area of act i v e research [ Thomas, 59; McClendon, 37, 39; Maunder, 33, 34, 35, 36; Mahowald and Tangora, 32]. §6.2. Construction of Induced Maps Between Postnikov Systems We s h a l l stay i n the category of countable 1-connected CW complexes (with base point) and based maps. The reason f o r us to consider t h i s " . category i s that i t i s closed with respect to t h i s construction. Consider the Postnikov decomposition we have given i n §4.6. I f we replace E by F and B by K(TT (F) , n ( l ) ) where TT (F) i s the f i r s t n i l ) n ( l ; non-zero homotopy group of F, that i s , F i s (n (1)-1)-connected. We then have the Postnikov system f o r F. 6.2.1. D e f i n i t i o n [43]. I f F i s (n-1)-connected f o r some n >_ 2, a Postnikov' system' f o r F i s a family of n-Postnikov systems, { ( F ^ , 9^)^' and morphisms, { n. .: F. y F. }. . , d i r e c t e d i n such a way that i f i/D i 3 X>3. i > j > h then TV. . = n . " , o n . ' . • i,k u,k 1,3 Furthermore n. , . : F. , —:—* F. i s a f i b e r space with f i b e r i + l 1 e KC 'T. , (F) , i+l) and n. , .og. , - g. . For 0 < i '< n -1 the F. consists i + l i + l , i i + l • i — — l of a s i n g l e p o i n t , the base point. Therefore K(HN (F), i ) f o r 1 <_ i <_ n -1 i s taken to be a si n g l e p o i n t by convention. 6 . 2 . 2 . Suppose we are given two spaces, F and F', with t h e i r Postnikov systems, {F., g., n. ,} and {FT, g7, .} and a map, f: F — 1 1 1 , 3 l " l 1,3 y F". Then by a map between the Postnikov systems induced by f, we mean a family of maps, { f ^ : F^ y F r } s a t i s f y i n g (1) f.on . = ru, .of. , and l i + l , i i + l , i i + l ( 2 ) gfof * f.og. . l i i Our goal i s to construct such an induced map. 6 . 2 . 3 . Suppose we are given two spaces, F and G; suppose also that F i s (n-1)-connected, G i s (m-1)-connected and n, m >_ 2 . I f f: F y G i s a map, then the following diagram i s homotopy commutative f F y G p(F) P ( G ) f# K(T T (F), p) — -> K(TT ( G ) , p) P P where i s the map induced by f : F > G, p = Min( m, n ) and p(F) represents e i t h e r the fundamental c l a s s i (F) ( p = n ) or the zero c l a s s i f (n-1) >_ p . The case that p < n or p < m i s e a s i l y seen by standard homotopy argument the other case f o r p = n = m follows by using the Hurewicz theorem and d u a l i t y . By the natural construction of 2.1.4 we may replace the above diagram by an equivalent diagram ( Covering Homotopy Theorem ): P ( F ) -> G' P ( G ) .. A. . . K(rr ( F ) , p) * K(TT ( G ) , p) p p which i s commutative and where P ( F ) ' and P ( G ) ' are f i b e r spaces. This diagram t e l l s us by the homotopy exact sequences of the two f i b r a t i o n s that the f i b e r s , F" and G " of p ( F ) ' and p ( G ) ' r e s p e c t i v e l y are p-connected. Let i P + 1 ( F " ) e H P + 1 ( F " ; TT I N ( F " ) ) and * P + 1 ( G ~ ) e H P + 1 ( G ~ ; p+1 ^ ^ ^ ( G " " ) ) be the fundamental classes of F" and G"" r e s p e c t i v e l y . Let k P + 2 ( F ) = T ( I P + 1 ( F " ) ) E H P + 2 ( K ( T T ( F ) , p); TT ( F ) ) and k P + 2 ( G ) = P P+1 T ( I P + 1 ( G " ) ) e H P + 2 ( K ( T T (G) , p) ; rr ( G ) ) be the f i r s t k - i n v a r i a n t s of p p+1 F and G r e s p e c t i v e l y . ( See 4.4.4. ) 6.2.4. Look at the diagram i n 6.2.3; i f we i d e n t i f y i r p + 1 ( F " ) with TT , (F) and TT , (G""-) with TT , CG) r then p+1 p+1 p+1 ( f | F ~ ) „: TT CF") -y TT CG") 1 # p+1 p+1 i s i d e n t i f i e d w i t h f„ . T H E O R E M . Suppose we l e t f ^ denote the c o e f f i c i e n t homomorphism induced by f„: TT ' CF) •> TT ,. CG) . Then f * k P + 2 C F ) = Cf») * k P + 2 CG) . # p+1 p+1 c ft Proof. See [25]. 101 6.2.5. By the above theorem, r e p l a c i n g maps i f necessary by equivalent f i b r a t i o n s , we have the following commutative diagram: <v . . f , K(TT ( P ) , p) -p+2 (F) -> K(TT (G) , p) P a. f, k P + 2 ( G ) K ( r r p + 1 ( F ) , p+2)- **" K ( ^ p + 1 ( G ) ' P + 2 ) C As before we use the ACHP property i f necessary to replace the homotopy commutative diagram by a commutative diagram. ) Taking f i b e r s of l c P + 2 (F) , ) c P + 2 (G) , we get a commutative diagram of f i b e r spaces: 'p+1 P+l/P K(rr (F), p) p+1 -> G p+1 n. p+i/P — £ > K'(ir (G), p) k P + 2 ( F ) K ( l T p + l ( F ) ' P + 2 ) ^ P + 2 ( G ) > K ( T T p + 1 ( G ) , p+2) F G where (F » n K(TT (F) , p)) and (G , n , K(TT (G) , p)) are p+1 P+1,P P .:- p+1 P+l/P P p r i n c i p a l f i b r a t i o n s and f = f,, . These are the second stages of the p # Postnikov towers and f ^ + x i s a map between them. 6.2.6. Since F , and G , are obtained by using the k-invariants p+1 P+1 as c l a s s i f y i n g maps, as i n chapter 4 we see that P ( F ) ' then l i f t s to F p + X and p(G)" l i f t s to G p + 1 ( Lemma 2.2.2 ). Thus we have the following diagram: p(F) p+l,p F . = K (TT (F) , p ) *• G p+1 n . p+i/P * K(TT (G) , p ) = G f P P where p CF) i s a l i f t i n g of p(F) and P(G) i s a l i f t i n g of p(G) main idea i s to a l t e r f , such that the diagram: p+1 The ~> G' p(F) p(G) P+1 •+ G P+1 p+1 i s homotopy commutative and that we s t i l l have f on , = 1 1 , . of ,., . P P+1,P P+l/P P+1 Let us pause f o r the moment, assuming that we know we can a l t e r f , as above. Then we can extend the technique to the whole system stage p+1 by stage and the proof works by induction. This i s the essence of the proof of the f o l l o w i n g : THEOREM. Let F and G be as above. Suppose f : F > G i s a map. F F ~ G G ' Then there e x i s t Postnikov systems, {F., g., r\. , .} and (G. , g.i n.,, .}, 1 1 i + l , i l l i + l , i and an induced map { f . : F. y G. }; i . e . , a diagram: i i l f K(TT (F), p) = F 2- y G = K(TT (G), p) P P P P where a l l the "rectangles" i n the f r o n t of the diagram commute and the r e s t of the diagram homotopy commutes. Also, f o r n >_ p+2 , * n * n f k <F) = f „k°(G). c n-2 6.2.7. With l i t t l e d i f f i c u l t y we can then use t h i s standard construction to obtain an induced map between two given a r b i t r a r y Postnikov * n * n systems f o r F and f o r G. Also, f k (F) = f k (G) f o r n > p+2 . c n-2 — Proof. See Kahn 125] f o r the proof of theorem 6.2.6 and 6.2.7. 6.2.8. Remark. A p p l i c a t i o n to Hopf-space structures. Consider the f o l l o w i n g s i t u a t i o n . Suppose X i s such that n ^ . j t X ) •+ 0, TT_. (X) = 0 f o r j ^ n ( i ) and n ( l ) < nC2) < n(3)' < Then take the Postnikov system f o r X and consider the following diagram • V I , P X V I , P y p p+1 V l ,P W E P + 1 x E P + 1 • E^xF? >• E P ^ K ( T f (X) , p) S K(TT (X) , p+2) P P+1 104 ^ p+1 Let be the' H - m u l t i p l i c a t i o n . Then E has a H-rnultiplication i f f y o(n x n ) l i f t s to E P + 1 i f f k P + 2 (X) oy o (n _ x n )• * * . p p+l,p p+l,p p P+1,P P+l/P which, when appropriately interpreted [25] ,means k P + 2 ( X ) i s T r p + 1 ( x ) _ p r i m i t i v e . Thus we have the following theorem. THEOREM. Hypothesis as above. X i s a H-space i f f i t admits a Postnikov decomposition f o r which each k-invariant, k n ^ ( X ) i s p r i m i t i v e with, respect to the i n d u c t i v e l y constructed H-space structures on the { E 1}. ( For a p p l i c a t i o n of t h i s theorem to H-space structure on spheres see 126] and a l s o the review f o r t h i s paper i n Steenrod's Review concerning theorem 2.2 of [26]. ) §6.3. The k-invariants f o r a L i f t i n g 6.3.1. Let F be an (n-1)-connected space f o r some n >_ 2 and l e t t h: F y Y = n i i n K ( Z , k ) q k=l a e ^T(F; Z ) 3 q q be a representation of "^(F) f ° r some t. Let X be the f i b e r of h. We *k would l i k e to be able to compute \> (X; Z ). Suppose the f i r s t non-zero q homotopy group of X occurs i n dimension n . The main d i f f i c u l t i e s l i e k i n showing that (1) { [ f ]; f : S »• X } , i s independent S a . a c f (X; Z) over Z and (2) { [ f a l g } _ k U ( [ t j ^ } v i s independent 3 q a E T ( X ; Z) a * aeSNx, Z ) q over Z q 105 Using the Barratt-Puppe sequence, i n the stable range, we have the exact sequence: """ * ... > H J(F) > H J(X) H j + 1 ( Y ) -^-»- H j + 1 ( F ) ( f o r j < 2n-2 ). A c l a s s x. , e tf? +±(Yi Z ) can be written i n the form, - 3+1 q x. , = ; a 3 + 1 ae 5 t(F) a a where <J> i s a primary cohomology operation, cl • • <j> : H k(Y; Z .) > H 3 + 1 ( Y ; Z ) . a(q,k) q q This i s the usual representation of H (K^Tr"*, k) ; IT) as primary cohomology operations i n the stable range ( see e.g. [44] ) . Hence if <$> . . i s a(q,k) non-zero then q" = q or e i t h e r one or both of q, q" i s zero. By the above exact sequence, J a e t ) t ( F ) j ~ * i s zero i f f there e x i s t s a c l a s s , xT e H (X) such that <S (x") = x. , . 3 3 3+1 6.3.2. D e f i n i t i o n s . We say an operation, 4>: H k(X; Z) »• H J + 1 ( X ; Z ) , q i k o f type (k, Z; j + l , Z^) i s p r i m i t i v e i f there e x i s t s a map, g: S > S , k j+l such that <j> i s n o n - t r i v i a l i n the complex, S e . C a l l an operation, g <T: H k(X; Z ) — * • H j + 1 ( X ; Z ) , : q q o f type (k, Z^; j + l , Z^) p r i m i t i v e i f (1) the image of ( J ) " under the natural homomorphism, * * H (K(Z , k); Z) y H (K(Z, k); Z ), i s p r i m i t i v e ; or q q — (2) (j>^  = , the Bockstein operator associated with the sequence: 2 1 " i Examples. (1) Sq , f o r i - 0, 1, 2 or 3 and f o r k > 2 ; and are p r i m i t i v e . 2 1 (2) Sq Sq i s also p r i m i t i v e . See Thomas [58, Theorem 4] and Mahowald [30] f o r an i n d i c a t i o n of the proof o f (1) and (2). j+l 6.3.3. THEOREM. Suppose x j + 1 £ H ^ ' Z ^ ^ s r e P r e s e n t e d as X i + 1 = 5! M O D + 1 ae (F) a a and q ^ 0 . Suppose also that f o r at l e a s t one a" e ^ (F) , <j> / i s t a j "* p r i m i t i v e . Then f o r any x e H (X) such that <5 x' = x j + ^ > there e x i s t s a map, f : S >• X, such that f x' generates H (S ; Z ), i . e . , x' i s q (Z ) s p h e r i c a l . q Proof. See [30] The following p r o p o s i t i o n gives a s p e c i a l case of t e l l i n g when k+2 some classes i n H (X; Z^) are s p h e r i c a l . 6.3.4. PROPOSITION. Suppose a" e ^ k ( F ; 7,^ f o r some k >_ 3 and k * k k i f f ' ^ : S > F i s a map such that f'^(a') generates H (S ; Z ), then a a 2 assume 2[ f'^] = 0 where [ f."A i s the homotopy c l a s s of , . Let x e a a a k+3 H (Y; Z^) be represented i n the form v 2 1 x = i 4> a such that tj> _ - Sq Sq . Then f o r any x' ae d t(F) k+2 '* H (X; Z 2) such that j x' = x , x' i s s p h e r i c a l . The proof of t h i s p r o p o s i t i o n uses Paechter's groups [46] and can be found i n Mahowald [ 3 0 ] . 6.3.5. Let E = (E, p, BG, P) be the un i v e r s a l bundle with F (n-1)-connected and l e t there be an MPT f o r E i n the stable range (6.1.7(2)): M l *• BG where x 1 = n n1 n . K ( Z , k) q k=l ae 3 ( X 1 ; Z ) & q =>k , i x K+X x—1 3c i. Let k e H (E ; Z ) f o r a E ^ (X ; Z ) be the k-i n v a r i a n t . An a q q ap p l i c a t i o n o f 2.1.3 to the f i b r a t i o n h^ gives the following diagram i - l where X, i s a homotopy equivalence, i . e . , u. i s equivalent to h, i - l i - l i - l A i Consider the induced f i b r a t i o n , (E' , p. E •, , M. , , X ) , i n the f i b e r square, i - l i - 1 Look at the diagram: i - i Since V 1 - ± a n ^ h are homotopically the same map we then have p.o(h.oX. ) - u • This then together with the f a c t that X. i s a i I i - i i - 1 i - i homotopy equivalence implies by the u n i v e r s a l property of the f i b e r square that (E , p. |E , M. , , X 1) has a cross-section, p. Hence y i " 1 1 ^ i - 1 1 - 1 E - M. . x X , and p i s chosen i n such a way that u. = u.op . We = 1 - 1 i i explain below. Assume we have shown that = M ( C BG ). Then we see that a l l other M.'s can be taken to be M. The new k-invariant• i i + l * i * — * k e H (E ) w i l l be a c l a s s i n Ker( p op, ) coming from the f i b e r i n g cl 1 h. : E l ! ~ i • ., ..i+l •+ E with X as f i b e r . Look a t the f i b e r square: E C .y V - l v-1 -y. M C E V - l v-1 where v = i or i + l V - l i ~ i o. The f i b r a t i o n , h.: X > X , induces a map, E ' > E , and "V • A a . p . , :" M y E induces p . : E C E 1 " 1 " i - l 1 " i V l 6.3.6. THEOREM. Suppose X1+^" i s ( j - l ) regular. Assume that a e oi-1 i+1 ^* 13 (X ; Z ) i s such that 6 a ^  0 where 6 i s the transgression i n q * ~* * i • , . + 1 <S ... h. . . HJ~l(XX) >• iP (X )• ^ ^ ( X 1 ) — H 3 ^ ) . * —* x+1 ' — ~ i Then i u.(k ) = 6 a , where i : X E ' 1 a . V l The proof i s j u s t simply p u t t i n g things together. 6.3.7. Since E £ M x X 1 , by the Kunneth formula, V i H*(E ; Z ) = H*(M; Z ) 0 H*(X X} Z ) . q q q Let p. , : M >• E be the cross-section. We may assume i n dimension x-1 y.^ 3 Ker(p ) 9£ I H J (M) & H (X ) . v=l j * So a e H (E^ ; Z ^ ) D Ker(p^ ^) i s a l i n e a r combination of m_. ^9 x^ _ t> i where m e H (M; Z ) and x, e H (X ; Z ) . a q £> q D e f i n i t i o n . Let e = (e e , •••) where a£(v,k) e t5 (X 1) and a l a2 t e . . e Hk+"*" ( E 1 "S Z ) ( & = 1, 2, • * •) . We s h a l l define an act i o n of a e H 3 (E Z ) r\ Ker(p ) f o r j < 2n on e . V l q t Now, X 1 = n . II n , . K ( Z , k). Let a . ,\-be the q k ae k ( X 1 ; Z ) A Q A ( Q ' K ) ' ... - ^ £ image of the fundamental c l a s s of K ( Z , k) i n X . a q 110 Since n < Dim a < 2n f o r a l l a , as i n 6.3.1 we can write — a a x = T • • a e H y ( X 1 ; Z ) y a e * (X 1) 3 a q t ,D+1 " i ~ i i '*—1 v i > J . Define (m. ® x )© e = (-1) y. ,m, ) . 4> e e H J (E ; Z ) , 3-y y r-1 3-y ^ ^ ( ^ j * * where Sty = <j> and S i s the suspension, y. i s c l e a r l y an isomorphism cl cl 1""X since j-y < n . Extend by l i n e a r i t y to a r b i t r a r y a e H (Ew" ; Z^)AKer(p ) With t h i s new action we have the following lemma. V i ' q 6.3.8. LEMMA. For f i x e d i l e t k = { k ; a e "5,(X ) }. Then a t * *_i * —<j_ j p . <5 a = a © k I i * for a l l a e H (E ; Z ) n Ker(p. ,) . y i - l q 1 - 1 Proof. I t i s enough to check t h i s f o r a = m. 0 x . The following D-y y diagram w i l l explain the maps i n the lemma. _* * * i H (E ) H ( E 1 _ 1 ) v. H (E ) V i H (M) H (E , E ) t V l 1 * i-1 —> H (E , M) * i-1 -> H (E ) * j—y * Just simply note that Q ( m. ® x ) = C-l) . m. "-6 x" , a f t e r i d e n t i f i c a -j-y y D-y y • * - i * * - i * - i t i o n with the preimage of m. . Therefore p. 6 (m. 0x ) =-y. _m. *p. x . D-y i D-y y 1-1 D-y i y * *_]_ * i Since j p. 6 4 a = d>k ,.. we see that the lemma i s true f o r a =. m. ®x i a a a a .-- J-V y i * and hence f o r any a e H CE ; Z ) r\ Ker (p . ) . - y i - l q 1 6.3.9. 6.3.8 applies to give the following: * * i - l * THEOREM. Suppose h. : H (E ) > H (E) i s an epimorphism i n i * dimension j . Then f o r each o e H (E ; Z ) A K e r f p ) there e x i s t s a V l q v E H (E ; Z ) O Ker(h.) such that y.v = a i f f a ® k = 0 . q 1 1 The proof i s easy; j u s t apply the previous r e s u l t and checking on the diagram above. i+1 * 6.3.10. THEOREM. Suppose X i s (j-2) regular, h^ ^ i s an _ A epimorphism and y. i s a monomorphism. Let k = (k , k , k ) be some X X s , vector i n H 3 (E 1; Z ) x H 3 ( E 1 ; Z ) ( s - f o l d product ) and q ^ 0 be q q f i x e d such that the following conditions are s a t i s f i e d . * — * (1) i y_^  k^ are l i n e a r l y independent. (2) I f i u. k = x = T <j> a , then f o r at l e a s t one a, i v . o »_v a a ae *3 (F) cj) i s p r i m i t i v e . (3) Let 01(q)^ be the y-component of the Steenrod algebra mod q, <31(q) . Then the vector space of H 3 1 (X^*^) generated by a=l has co-dimension s i n H 3 1 (X 1*^) . (4) y,. k^ e Kert P ) • Then there e x i s t s a c l a s s t ^ 3 ( X 1 ^ ; Z ) such that k i s the c o l l e c t i o n , q { k i + 1 ; a E * 5 j - V + 1 , Z ) }. a q Proof. See Mahowald [30, Theorem 3.4.2]. 6.3.11. Remark. Thomas [58] gave a computation of some of the s p h e r i c a l sets of the f i b r a t i o n , 112 V -> Bso(n) -*> BSO(n+2), f o r n > 2 , using n+2,2 Paechter's groups [46]. When n = 4s+l , the computation there i s much * simpler. He a c t u a l l y computed Ker(p JnKerd:^) to f i n d Ker(h ). Since 2TT ( V ) = 0, an a p p l i c a t i o n of 6.3.4 gives the required sets, i s the modified transgression i n 2.3.5(5). §6.4. Obstruction Theory f o r Orientable F i b e r Bundles We now consider the question of how a MPT can be used to give an obstruction theory f o r l i f t i n g s of or i e n t a b l e f i b e r bundles. Let £ = (&, p, BG, F) be the u n i v e r s a l bundle and £ = (E, p, B, F) be a G-bundle with F as f i b e r and ^  : B > BG as c l a s s i f y i n g map. Suppose B i s a CW-complex of cohomology dimension t . Suppose further that F i s (m-1)-connected. Consider the MPT f o r £ of dimension t i n the stable range: The f i b e r of h.. x i\, O i l - 1>i E *• E i s t-connected. i f E i s the l a s t space i n ^i" — i " the MPT. Then any map, <J>^ : B >• E , from B to E can be l i f t e d to a map, <j> '* B y E. 6.4.1. £ has a cross-section i f f <j)^ : B >• BG can be l i f t e d to „ ^ a>l ^2 a map, <j> : B > E, which i s equivalent to l i f t i n g of ^  to E , E - i " and so on upto E . The following i s well known: THEOREM, b^^'- B • E 1"" 1 can be l i f t e d to a map ^ : B »• E 1 i f f (. k 1) = 0 f o r a l l a e 5^ ' (X 1) i - i a t. For the f i r s t l i f t i n g we have the following c r i t e r i o n : * * * 6.4.2. THEOREM. Suppose 6 : H (F) > H (E, F) i s the coboundary homomorphism. Then <j> : B > BG can be l i f t e d to a map <|>: B > E -1 i f f 6 (a) = 0 f o r a l l a e O (F). 1 * ]_ * * * Proof. Just use the f a c t that p k = 6 i a whence S a = 0 a i f f k^ = 0 ( r e l a t i v e Serre Theorem ). Here i i s the i n c l u s i o n of the a f i b e r , F, i n E. We have the following diagram: •+ E BG 114 By 6.1.5 <j>^  l s so chosen that <J>^  = ^ i ° p w ^ e r e " P 1 5 a cross-section. Then 1 "1 — — E • = B x X . . Thus i f p i s chosen. so that V^op = P°y Q i then i n the _ i * * —* stable range y^ i s an isomorphism between Ker(p ) and Ker(p ) . Suppose B i s a CW-complex. Then * * 2 * —* * 2 P *I k a = " h l 11 \ +1 k a [30, §5.3.3], where U i s the inverse map ( defined only i n the stable range ), H: Ker(p ) >• Ker (p ) , to y |Ker(p ) . We thus get 4c 1 4c 4c H (E ) SsL Ker (p ) 0 Im(p ). —* * 2 * * fly <j> (k ) e Ker ( p ) implies there e x i s t s a c l a s s z e H (B) 1 1 a * _* * 2 * 2 * * * * 2 such that p. (z) = If y_ cj>. (k ) - <j>, (k ) . Thus z = p p (z) = - p <i> (k ) 1 l l a l a l i a *2 * * 2 * * * = - cb" (k ) . So since h cj> (k ) = 0 and h p = p , we thus e s t a b l i s h X cl X X cl X X 4c 4c O 4c 4c 4c 4c ""4c 4c O -p A- ( O = p (z) = h. p. (z) = h 1f y. 4> (k ) . l a l l i l i a The proof of the next theorem uses the above formula 6.4.3. Suppose we have a l i f t i n g a l l the way up to E of the c l a s s i f y i n g map 4>^ .: B *• BG. *i-2 B y BG Then define f o r each a e *£ _ ^ ( X 1 x ) , i i * k (cb,.) = { z e H (B) ; there e x i s t s a l i f t i n g , cj>r . / a £ i - l o f cj>R with ij>' ,K = z } 1T2 i - l a 115 k Let z = ( z ., z • • •, z ) c H (B; Z) be indexed by each a e a l a2, a* ^ (X 1 x) , i . e . , z = { z , Consider the action we described i n t a(v,k) i * 6.3.7.. Analogously we define an act i o n of a E H (E ; Z ) n Ker (p ) on z ( j < 2n ). F i r s t define action of m. 0 x on z by * * - l v D-y y r-2 i-2 D-y ^ o ^ 1 " 1 ) a a j * and extend by l i n e a r i t y to a r b i t r a r y a E H (E^ ; Z^)n Ker(p^_ 2) . 6.4.4. THEOREM. Notation i s as above, k 1 (<j>r) i s a coset of the a c, — k group, a" © z where z . , . ranges over a l l of H (B; Z ) and a' = a a(v,K) V a V* , k 1 e H*(E ) f o r a e ^ ( X 1 - 1 ) . i - l a y. „ t i-2 Proof. See [30, 5.4.3]. 6.4.5. Remarks. (1) Mahowald has s u c e s s f u l l y used t h i s theory i n determining the regular s p h e r i c a l sets f o r V ( ori e n t a b l e ) bundles. Notice he n+m,m used cohomology operations without twi s t i n g . In general one would expect * the set Ker(p ) f l Ker(t^) to have a tw i s t i n g a c t i o n . Mahowald [30, §6] gave a computation of the k-invariants f o r V bundles. For the non-n+m,m or i e n t a b l e case see McClendon [37; 38; 39], Thomas [59] and Maunder [33]. (2) For ap p l i c a t i o n s to questions of embedding of r e a l p r o j e c t i v e spaces i n Euclidean spaces see Mahowald [30, §7] and Schwarzenberger and Epstein [15]. (3) For an a p p l i c a t i o n to the number of ( l i n e a r l y ) independent vector f i e l d s on manifold, f o r a l e a s t lower bound of such a number see [59] 116 (4) I f M i s a spin manifold of dimension n , and n = 3 mod 8 , 2n-4 n > 3 , then M immerses i n R [ 59, Theorem 1.6]. §6 . 5 . An I l l u s t r a t i o n 6.5.1. Remarks. (1) Let £ be a stable vector bundle over a CW-complex, X . Following Atiyah [5], % i s said to be of geometric dimension n , i f n i s the l e a s t p o s i t i v e integer k such that the following l i f t i n g problem i s solvable: ^»BO(k) X > BO where BO i s the c l a s s i f y i n g space f o r the i n f i n i t e orthogonal group and <j>£ i s the c l a s s i f y i n g map of £ . One can use the method o u t l i n e d i n the l a s t sections to f i n d a bound f o r the geometric dimension of £ . Following Thomas' l i n e , some r e s u l t s concerning the geometric dimension of stable vector bundles over lens space have been obtained together with an immersion theorem f o r lens space i n Euclidean spaces I D. Sjerve, 50]. (2) For other a p p l i c a t i o n s , see D. Sjerve 151], to geometric dimension of stable vector bundles over o r b i t manifolds, to mod p connected K-theory, see J.C. Alexander 14]. 117 6.5.2. Consider the question of l i f t i n g a map, E: X >• BSO(n+l), to BSO(n), i . e . , the problem' of. obtaining ."a commutative diagram: BSO(n) P X ->- BSO (n+1) Observe that BSO (n+1) i s 1-connected so that the Serre Theorem applies to give the exact sequence: - H n(S n) -»• H n + 1(BSO(n+l)) -*• H n + 1(BSO(n)) -> 0. Thus Ker(P ) = Im(t) i n dimension n+1 i s generated by the Euler-Poincare c l a s s , X n + ]_ • We s h a l l construct a Postnikov system f o r the f i b r a t i o n and compute some o f i t s k - i n v a r i a n t s . We know of course that the map, E, l i f t s to BSO(n) i f and only i f a l l the classes k X(£) vanish successively, where 'k.1 (E) i s defined as i n 6.4.3 when k 1  x (E) vanishes. We s h a l l attempt to c a l c u l a t e the k 1 f o r i <_ n+5 . So we have the following diagram: . . ^ , n S S l K(Z, n) -> BSO(n) -*• E BSO(n+1) 'n+1 -»• K(Z, n+1) i . e . , the f i r s t stage of a Postnikov system i n which we have T(S.) = ± x • 1 n+1 118 Ker(p*) = Ker(p*) as x -, ~ " •, ( mod 2 ) ( (n+l)-th Stiefel-Whitney 1 " n+l n+l clas s ). By lemma 2.4.3(1) we see that the following i s a f i b e r square: uo'(lx q )=v K(Z, n) x BSO(n) > E P l BSO(n) y BSO(n+l) 1 1 1 where y: K(Z, n) x E >• E i s the action of K(Z, n) on E . The Serre Theorem applie s (2.3.5(5)) to give an exact sequence: T 0 y H t(E 1) •+ H f c(K(Z, n) x BSO(n)) >• BSO(n+l) fo r t < 2n+l . * Our aim now i s to compute K e r ( T ^ ) ^ Ker(s^) . Let l ® b e * a H (K(Z, n)xBSO(n)) . Then T (l®b) = 0 f o r a l l b e H^(BSO(n)) and q <_ 2n , * since T^(1®1) = 0 and i s an H (BSO(n+l))-morphism. Since q^ = vos^ where s^ i s a cro s s - s e c t i o n of K(Z, n)xBSO(n), we see that, by d e f i n i t i o n * * of v , to f i n d Ker(q 1) i t s u f f i c e s to compute K e r ( t 1 ) / i K e r ( s 1 ) . In the 1 f i b r a t i o n , K(Z,n) y E  y BSO(n+l) , we have T ( I ) = ± x ., where n n+l i e H°(K(Z, n); Z) and T i s the transgression i n the f i b r a t i o n . Let n . i be the fundamental c a l s s of s n . I t i s well known that T ( I ) = OJ e n+l n+2 — H (BSO(n+l)). Thus T (I 01) = tii . Hence i n mod 2 c o e f f i c i e n t s .1 n n+l T ^ t U l ) = S q S ^ l ) = S q \ + 1 = V l . U i * using Wu's formula. Since i s a H (BSO(n+l))-morphism, we have T_ ( i ®p a) = io 'a I n . n+l * ' *v f o r any a e H (BSO(n+l)) where i i s the mod 2 reduction of i n n 6.5.3. Computation of Ker(q^). Here i denotes the mod 2 reduction of the fundamental cl a s s of K(Z, n) . . '. Dimension n: T_ (x0l) = to ,. . 1 n+1 * Ignore t h i s dimension as KerCs^) = 0 . * Dimension n+1: Ignore t h i s dimension as Ker(s^) = 0 , since H n + 1 ( K ( Z , n)xBSO(n)) H n + 1(BSO(n)) . 2 2 Dimension n+2: T . , Sq = Sq to . = to , _ • to_ and 1 ^ ^ n+1 n+1 2 T 1 ( 1 0 U 2 ) = Un+l'M2 • 2 Thus Sq (i®l) + \®to2 e Ker(r^) . 3 Dimension n+3: T.Sq (\<81) = to ., *to_ and 1 ^ n+1 3 T. (i®to_) = to '10 . 1 3 n+1 3 3 Thus Sq (iSl) + i©to3 e Kerd^) . 4 Dimension n+4: T,Sq (i®l) = to , *to. ; 1 ^ n+1 4 Tl<l0V = Vl*U4 ; T l S q 2 ( T 0 ( O 2 ) = ( « n + 1 ' » 2 ) ' « 2 - 0 ) ^ . 0 . 2 ; T l< l« U2 > = V l' U2 * 4 2 2 Thus Sq \®1 + i®to4 , Sq i®to2 + \®u)2 e Ker(x 1) . 5 Dimension n+5: T,Sq (i81) = to , *to ; 1 n+1 o • r l S q 3 ( i ^ 2 ) = V l ' V s •V^da^) ° V l , U 3 U 2 ^•(1*0)5) - a . n + 1 . W s ; ' T l ( l 9 w2 U3 > = V i ' V a '* T l (;®to 3to 2) = • 5 3 So Sq + \ ® t 0 5 , Sq (i0to 2) + x&m^bi^,. a n d 2 Sq (i®to3) + t®" 3to 2 e KerCx^) . 120 As a summary we l i s t a table of Ker(t^) upto dimension n+5 , ignoring those not i n Ker(s ) . * Dimension Ke r (T ^ ) n Ker(s^) <. n+l 0 n+2 A = Sq 2i®l + x®to2 n+3 1 3 Sq A = Sq 101 + l©w 3 n+4 2 4 Sq A, B = Sq 101 + n+5 3 1 2 3 2 Sq A = Sq Sq A = Sq (xBt*^) + Sq 100^; 2 1 5 Sq Sq A = Sq 101 + l©(w_ + to ' ( 0 b 2 3 2 + Sq ;©co3 ; 1„ 5 Sq B = Sq 101 + 10U),. 6.5.4. Apply the previous discussion to the f i b r a t i o n , Vn+2 2 ^ BSO(n) >- BSO (n+2) f o r n _ 2 . Using Paechter's groups we get when n = 4s+l ( and so n = 1 mod 4 ) and s >_ 1 the following t a b l e : * Dimension Ker( S ; L) o K e r ( T I ) <_ 4s+2 0 4s+3 2 A = Sq 101 + I 0 u 2 4s+4 1 1 2 1 Sq A = Sq Sq 101 + Sq 100^ ; 2 1 1 B = Sq Sq 101 + Sq 10U)2 4s+5 Sq 1B = S q 2 S q 2 l 0 1 + Sq 1!©!^ 2 = Sq A + A*0) 2 * 1 * Hence there e x i s t k^,. e H (E ) such that v k^ 1 2 and so Sq k 4 + Sq k + k ' i i ^ 8 3 0 . A and v k, * Dimension KerCq^ < 4s+2 0 4s+3 V 4s+4 Sq 1k 3 , k 4 4s+5 Sq k 4 , Sq k 3 These agree w e l l with Mahowald's computations. So E l i f t s i f f a l l the k - i n v a r i a n t s , k^(tj), vanish. The above shows that the main d i f f i c u l t i e s l i e i n computing them. §6.6. Some Examples of The Use of Postnikov Towers 6.6.1. Suppose X i s a f i n i t e dimensional CW-complex. I f ^ i s the fundamental cla s s of S P, then, given any v e H P ( X ; Z) , there e x i s t s an integer N such that there e x i s t s a l i f t i n g of N*v to S P, provided .N'V •* K(Z, p) (1) p i s odd, or (2) p i s even and v = 0 Proof. Case 1. Consider the standard Postnikov Tower f o r 8 P : S P 122 -> K(Z, p) : V hk . T k+l p+1 F p = K(Z, p) Sq -> K(Z 2, p+2) . Since ( S ^ ) i s always f i n i t e f o r k+l > p and p odd [24], we see that f o r some integer n' , i f n»v i s l i f t a b l e to F » •(n«v) i s l i f t a b l e to . Thus f o r p odd and f o r s u f f i c i e n t l y large M, i f N*v i s l i f t a b l e to F ' , then N-v can be l i f t e d a l l the way and hence i t l i f t s to S P . M 2 Case 2. p i s even and v = 0 . I f p i s even then TT ( S P ) f o r m > p > 2 i s always f i n i t e except m — f o r m = 2'p-l . As before there e x i s t s an integer N such that N «v l i f t s * 0 0 to F„ . Now TT _ ( S P ) i s isomorphic to a d i r e c t sum of an i n f i n i t e 2p-2 2p-l c y c l i c group and a f i n i t e abelian group [24] ..••As before i f we can somehow a l t e r N «v so that i t l i f t s past F^ _ , then i t l i f t s a l l the way. o 2p-2 Consider the Postnikov Tower f o r S P . Let 6 P e H P ( S P ; Z) be the p p fundamental c l a s s . P l a i n l y G = 0 . Suppose Vwv = 0 ; we see that N vwN v = 0 too . We can even take k„ „: F„ „ 0 o 2p-2 2p-2 *" K ( i r (S*'), 2p) to be induced by the cup-squaring operation. So t h i s means the i n t e g r a l 123 F 2p-l h2p-2 2p-2 S P , . F ^ K ( ( S P j ^ 2 p ) /" 2p-2 2p-l / 0 S q 2 X y K(Z, p) — : y K(Z , p+2) pa r t of k 2o(N o»v) i s homotopic to Zero. The r e s t of the proof proceeds as before to obtain a l i f t i n g of N*v f o r some p o s i t i v e integer N. 6.6.2. STEENROD CLASSIFICATION THEOREM [55]. Let X be a CW-complex and 6^ e H n ( s n ; Z) be a generator where * n n n > 2 . Then the map induced by 6^ , 8^: [X, S ] y H (X; Z), maps onto { u E H n(X; Z) ; Sq 2u = 0 } i f Dim(X) <_ n+2 ; and 6^ 1 (u) i s i n one-one correspondence with the quotient abelian group, H n + 1 ( X ; Z 2 ) / S q 2 H n _ 1 ( X ; Z) , i f Dim(X) <_ n+1 . Proof. Looking at a Postnikov tower f o r S n, we see that i f Dim(X) <_ n+2 the only obstruction to l i f t i n g a c l a s s u E H n(X; Z) to S n i s Sq 2(u) e H n + 2 ( X ; Z 2) . The second statement can be proved using c l a s s i c a l o b s t r u c t i o n theory f o r K(Z 2, n+2) p r i n c i p a l f i b r a t i o n s ( i . e . a f i b r a t i o n of type K(Z 2, n+2) ), noting that the higher d i f f e r e n c e obstructions a l l vanish i f Dim(X) <_ n+1'.."'( See Spanier [53, Chapter 8] f t f o r details.- " For a ge n e r a l i s a t i o n to H-spaces see B. Drachman 124 "A g e n e r a l i z a t i o n of the Steenrod c l a s s i f i c a t i o n theorem to H-spaces", Trans. Amer. Math. Soc. 153 '(1971), pp.53-88 .) 6.6.3. Remarks. (1) Thomas [60] using h i s method proved the follo w i n g : THEOREM. I f M i s a closed, connected, smooth manifold of dimension 4s+3 and s >_ 0 and i f u, (M) = to (M) = 0 , then span M [59; 60] >_ 2 . Generalising the technique by using cohomology operations he i s able to show that i f furthermore 6 t o ( M ) = 0 or k i s odd, then 4k span M > 3 . (2) For a S t i e f e l manifold f i b e r i n g ( c f . 6.5.4 ) McClendon i s able to generalize the technique used by Thomas [59] to construct a Postnikov system f o r such f i b e r i n g ( not n e c e s s a r i l y o r i e n t a b l e ) and obtain a l i f t i n g theorem [38, Theorem 6.1]. (3) There i s a Postnikov system defined f o r spectra [4]. This f a c t does not seem to have been exploited u n t i l only rece n t l y . BIBLIOGRAPHY' [1] ADAMS, J.F., Vector f i e l d s on spheres, Ann. of Math. 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