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UBC Theses and Dissertations

H* and some rather nice spaces Body, Richard A. 1972

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\6 I ^ H* AND SOME RATHER NICE SPACES by Richard Body A THESIS PRESENTED TOWARD FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1972 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g 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 g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a l i , Supervisor: Dr. R. Douglas ABSTRACT The Problem The integral cohomology algebra functor, H , was introduced to algebraic topology i n hopes of deciding when spaces are homotopy-equivalent. With this i n mind, l e t T(A) «• { XJ H (X) «• A} , the collect ion of a l l simply-connected f i n i t e complexes X , for which * the cohomology algebra H (X) i s isomorphic to A . We ask: when are there only a f i n i t e number of homotopy equivalence classes i n T(A) ? The Result Let A sat isfy the condition: k n At$Q - <X> Q[x . ] / (x . x ) . i » l 1 1 Then there are only a f i n i t e number of homotopy-equivalence classes i n T(A) . The Ilethods For a given A we construct a "model space" M and show that for any X e T(A) there exists a continuous map X t 15 "within N" . The concept of a map within N i s less res t r ic t ive than that of a homotopy-equivalence, but more res t r ic t ive than the concept of a rational equivalence. N We then show that i n the category T /M , whose objects are £ M , maps within N , having range M , there are only a f i n i t e number of equivalence classes. This i s proved with the use of Postnikov Towers and algebraic arguments similar to Serre's mod C theory. The result then follows. Applications The r e s u l t applies to spaces having r a t i o n a l cohomology isomorphic to the r a t i o n a l cohomology of t o p o l o g i c a l groups, H-spaces, S t i e f e l manifolds, the complex, and quaternlonic p r o j e c t i v e spaces and of some other homogeneous spaces. i v . TABLE OF CONTENTS CHAPTER TEE FIRST: Containing as Much as is Necessary or Proper to Acquaint the Reader with the Beginning of this History page 1 CHAPTER THE SECOND: Introducing Spaces and Rings Rather Nice; also a sketch of occurrences and transformations there among page 7 CHAPTER THE THIRD: Deserving of the Accolade: Hurewicz (within N) page 19 CHAPTER THE FOURTH: In Quest of the Uomotopy Type . . page 29 CHAPTER THE FIFTH: The Opination of our Discoveries; a Discourse page 39 A GLOSSARY OF TERMS page 44 A CITATION OF RELEVANT LITERATURE page 45 ACKNOWLEDGEMENTS The author i s happy to acknowledge the help and sometimes decisive encouragement of Dr. Roy Douglas during the preparation of this thesis. Gratitude i s extended to Drs. K. Lam, D. Sjerve and U. Suter for their kind suggestions of improvements in the presentation. The author gives special thanks to Dr. Caspar Cur j e l for many enjoyable and profitable discussions. v i . The Author expresses his Dedication to Kemil Atatiirk Janis Joplin Pallas Minerva Athena The Jefferson Airplane Joan Baez Choderlos de Laclos Alessandro Botticelli Doris Lessing Chiron C.S, Lewis Judy Collins Rene Magritte Cybele John Stewart Mill Angela Davis Vladimir Nabokov Bacchus Dionysius Kore Persephone J.P. Donleavy Jerry Rubin Isadora Duncan Bertrand Russell Douglas Fairbanks Jr. George Sand Henry Fielding Elizabeth Tudor Jane Fonda Tina Turner E.M. Forester Simone Weil Evariste Galois Virginia Woolf Greta Garbo Bob Dylan Zimmerman David Hume Glenda Jackson CHAPTER THE FIRST Premises what is necessary to be known: concerns a few common-places of the cohomology Functor H , its early successes followed closely with a somewhat melancltolic discourse on its later disappointments; finally a revelation of some bold aspirations for H s and a promise of a satisfactory conclusiony tho' the Reader perhaps will guard sundry doubts-The early pioneers of algebraic topology Alexander, Eilenberg, * Whitney and Zilber noticed that the integral cohomology functor H was equipt with a natural multiplicative structure, the cup product. It * was hoped that H could thus distinguish homotopy equivalence classes of spaces, or homotopy types. * Let us agree that H i s a functor of the topological-homotopy category T^ to the category of integral algebras. Thus we consider the category, T^ with objects spaces having the homotopy type of a s imply - connected, f i n i t e CW complex, with basepoint, and with transformations which are homotopy classes of continuous maps. Let A„ be the category with objects which are associative, graded commutative algebras over the integers, and with transformations vihich are graded algbra maps. H gives a representation of the spaces of T^ in terms of the algebras of A^ . The early pioneers noticed, that i n some cases, i t does this well: Case 1: Moore spaces Y ° have a nontrivial reduced integral homology VI group G i n only one dimension n . As always, Y^ i s simply connected, n > 1 1.1 Lemma If H*Y - H*Y^ , then Y - Y^ . ————— Or (j Pf. by the universal coefficient theorem and the Hurewicz isomorphism iv (Y;G) - Hom(G,G) + Ext (G,TT • Y) - [Y^,Y] and id, n <J n+l <j i* i induces' an isomorphism of integral homology. U J 3. Case 11: Complex projective n-space CP(ti) i s the 2n dimensional skeleton of K(Z,2) . 1.2. Lemma If H*Y - H*CP(n) then Y - CP(n) . 2 Pf. The generator of H (Y) is a map i n [Y;K(Z,2)] . It induces a cohomology isomorphism on the 2n-skeleton. 11 * In such cases, we shall say that the integral cohomology H uniquely  determines the homotopy type of the space. Recently i t was discovered that an associated functor H ( ;Q) gives quite a good representation of another class of spaces: Case 111: Finite dimensional H^spaces. Let G be a space which supports a homotopy multiplication. 1.3. Theorem If H Y < $ Q - H G $ Q and Y , an object of T n , supports a homotopy multiplication then Y must belong to one of a f i n i t e number of homotopy types. Pf. [Curjel-Douglas]. 11 In this situation we may say that the underlying homotopy types of H-spaces are f i n i t e l y determined by H ( ;Q) Theorem 1.3, may be combined with another result of [Curjel] which states that a f i n i t e CW complex admits at most a f i n i t e number of mutually non-isomorphic structures as a group in T^ . The result i s that on the * category of group objects of T h H ( ;Q) f i n i t e l y determines fi-type. 4. 1.4, Definition Let F : C-»• C' be a functor and X an object of C . Then F f i n i t e l y determines C-type at X i f the preimage of the C 1-equivalence class of FX is a f i n i t e set of C-equiva-lence classes. Despite these early success of H , as exemplified in Lemmas 1.1 and 1.2, pioneers of topology noticed that the functor did not give a com-plete picture of some tangibly evident spaces. Representative examples ft w i l l demonstrate some of the weaknesses of H Case IV: Let RP(n) denote n-dimensional real projective space and "v" the wedge product. 1.5 Lemma K*RP(3) =• H*PP(2)vS3 , but RP(3) ? PP(2)vS3 . Pf. see [Hilton and Wylie], This i s - a particularly galling example, since even II ( ;Z/2Z) gives a better showing. 1[ 3 Case V; Denote a Whitehead product of the three inclusions of S In S 3vS 3vS 3 by k e ir 7<S 3vS 3vS 3) . For n e Z + , denote the g cofibre of nk by W (obtained by attaching e with attaching n map nk ). 1.6 Lemma: H*w H^^ W but W f W i f m ^  n . n m n m ft Pf. The cohomology groups of H W may be determined from the 3 3 3 i 8 exact sequence for the cofibration S vS vS •» W •+ S . The n gradation shoxtfs that a l l multiplication except by the identity i s t r i v i a l . » 5. k i s of i n f i n i t e order [Hilton 2]. Moreover * ,W - Tr,(VS3)/(nk) / n / 3 where (nk) i s the subgroup of TT ^ VS generated by nk . In ir ^W^  , j„(k) i s of order n . Thus W } W i f n 4 m . V n m {Wjc i s an i n f i n i t e family of spaces of mutually distinct homotopy ft types, but with isomorphic cohomology algebras. In this situation, H does not f i n i t e l y determine homotopy type. Nevertheless the main result of this thesis will be that Jfox many of * the spaces of T, , H f i n i t e l y determines homotopy type, h 1.7. THEOREM Suppose that A is an algebra of which satisfies - k 0[x ] the condition A^pQ - ® ~— . Let ITT (A) be the 3et of home-l y , i . (x ) topy types Y with H Y - A . Then RT(A) is a f i n i t e set. Pf. see THEOREM 4.3.4. The cohomology algebras of Cases 1, 11 and 111 satisfy the conditions of 1.7. For more examples see 5.3.0. The proof of the THEOREM w i l l progress in three stages. In Chapter Two we give a description of a "model space" M which has the desired -k rational cohomology. For any space X with H X - A , a map <|> : X •*• M i s constructed so that X i s "near" to the model space. 4> i s "almost" an equivalence. With the help of the transformations $ , we show i n Chapter Three that the homotopy groups of any space X with H X - A must also be "near" to those of M . Then in Chapter Four, again with 6. the help of the maps $ x > we show that the k-invariants of X must be "near" to those of M . Finally we remember that k-invariants provide a complete description of homotopy type, to affirm the THEOREM. For a more complete discussion of the result, see Chapter Five. 7. CHAPTER THE SECOND In which the quest commences H 's rationalizations give rise to an abstract model to which it can give the appropriate ring structure (at least rationally I). However such is the vexatious estate of the topological universe that an abstract model space can seldom satisfy our functor's integral requirements. Frovidently3 and at wondrous length, the Author * recites how H begins to take stock of integrally satisfactory prospects, to discern the Similarities and Particularities between them and its rational model. The recitation includes an account of calculations of so BASE a character3 that some may not think it worthy of their notice. The Header is encouraged to approach the chapter with Charity and Eclectic diligence. 8. 2.1.0 MODEL SPACES Let us agree on some nomenclature: 2.1.1 Definition An algebra of A^ over Q is rather nice i f k Q[x 1 A - <2C . Here Q[x] i s the (graded-commutative) i=l , n i . polynomial ring on one generator x , and (x ) i s the ideal therein generated by x n Colloquially, we shall eay an algebra over the integers A i s rather nice i f A Q i s ; a space, or homotopy type, or even rational * homotopy type i s rather nice i f H ( ;Q) i s . ft We shall now construct a space H for which H (M;0) - A . This particular space shall be called the model space for A 2.2.0 LOOP SPACES OF SPHERES AND THEIR SKELETA The loop space of a space Y in T, i s the function space of a l l continuous basepoint preserving maps a : -*• Y , with the compact open topology. It is denoted ft(Y) , n i s a functor. There exists a natural isomorphism ^ .00 - ir ^(&Y) The loop space of an odd dimensional sphere S1**"^ - (m even) is mil i n f i n i t e dimensional, with rational cohomology H QS <$, Q - Q[x] the free polynomial ring of one generator of dimension m , over 0 The integral cohomology is a divided polymomial algebra on one generator (see 2.5.6 Example). We shall denote the loop space of S1"^" as (S111)^ for m even, cf [James]. For m odd, let (S1"1) denote S m . In 9. either case, (S ) has rational homotopy groups t r i v i a l i n a l l dimensions OO except m [James] gives a canonical skeletal decomposition of (S M) OO Denote the mn dimensional skeleton of (S m) by (S m) . Then i t s oo tl rational cohomology i s H*(S m) n $ 0 - Q [ X ] / ( x n + 1 ) , where x i s an m dimensional generator. More recondite i s the fact that ( S m ) n (m even) has exactly two non-trivial rational homotopy groups: ir (S m) 8? Q - Q m n and ir(n+l)m-l(S m) n ® Q - Q [Toda]. 2.2.1 Definition Let A be a rational, rather nice algebra, i . e . k Q[x ] A z. <£) — _ — t h e degree of x J i s called m, . Then i-1 (x.V1) 1 1 1 k m the model space for A i s M = X (S ) , where X indicates i=l n i topological direct product. Ii may also be called the rational model space for any Z algebra A1 with A* ® Q - A . A simple calculation involving the Kuenneth Theorem shows that H*(M;Q) - A . Another fact of prime importance for later theorems of this chapter i s a result of [Mimura-Toda; Lemma 2.4]: If i 4 (n+l)m , there exists an endomorphism of the relative complex ((S m) , (S m) ) which induces oo n a t r i v i a l map on the i t h relative homotopy group ^( ( S 1 1 1 ) ^ , ( s m ) n ) > while maintaining an induced automorphism on a l l rational relative homotopy 10. groups 7rk< (S111)^ , (Sm)n)<£>Q . This i s why rational model spaces are useful. They have internal transformations which preserve their rational structure, yet which ignore torsion i n their cohomology and homotopy groups. 2 . 3 . 0 HOW H* COMPARES SPACES 0/tIN SEARCH OF A WHITEHEAD THEOREM WITHIN N 2 . 3 . 1 Definition Let <|> : A -»• B be a homoniorphism of f i n i t e l y -generated abelian groups, f i s a (group)map within N i f "Ker(<j>) and Coker(o) are f i n i t e groups of order a factor of N. A map <J>^ : A n -»• B n of graded abelian groups i s within N i f the order of the kernel of ~tyn = o(ker cj>n) raised to the n t n power is a factor of N, for a l l n. An!:.fiilgebr'a map of A z is'within N i f i t induces a map within N ' on the underlying graded abelian groups. A map of T i s within N i f ll"(ty) i s within N . Finally, two spaces Y and Y' are within N i f there exists a morphism within N between them. Group maps within N enjoy some elementary properties: 2 . 3 . 2 Lemma Let A ^  B , B C be group maps within N and N* resp. Then ty ° ty is within N'N . 1T 2 . 3 . 3 Lemma Let <j> : A -*• B be an abelian group map witliin N . Then 2 2 there exists a map ty : B -»• A (within N ) such that ty o ty=> K and f°<p " N 2 . i s a commutative diagram. A > B 11. 2.3.4 Lemma Let 4> : -»• C^ . be a map of chains of abelian groups within N . Then the maps induced on the homology of the chain 2 complexes, Hn(<|>) are maps within N Pf. Diagram hassling. 11 Suppose that A i s a given, integral, rather nice algebra. The main result of this chapter w i l l be to show that a l l spaces Y with * H Y - A are within a uniform N of the model space M . 2.4.0 OBSTRUCTIONS To construct maps within N between rather nice spaces and the model, we rec a l l some of the results of obstruction theory. See [Hu; Chapter VI, i n particular E]. i 1 i 2 Let * -—> Y c—> Y c—> ... c—> Y be a skeletal CW decomposition + for Y . Suppose a map has been defined f : (Y,Y ) -»• (Z,Z') has been t+1 defined and we wish to extend this map to a map f... : (Y,Y ) -*• (Z,Z') •fc+1 so that ( y , y t + 1 > - l ± i > ( z , z ' ) t / ( l , i , ) j / i s a homotopy commutative. x j y It i s a standard result of obstruction theory [Ru] that an obstruction class Y(f ) e B.^+^(Y;-n. ,, (Z,Z') measures the obstruction t t+1 12. to this extension. The map f, , can be extended to f, , , i f f t -1 t+1 Y ( f t ) = 0 . The obstruction class i s defined naturally in the following two senses. 2.4.1 Lemma Consider the following diagram i n the category of excisive pairs of T (u,wn + 1) — ( x , x n + 1 > (W,Wn) > (X,Xn) g > (Y,Y«) 5 > (Z,2») n+1 n# Then a) h induces a coefficient map : H (X;TT , - (Y,Y' ) ) > n+1 H n + 1 ( X ; 7 r n + 1 ( Z , Z » ) ) and h # ( 3 T ( g ) ) = y ( h o g ) . b) f induces a cohomology map H n + 1 (X , i r (Y,Y')) — > H n + 1 ( W , i r . (Y.Y 1)) n+J. n+i. and f*Y (g) = Y ( g o f ) . Pf. from defini t ions . Ii 2.5.0 MEDITATIONS ON THE INTERNAL STRUCTURE OF A Z-ALGEBRA Let A be an integral rather nice algebra i n A^ • Then k Q[x 1 A '% Q - 3& -r— with degree x, = . , f— n.+l. ° i i 1=1 (xjL 1 ) Let i : A A :X» Q be the rationalization of A . For each Q _ _ V 1 i « 1,2, ,k we have the inclusions Q [ x i ] / ( x i ) > A iH} Q . 13. 2.5.1 Definition The i t h faculty of A , Fac^A i s the subalgebra -1 n i + 1 i Q CQEx±]/Cx± 1 )) til In dimension m^. the i faculty, considered as an abelian group, has free rank 1. Choose a non-torsion generator of this group, called X i • 2.5.2 Definition The power subalgebra of x , Pow(x) is the subalgebra 0 2 of a generated as a Z-module by the powers of x (x = l,x,x etc.) For the next definition, we consider only the torsion-free graded quotient algebras of the two subalgebras. Call them FreePow(x^) and FreeFac^A . 2.5.3 Definition The depth of Fac^A i s the order of the cokernel of the inclusion FreePow(x^) > FreeFac^A , considered as a map of graded abelian groups. It can easily be shown that definition 2.5.3 is independent of the choice of x^ : any too choices have + the same representative i n FreePow(x) . Similarly for more than one generator, 2.5.4 Definition The power algebra of (x^x,,, .x^) , denoted Pow(x^,,x^) i s the subalgebra of A generated as a Z-module by a l l products of the x^ The algebra Pcn^(x1,,x^) modulo the ideal of torsion elements w i l l be denoted FreePow^, ,x^) . Similarly for Free A . 14. 2.5.5 Definition The depth of A , dA is the order of the quotient of the inclusion FreePow(x^, j X ^ ) > Free A considered as a map of graded abelian groups. An example w i l l surely not further mystify the situation. Perhaps i t w i l l clear up most of the fog: 2.5.6 Example The free divided polynomial algebra on one generator of height n+1 may be denoted I"n(x) . It i s a free Z-module of rank n+1 with generators (x._. . ,,x. s) . It has a (v) W (n; multiplication defined by the relations x, * -x, » = ( n ) x, . v • (m) (n) (n+ft) (^ ) means "n choose k" . In particular, l e t x = x(-Q Then x = k!x^^ . The power algebra of x i s the free module generated by the powers of x . There i s exactly one faculty and i t s depth i n the lc*"*1 dimension i s k! . The depth of ^ ( x ) i s l!2!3!..n! One more definition and we can start demonstrating something. 2.5.7 Definition The torsion number of A in dimension k i s the order t (A) n of the subgroup of torsion elements of degree k 2.6.0 HOW THE OBSTRUCTIONS ARB OVERCOME; A COMPARISON IS POSSIBLE , i We consider the class of a l l spaces Y such that II Y - A , a given particular, rather nice algebra. For each such space, we wish to produce a map $ : Y -•• 14 . While we shall be doing this, we shall take care that a l l maps ^ are maps within some (uniform) , an integer defined solely i n terms of A . A sometimes delicate quantification puzzle presents i t s e l f at this time. Think i t through! The result of this chapter w i l l be 15. 2.6.1 Theorem Pick a rather nice algebra A . Then there i s an (exp l i c i t l y stated) integer N so that a l l spaces Y with H Y - A are within of A's rational model space II . Pf. The discussion w i l l largely follow [Ilimura and Toda; Lemma 2.5] Since A -is an integral rather nice algebra in A^ k _ _ n+1 k m A O Q « <g> Q[x ]/(x. 1 ) ;M- X (S *) 1=1 1 1 1=1 n i ... where m^  i s the degree of x^ a) We can consider the model space one factor at a time: ..To .ohtain-a map-within i ^ - * ^ . ': Y ' 1 1 , - i t "is sufficient to i m i construct a map within (a uniform) N . <J> : Y (S ) . Indeed i f l i n^ i * m-f <J> * : H (S ) -»- FreePow(x ) i s within N for a l l Y , set k n H A s (<U n ( N / ' t . ^ ) .. Then set = ( " f r 1 , * 2 , tfy and the i=l result w i l l follow . . Let us now agree to suppress the index i from x^,m^,n^,N^,Fac^A,c}>* b) If m i s odd: Then n(=n^) = 1 and [Serre] demonstrates there exists ty : Y -*• S m ~. One observes that,-upon the required restr-tction, dimA-m d ± m A m d l n ^ this induces a map within N = 2 n (.(order of TT (S ) ) j=m+l 3 m ' •: Since the order of ir^S i s f i n i t e for j > m ,. H is-* aiwell defined integer, an invariant of A 1 6 . c) I f m i s even: the demonstration i s more complex. [Berstein] i s c r e d i t e d with e a r l y work i n this d i r e c t i o n . The idea i s to produce a map 6 : Y ( S m ) o o . By c e l l u l a r approximation, 6 may be viewed as a map (Y.Y™ 3) •*• ((S m) , (S m) ) . We then t r y to extend t h i s * t o a map (Y,Y) -> ((S m) , (S m) ) . This almost works, but not quite; we must modify % by an endomorphism of ( ( S m ) , (Sm) ) to get a map which does extend, and th i s map w i l l be the n required fy^ Choose a tor s i o n - f r e e generator x of dimension m i n FacA Without loss of generality we may assume ^ ( x ) = x (see 2 . 5 . 1 ) x i s -< an m + 1 dimensional cohomology generator of Y . Let us c a l l the generator of HTn+"^ Sm+''", u , and l e t u be a generator of H m(S m) Again invoking the work of [Serre], we know there e x i s t s a map 8 : £Y -*• S S1""4"1 such that 6"*'(u) = N'£ x where N' i s given by the formula i n b) su b s t i t u t i n g m + 1 for ra , and dim A + 1 f o r dim A . The adjoint map 8 : Y fiSID+1 = (S m) o o has the s i m i l a r property 8*(u) = H*x . We thus have a map (Y,Y m n) -*• ((S m) , (S m) ) . A moment's r e f l e c t i o n on the c e l l structure of (S m) indi c a t e s that this CO may automatically be extended to a map ( Y , Y m ^ n + ^ 1 ) ( ( S m ) O T , ( S m ) n ) Thus the f i r s t (and most d i f f i c u l t ) obstruction to extending 8 to a map (Y,Y) + ( ( S m ) a , ( S m ) n ) occurs i n H M ( N + 1 ) < Y ; i r m ( t t f l ) « s \ , ( S m ) n ) ) . Now i r m ( n + 1 ) i s e a s i l y c a l c u l a b l e : % ( n + 1 ) ( ( S ^ , ( s " ) ^ H m ( n + l ) ^ s m ^ ^ (s m)^x z . The obstruction cocycle c(8) Is also r e a d i l y 17. calculated: Let w - m(n+l). c(e) i s equivalent to 0 . : C (Y) = H (Y", v * " 1 ) - H ((Sm) , ( S m ) ) - H ( S \ = Z . Y ( e ) = e*(u, ..*) , where w °o n w 0 0 (,n+l^  U(n+1) * S t * i e S e n e r a t o r ° f t * i e w degree of r (u) = H (Sm) . We now note that (n+1) ! Y (e ) = 6 * ( u n r l ) which must be a torsion element: e (u 1 1 ^) = (N* ) n + 1 and x1*^ ±s torsion for J C * 1 * * = 0 . Thus Y ( 8 ) i s a torsion obstruction. In a l l other dimensions, the obstruction to extension of any map i s a torsion cohomology class since a l l the coefficient groups TT , ((Sm) , (Sm) n) are torsion groups. Let us denote the order of the group ^^((S111)^ (Sm) ) by p. . n *• [lUmura Toda; Lemma 2.4] show the existence of endomorphisms of ((S111)^ , (Sm) n) which t r i v i a l i z e torsion homotopy groups. We compose theta with a sequence of these endomorphisms, and c a l l the composite <j> From Lemma 2.4.1 we confirm that the obstructions to extending to (Y,Y) a l l vanish and the theorem 2.6.1 shal l be proven. [Mimura Toda; Lemma 2.4] supplies us with an endomorphism h U J - J J r J n+1 _Tm(n+l), ,.a> v „ whicn induces a map of degree q on \\ ((S )oo , (S ) ) . For this reason i f we let q = t (A) w * * „ n+1 , Y(h o 6 = 8 h u , , = q Y ( 8 ) = 0 . h o Y extends to the next q q (m(n+l) ^ q dimension. [Ilimura Toda; Lemma 2.4] also shows the existence of l i , v an k m endomorphism which induces a map of degree p. on n, ((S ) , (S ) ) ic k oo n when k ^ m(n+l) . Hence 18. h (P oh (P. dimA' dimA-1 m(n+l)+l has no obstructions and extends to a map ty : (Y,Y) - ( ( s V , (s") ) » n A laborious calculation shows that h^ p j induces k' m k * a map of degree p^ " in H ((S ) ) . So f i n a l l y ty (u) = Nx where m ( n + 1 ) j=m(n+l)+l J The main result of Chapter Two has been proven: A l l spaces Y with H Y - A , a given rather nice algebra, are within some uniform N of the model space M . Recall that two spaces are within N i f there exists a map with induced cohomology map within N between them. We shall see that ft this kind of comparison "H - within N" is important in discovering how ir compares spaces. 19. CHAPTER THE THIRD Containing such serious matter that the Reader cannot laugh once through the whole chapter, unless peradventure he should laugh at the author. The Reader is cautioned that without Familiarity of the Riggings of the language of the following introduction, or without Sensibility of the direction of the demonstrations he will soon welter in a desolate Sargasso of minutiae and conundra. HERE BE SERPENTS !! To the Intrepid Reader, the Author now pledges Solemn Promise of Prodigious Treat upai completion of his Endeavours* To the Prodigal Reader, the Author imparts begruagingly that hereto-fore bespoken Treat might perhaps be ferreted out at 4.3.5. Of the Disciplined Reader, the Author commends Zeal and Patience, and invites him to prove his Mettle forthwith. 20. 3.1.0 A NOVEL CATEGORY AND A QUESTION OF INFLUENCE. Let us agree that A denotes a rather nice algebra i n A , M la i t s r a t i o n a l model space as defined i n 2.2.1 and N the integer defined i n Theorem 2.6.1. I f Y i s a space of T^ and II Y - A , we have already constructed a map ty : Y ->- H which i s within N . Let us aff i r m the importance of the role of ty with a change of viewpoint (and a dextrous modification of our categories). N 3.1.1 D e f i n i t i o n The category of spaces within N over M , T /M has objects which are homotopy classes of continuous maps : Y £ M within N . Maps are (homotopy) commutative t r i a n g l e s of T^ N , On T /M l e t us introduce a u x i l i a r y functors D and Fb which d i s t i n g u i s h r e s p e c t i v e l y the domain f o r ty , and I t s f i b r e . There i s a f i b r a t i o n Fb(<j>) -»• D($)-»- M . We s h a l l also require a notion of when a functor ty influences a functor ty 3.1.2 D e f i n i t i o n Let ty : -*• Cv, and ? : C^ -*• C 3 be two functors. Denote the class of equivalence classes of a category C by C . Then the functors ty and 4* induce maps ty : C^ -*• C2 and f : C 1 -> C 3 . ty f i n i t e l y determines ty i f Y ty"1 x i s a f i n i t e set, f o r a l l x i n C£ & ie 3.1.3 Example On T^ , H f i n i t e l y determines H ( ;Z/pZ) . S i m i l a r l y A ie 0 it H ( ;Z/pZ) f i n i t e l y determines H ( ;Z/p Z) . Thus H f i n i t e l y ie 2 ^ determines R ( ;Z/p Z) . H uniquely determines H ( ;Q) 1.5. Similarly for several functors: 3.1.4 Definition Let ¥: C -*- C be a functor and {m : C C.} be a o r 4. 1 . set of functors. Then {<(>.} f i n i t e l y determine V i f ? ( H *7" L((I )) i j[ I i i s a f i n i t e set, for a l l in 3.1.5 Examples II . ( ;Z) and H ( ;Z) f i n i t e l y determine H n( :Z) . n-1 n ' Indeed by the universal coefficient theorem, they uniquely determine H n( ;Z) . More d i f f i c u l t : see Theorem 3.4.1 {TT (Fb) n , (Fb) n * n-l N and 7rnM} f i n i t e l y determine irn<D) on T /M . 3.1.6 More Examples Consider the category Z + whose objects are the non-negative natural numbers. If m f n Map W (m,n) has only one element, * . If m « n , Map ' (m,m) has exactly two elements + z * and i d ^ . Composition with the i d ^ does not change any map. A l l other compositions = * . This might be called the discrete + + category on the set Z . The functors C -*• Z are exactly the integer invariants of C Let At' be the category of f i n i t e l y generated abelian groups, Rk : Ab •* Z+ be such that Rk(G) i s the minimal number of cyclic direct summands of G . It i s an Invariant of Ab-isomorphism class, and hence a functor. Similarly Tn : Ab •*• Z + i s the order of the torsion subgroup. Then Rk and Tn f i n i t e l y determine the iden-t i t y functor on Ab , or i n the language of 1.4, Rk and Tn f i n i t e l y determine Ab-type. 22. 3.1.6 Exposition Thz filnAt aun oft Chaptzn. ThAzz shall bz to dzmovu\th.atz * that on thz catzgoAy o£ Aathzn. ru.ce. ipaczs, H filnltzly dzteAmlnzi TT^ , ion. all n . I n otke/i ioondt> Ifi A <U Aathzn. ni.cz, we shall dzduzz that thz class t ^ ' ^ \ H Y • nzsolvzs Into a finite. nimbe/L o& IbomoApiilbm classes ofi gftoups. Viz shall approach this nzsu.lt {Thzosizm 3 . 4 . 2 ) cxm.z{jxlly, In thne.z stages ( 3 . 2 , 3 . 3 , 3 . 4 ) . Task thz ViASt: Lzt Vb[$) bz thz ^Ibnz ofi thz map $ : V •+ M givzn In Th.zon.zm 2.5.1. With thz help o{ thz SzfiAz spzztAal sequence, H (P) finitely dztznmlnzs H^(Fb) fioA all n . ttwizovzh, TTJ(FD) lb ilnltzlij dzteAmlnzd by H (P) [Hitizwlzz) Task tliz Szcond: Lzt CC (Fb) bz thz n zonnzctlvz ca/znlxg o<5 Fb . TheAZ Is aviation K[ir F6,n) CCn{Fb) + CC^' jFb) Applying thz Sznxz spzctAal szcfizncz again {TT (F6) , \{.[CCn~^) tfJCCn_J(Fb) )[t <_k} ilnltzly izWunlnz Hfe(CCn(Ffc)) . Ta&k. thz ThlAd: ir (F6J = H (CC n ~ ' ' ( Fb) ) ^fiom ( t fttewicz) . We can zalaxlatz thz homotopy gn.ou.pt> o& V {,Aom thosz o& Fb and M . 23. 3.2.0 THE FIBRE SPACE OF A MAP WITHIN N Given a map ty :Y -»• M of T^ , within il we may construct the fibre of ty , Fb(<{>) , so that Fb(4>) Y •* M i s a fib ration. There are two things to notice i n i t i a l l y . Fb(ty) is not in general f i n i t e dimensional, though for our purposes this w i l l not matter. irn(Fb(<j>)) and Hn<Fb(<{>)) are fi n i t e groups for n > 0 , by the use of "mod C" theory [Serre]. We shall limit the cohomology groups of the fibre by applying the Serre spectral sequence for cohomology with integral coefficients.[Serre 2]. The E£ level of the spectral sequence has the property E 2 P ' q - HP(M;Hq(Fb)) © GP>q p+q=»n *,q * where i s the graded group associated with H D(<{>) with respect to the f i l t r a t i o n ( F * ' q - 1 > F* , q) of li*V(ty) . ty ty 3.2.1 Lemma E ^ , q is f i n i t e l y determined by E*D (on the category TlJ/M) Pf. Consider % = G << CO e cokliq(<|>) < Show that 8 exists and is epic. Thus the number of direct irreducible summands Rk(E°'q) ^  RkHqD , and the torsion order T n(E° , q) 5 N . 1 1 HqD <- Hq(>},) 1 nqM 24. 3.2.2 Lemma { ^ ' ^ ^ q a n d H * D f i n i t e l y determine E ^ , q Pf. The number of isomorphism classes of subgroups of a f i n i t e l y generated abelian group i s f i n i t e . The number of isomorphism classes of quotient groups of a fi n i t e abelian group i s f i n i t e . E*' 3 - H/^MjH^Fb) - nXli $ HjFb Tor(H^+^ll,H^Fb) - H^l® E02'i + TorCH^ V E j ' " 3 ) i s uniquely determined by E^'^ . It i s also a f i n i t e group (-valued functor) when j ^ 0 , because E^'-* i s , when j ^  0 E*»^ , a sub-quotient of E*'-* i s f i n i t e l y determined by E^'^ i f s /. z j # 0 . Let d : E ° ' q > EB^-0+l b e a n s t h d i f f e r e n t i a l . When s s s s ^  q + 1 Im(d ) is f i n i t e l y determined by E * q S + * s £ When s = q + 1 consider I m d ,.Eq+1.0 > > Eq+l,0 m Eq +1,0 S S S+l <o 6 k e r l i q + l > H q + 1 M > H q + 1D 25. Show that 6 exists and i s epic. Then Tn(Imds) < Tn(kerH q + 1) ^  N. Hence Im(d ), a fi n i t e group (-v.f.) i s f i n i t e l y determined s by E^' q for a l l s . We have a short exact sequence E s + I E s ' q — I ^ d s > f o r a 1 1 S Since the isomorphism class of E u' - i s uniquely determined by the extension class of d which is i n Ext(Im(d ), E°' q) , a fi n i t e group (v.f) i t might not be a bad idea to notice that Ext( ) is f i n i t e l y determined by E ° ? q and {E?^}. . E°' q i s s+1 2 j<q s fi n i t e l y determined by E°' q and {E^ *-'} . . By the convergence S T J . Z J < q properties of the Serre spectral sequence E°' q - E°' q , and by q+z 0 0 Lemma 3.2.1, Lemma 3.2.2 i s proved. IF 3.2.3 Corollary H*D f i n i t e l y determines E°' q = HqFb , for a l l q . Pf. Induction on q of Lemma 3.2.3. 11 For the next task, we shall find i t somewhat more convenient to restate this as 3.2.4 Corollary H D f i n i t e l y determines H^Fb . Pf. Universal Coefficient Theorem. U 26. 3.3.0 THE n t h CONNECTIVE COVERING OF THE FIBRE The homotopy exact sequence includes the sequence TT2" > ir2M — » TTj^ Fb > TTjP = 0 This implies that ir^Fb i s abelian and there i s a natural isomor-phism Tr^Fb * H^Fb . Comparing this with Corollary 3.2.4 we get 3.3.1 Lemma Tr^Fb i s f i n i t e l y determined by n D (on We now perform the Cartan-Serre-Whitehead method of k i l l i n g higher homotopy groups [Mosher--Tangora] with a map i ^ : Fb > K(Tr.jFb,l) . The fibre of this map w i l l be denoted CC1(Fb) and i s called the simply-connected covering of Fb til 3.3.2 Recursive Definition The n connective covering of a space X , denoted CCnX i s the fibre space of the Whitehead-Serre-Cartan map ^ : CC n - 1X K(ir X,n) As i t s name implies, i t i s n-connected, Furthermore TT CC^X * TT.X i f j > n 3.3.3 Lemma Let 3 be the set of functors {TT Fb,H CC 0 - Fb.H.CCpFbjj<q} - q n q J Then J' f i n i t e l y determines H CCnFo . q q Pf. Note that a l l are f i n i t e abelian (group valued functors), from "mod C theory". 27. Consider the Serre spectral sequence of CC^Fb -»• CC Fb -> K(ir Fb,n) . The E 2 level: E 2 _ - H ( K(ir Fb,n) ; H (CC^Fb)) n s»t s n t - HgK(irnFb,n) ® H^C^Fb © Tor(Hg_^K(irnFb ,n) , H CC°Fb) i s uniquely determined by 3* i f t < q . r r 2 On the E level E i s a sub-quotient of E ^ and hence i s s,t s,t fi n i t e l y determined by i t . If t < q ^ f i n i t e l y determines s,t Consider the r t h differential d r : E r _.,-»• E* . The r,q-r+l 0,q r r image of d i s a quotient of E r q . ^ j aad s ° i s f i n i t e l y deter-mined by J • q r r x'l 1 We have an exact sequence Ira(d ) > E- — E _ . From 0,q 0,q the properties of Ext, we have that i s f i n i t e l y determined q 0,q By the convergence of the Serre spectral sequence, and by the use of the edge homomorphism we have E q + 2 = E™ > p = JJ cc 1 1 "^Fb 6 0,q 0,q o,q q q+2 n-l —r Hence E^ i s f i n i t e l y determined by H CC Fb , and hence by vjf 0,q ' q ^ q Lemma 3 . 3 . 3 follows by downward induction on r , of the previous paragraph. % ft 3 . 3 . 4 Corollary ir^Fb) i s f i n i t e l y determined by H D . Pf. Remark that IT Fb - H CCn~1Fb . Apply Lemma 3 . 3 . 3 inductively n n f i r s t on q , then on n , to show that E^CCmFb i s f i n i t e l y ft determined by H D for any p,m . % 28. 3.4.0 THE HOMOTOPY GROUPS OF A RATHER NICE SPACE W * 3.4.1 Theorem On T /M TT D is f i n i t e l y determined by H D . n Pf. The homotopy exact sequence includes the sequence i , TT Fb -S—>TT D -^ -> TT H >TT , Fb and so n n n n-1 Im(i ) > TT D > Im(d>) i s short exact. n n Considered as a quotient of ^Fb , ImCi^) i s f i n i t e l y determined ft by H D (Lemma 3.3.4). Im(<j>) i s a subgroup of TT M , and there are only a fi n i t e number of extensions, up to isomorphism, 11 3.4.2 Corollary Let A be an integral, rather nice algebra. i * Then {u^YlH Y - A} resolves into only a f i n i t e number of group-isomorphism classes. 3.4.3 Lemma There exists an integer N7* such that TT (<j>) i s within n n for a l l objects <f> of T J ; I /H . Pf. Let N be the least common multiple of the fi n i t e set n {Tn(nnFb((j,)) , Tn(rr^Fb (<),)) |<J> e obj TN/M} . « CHAPTER THE FOURTH A wonderful long chapter concerning the marvellous; containing much clearer matters, but which flow from the same fountain with those in ihe preceding Chapter. The memorable Transactions which occur within may encourage the Reader felicitously to adapt some of his Categories. In any case, these Transactions are indeed integral in a happy resolution of our Recitation. 30. 4-1.0 n-TYPE AND POSTNIK0V TOWERS 4.1.1 Defini t ion: Two CW complexes K,L have the same n-type i f there exist maps f : K n ->• L n and g : L n F.n wh^re L n - , K n are the n1"*1 dimensional skeleta of K and L . The maps sa t i s fy : a) For every map from any (n-l) dimensional Cl complex h : If -»• K n gof »h - h . b) For every map from any (n-l) dimensional CW complex h : M L n f <>goh S h . One of the most important observations about n-type i s 4.1.2 Lemma Let K and L be n-dimensional complexes. Then they have the same (n+1) type i f f they have the same homotopy type. 11 Given a CW complex Y there is a canonical space P^Y with the same (n+1) type as Y : The Cartan-Serre-Hhitehead method of k i l l i n g higher homotopy groups constructs a map i : Y P^Y , with the property that i : TT.Y - TT.P Y for i £ n , and r wff l x n TT.P Y = 0 for i > n . i n 4.1.3 Lemma X and Y have the same (n+1) type i f f P X - P Y . ———— ii n Pf . Whitehead theorem. If If we now apply the Cartan-Serre-TJhitbead method of k i l l i n g the (n+1)st homotopy group of P n + ] _ Y > w e obtain a map p n + ^ • Such maps form a Postnikov tower for Y In the sense of [Mosher-Tangora]. 31. 4.1.4 Definition: If X i s a simply connected complex, the diagram • is called a Postnikov Tower for X i f i t satisfies four conditions: a) the diagram i s homotopy commutative b) TT.X = 0 for i > m i m * c) p induces isomorphisms TT .X - TT.X for i < m m * i i m d) p™ , i s a principal K(TT X,m) fibration. m—X m Happily, there i s a classification theory for princiapl fi b r a -tions [Keyer] and ^ i s classified by a homotopy class k,- .,N:PX , •*• K(TT X,m+1) . This homotopy class i s often called the {tar-1) m—1 m (m-l)st k-invariant of the Postnikov tower. Postnikov Towers enjoy some functorial properties: 4.1.5 Lemma Let X and Y be simply-connected CW complexes, and l e t f : X -*• Y be a continuous map. Suppose P^ X and P^ Y are Postnikov Towers for X and Y . Then there exists a family of maps P f: P X .* P Y with the following properties: n n n 3 2 . a) k x P .X N >K(TT X,n+1) n-1 n n-1 k Y f# is homotopy commutative P nY n-1 -*K(ir Y,n+1) n b) P f P X n n ->P Y n n-1 p , i s homotopy commutative n-1 P .X- p>P .Y n-1 P .f n=»l n-1 c) X >P X n P^f i s homotopy commutative >P Y P n n Pf. [Kahn, Mosher-Tangora]. U It w i l l be important to know when two spaces having the same type have the same (n+1) type. 1.6 Lemma Consider the following diagram: n . x p , k P X > P .X — > K(TT X,n+1) n n—x n n P Y n n - 1 \ 1- Y •> P . Y n-1 n -> K(TrnY,n+l) There exists a homotopy equivalence, allowing the diagram to commut«-', on the l e f t , i f f there exists one on the right. 33. Pf. If there exists a homotopy equivalence on the right (commutative) the induced fibre map i s the required commutative homotopy equivalence on the l e f t . If there exists a homotopy equivalence on the l e f t (commutative) say P f: P X * P Y , note that {P X} , {P Y} for m < n form n n n m m " Postnikov towers for P X and F Y . Let the map on the right be n n P^f^ . Property a) of Lemma 4.1.5 shows that the diagram commutes on the right, and the homotopy exact sequence for fibrations shows that i t i s a homotopy equivalence. 11 We shall now begin to prove the main result of this thesis: * _ N . THEOREM: H D f i n i t e l y determines homotopy type on T /M . N Recall that an object of T /¥>. is a homotopy class ty : Y 11 which i s within N , and that a map of T /!* is a (homotopy) commutative triangle. 4.1.7 Definition If ty and ty1 are objects of T^/ll , they have the same (n+1) type (over II ) i f there exists a homotopy commutative diagram Pn(D*) — > ?n(nj,') P II n Since everything i s simply connected, a l l objects i n T / ! ! have the same 1-type. 34. .8 Explication Su.ppo&z we hauz dzmont>th.atzd that H V fiinitzly * dztzhtninzs n-type.. We tuu/i to &h.m that {n-type., v^V and H V} fiinitzly i ztzminzi (n+1) typz. Con&idzn the following diagram , n-1 y ? .Vi n-1 * feA» n-l K(*M,n+1) <f> and <(>' hcojz the. &amz [n+1)-typz ififi thzh.z zxibtA a. homotopy zquixfoLznce. a , indicated ab a dottzd tlnz, which allow tkz zntixz AX.OQh.am to commxtz. Thz zxAAizncz o$ a homotopy zqxi.vatzncz alZoiaing tkz light-hand tftuianglz to commitz ii, a pxhzty ghxup th.zoh.ztic question about thz iAomoh.pfaum& o{, -»- ^ n ^ * ' • It &hall bz dzaZX witli in 4.2. Thz question oft thz homotopy comrrutativity o& the whole diagram shall be considered in 4.3, with our knowledge of IT A . Thus shall we perform the induction step* 3 5 . 4.2.0 THE CATEGORY Ab/C 4.2.1 Definition The category of abelian groups over C» Ab/C has objects which are group homomorphisms B ^  C , where B and C are f i n i t e l y generated abelian groups. Morphisms in Ab/C are commutative triangles. The f u l l subcategory of groups over C and within N , denoted K Ab /C has objects which are group maps within N N Of course, two objects <J> and ty * w i l l be equivalent i n Ab /C I f f 3 *• B C i s commutative. 4.2.2 Theorem There are only a f i n i t e number of equivalence classes i n AbH/C . Sketch of proof: Let D(<j>) denote the domain of $ . Then the .free rank of DGj,) - free rank of C and the order of the torsion subgroup of D(<j>) must divide N times the order of the torsion subgroup of C . Thus there are only a f i n i t e number of group isomorphism classes for D . Since Hom(B,C) =" Hom(FB,FC) ® Hom(FB,TC) Hom(TB.TC) where B - TB 3? FB , the torsion and free summands of B , we may re s t r i c t our attention to the case when B and C are free groups. The result then follows from elementary observations. II 4.2.3 Definition The category Ab,.\C, abelian groups under C and within N u has objects which are group maps C B within W A.2.4 Theorem There are only a f i n i t e number of equivalence classes i n Ab^C 3 6 . 4.3.0 THE MAIN RESULT ; THE WHITEHEAD THEOREU WITHIN N ft Let us review: with the assumption that H D f i n i t e l y determines the n-type of ty , we consider the following diagram for two objects ^ and (J>q of TN/M . 4.3.1 Diagram P n - 1D(*) — > K(irnD(<j>),n+l) That we can consider side 11 to be (homotopy) commutative is a consequence of the induction hypothesis. (Definition 4.1.7). Sides W and E are homotopy commutative, from the properties of Postnikov towers. (Lemma 4.1.5 a)). That we can consider side S to be commutative i s the result of 4.2.2 . 37. To decide when the cover C is homotopy commutative, i s to decide when ty and <j> have the same (n+l)-type. (Take fibres "to the north" and compare Definition 4.1.7). The direction of the proof should now be evident. 4.3.2 Lemma: Under the assumption that H,E,U,S are commutative, the torsion elements of H n + 1 ( P ,D(<j> ) , TT D(rf> )) characterize commuta-n-1. r o n Yo ' ti v i t y of C . N Pf. For any ty , an object of T /li for which there exists a Diagram 4.3.1, let |(<J») • OL.,0 (k 1 D d ))«a~ 1 - k ,D(h , where each of the V n-l T n-l To terms i s considered as an element of H n +^(P .D A , ir DA ) . If n-l r o n T o t((j)) = t(<(,') , then k Jjty Pn-1 ( D ( < { , ) ) > K(TrnD(*),n+l) \L a ' " 1 * a d a f " 1 0 o„ i s commutative. k V V W 1F P n - l D ( * f ) ~ S Z : ^ > K < % D ^ ' ) ' n + 1 > To see that t(<j,) i s a torsion element, note that 4 ^ i s a rational equivalence. From the commutativity of N,E,W,S, t(*)o<l>o# = 0 ' "f 4.3.3 THEOREM ("Whitehead Theorem within N") : There are a f i n i t e number N of homotopy types of objects in T /M 38. * Pf. Lerrnna 4.3.2 shows that II D f i n i t e l y determines n-type for a l l n . Apply Lemma 4.1.2 to show that H. D f i n i t e l y determines homotopy-(equivalence) type. Then note there are only a fi n i t e ft i N number of isomorphism classes of {H B(<j))|<j> e objT III } . H 4.3.4 COROLLARY Let A be a finitely-generated, associative, graded-commutative simply-connected algebra such that A X Q 'W — ~ i-1 , \ (x. ) . * Then {Y|H Y - A,Y has the homotopy type of a simply connected f i n i t e CW complex) resolves into only a f i n i t e set of homotopy types. 4.3.5 TREAT Now define two fi n i t e CW complexes X and Y to be within N i f there exists a CW complex Z such that there exist maps f f : Z -*• X within N and g ; Z •*• Y within N . Dualize the argu-ments of Chapter Three and Pour to show that there are a f i n i t e number of homotopy types of TV.: . Choose an object X of a N rational homotopy1 type (5.1.1).. Define \T to be the class of a l l spaces within N of X . Sho-J there are only a f i n i t e number of homotopy types of wj* . Finally, lim W^  » W* = the rational homotopy type of X For rather nice spaces, Theorem 2.6.1 shows there is an easily ft calculable invariant, namely H Y which guarantees that a l l such N spaces are in some W f i l t r a t i o n of the rational homotopy type. Can the Reader think of a larger class of spaces for which the same result i s true? 39. CHAPTER THE FIFTH Showing what kind of History this is; what it is like and what it is not like. Lastly a heartfelt farewell to the Reader. 40. 5.0.0 DISCUSSION 5.1.0 THE HISTORICAL SETTING 5.1.1) In 1969 [Quillen] introduced the concept of rational homotopy theory With apologies to that author^ a rational homotopy type may be considered a subcategory of T^ , where the morphisms are 0-equivalences. ( i . e . within N for some N ) . Moreover, for any two objects X,Y there exists another object Z and morphisms Z -> X and Z -»- Y . 5.1.2) In 1970 [Mimura and Toda] announced that for rather nice spaces, the rational cohomology functor uniquely determines rational hcmotopy type. In particular , given a rat ional , rather nice algebra A , one can construct a space Z such that 1) ll'Vz ;0)"lT A and o o 2) for any other space X with II%X:Q) - A there exists a 0-equivalence X ->• Z 5.1.3) In the same year (Curjel and Douglas] succeeded i n showing that there are a f i n i t e number of bomo-topT- l y p c o jimonoioo y «<4*i-str support an H-space multiplicat ion. In other words, the dimension f i n i t e l y determines the underlying homotopy type. 5.1.4) Then i n August of 1971 [Curjel and Douglas 2] asked: Let A be a candidate for H (X-Z) i.e. an associative, graded simply connected algebra over the integers Z . Let T(A) be the set of homotopy types of simply connected finite complexes having integral cohomology ring isomorphic to A . Question For which A is T(A) a finite set? 41. [Curjel and Douglas] then announced a proof of a positive result when A^pQ i s an exterior algebra on odd dimensional genera-tors. In some Sense, the result i s the easy case of Theorem 4.3.3; m i the obstructions to getting a map from Y to XS are torsion i due to the work of [Serre]. lloreover a l l k-invariants must be torsion. Thus this thesis i s a generalization of the announced result of [Curjel and Douglas 2], and in the same s p i r i t . 5.2.0 COGHOMINA AND METHODS 5.2.1 [Larry Smith] discusses "nice" and "super-nice" algebras i n a paper which introduced the author to the conept of a space over a space. Since the algebras we are concerned with in this thesis are a l l nice, though somewhat less free than super-nice algebras, they merit the appellation "rather nice' ' . Horeover this thesis demonstrates that they permit a rather nice dist inction of homotopy types. Some of the methods of demonstration have been commonplace and sometimes perhaps elephantine. The Serre spectral sequence, Postnikov Towers and the higher connective coverings are heavyduty methods of producing results ; the author is certain the Reader i s well aware of their manifest and wide-ranging appl icabi l i ty . Three novel concepts are central to this thesis and deserve comment. 4 2 . 5 . 2 . 2 ) Categories over a base. The question of when two spaces are homotopy equivalent has long puzzled topologists. Yet the problem we consider i s apparently much more d i f f i c u l t : When are two spaces over a model space homotopy equivalent? i . e . when does there exist a homotopy commutative triangle X —=—> v M That this apparently more d i f f i c u l t problem sometimes admits a solution rests perhaps on the fact the category of abelian groups over a group, has more structure than the category of abelian groups considered i n i so la t ion . Iience when a c lass i f ica t ion i n Ab/C is possible, there may be more chance that i t w i l l carry over to the topological s i tuation. 5 . 2 . 3 ) Maps v i t h i n H Consider a (Serre) congruence class of groups "modulo f i n i t e groups". There i s a free group i n the class . Cal l H i t C . Then Ab /C may be considered to be a f i l t r a t i o n of the 00 congruence class A b / C . Moreover there are a f i n i t e number of isomorphism classes i n each A b / C 43. The result of this thesis i s that we can perform a similar procedure on rather nice rational homotopy types with the added feature that such a. f i l t r a t i o n i s compatible with integral cohomology. When this happens we may conclude as in Theorem 4.3.3 that HT(A) is a f i n i t e set. 5.2.4) Category theory seems sadly lacking i n language to describe how well a functor functions. Think of a homotopy functor as drawing a likeness of a space. Can we recognize the or iginal from i t s likeness? If we can, le t us say that the functor uniquely determines the (equivalence class of the) space. It i s with this i n mind that the author introduces the notion of " f i n i t e l y deter- mines". Hie name however seems fraught with unwonted ambiguities, and the Author w i l l welcome emendations. 44. 5.3.0 APPLICATIONS AND EXAMPLES 5.3.1 By an early theorem of Kopf, f i n i t e dimensional rational Hopf algebras are mult iplicat ively exterior algebras on odd dimensional spheres genera-tors. Thus Lie groups, topological groups„ I-I-spaces and H-spaces mod F are a l l rather nice . 5.3.2 From the work of [Borel] and [Cartan], real and complex St iefe l manifolds also have rational cohomology algebras which are exterior algebras. In general, homogeneous spaces are often rather nice . See [Atiyah-Hirzebruch] and [Baum] for some interesting examples. 5.3.3 By the Kuenneth Theorem, a product of spaces is rather nice i f f each factor i s . More generally 5.3.4 Lemma If X,Z e J a n d X is a retract of Z a rather nice space, then X is also rather nice . Sketch of proof: Let q : A—>>QA be the quotient map onto the - ~2 (Q-vector space) of indecoirrposables. i . e . OA - A/A . An endomorphism a of A is an automorphism i f f the induced map a : OA. QA is also. Characterize rational rather nice algebras as follows: *) For any set of elements x ^ j X ^ , , sx^ z A such that q(x^), q ( x 2 ) » , qCx^) k m. are l inearly independent, then 7T x. 1 4 0 s where m. i s the largest i - 1 1 A m. integer such that x^ 1 4 0 *) i s equivalent tc the statement that A i s rather nice . If X i s a retract of Z Q(H*X & Q) c > 0(H*Z $ Q) and tl"x ® Q > H Z Q are both monomorphisms. Apply *) to show that H X (85 0 i s rather nice . 11 45. CITATION OF RELEVANT LITERATURE [Atiyah-Hirzebruch] "Vector Bundles and Homogeneous Spaces", Proc. Sympos. Pur,;Math. V o l . I l l (Providence R . I . A.M.S. 1961), pp. 7-38. [Baum], Paul "On the Cohomology of HoBiogeneous Spaces" (thesis) , Princeton University 1961. [Berstein], Israel Commentarii Hathematici Helvet i c i , V. 35:1 (1961), pp. 9-14. [Cartan], E l i e "Notions d'algebre d i f f e r e n t i e l l e ; applications aux varietes ou ope*re un groupe de Lie t : ,Collogue de Topologie  (Espaces fibres} (Liege: Centre Beige de Recherches Mathema-tiques 1950) , p. 25. [Curjel] , Caspar "On the H-space structures of f i n i t e complexes", Commentarii Mathematici Helvet i c i , 43:1 (1969), pp. 1-17. [Curjel-Douglas], "On H-spaces of f i n i t e domension",Topology 10 (1971), pp. 385-389. [Curjel-Douglas 2] 'On Stasheff's Fif th Problem", Notices of the A.M.S. 18:5, (August, 1971), p. 637. [Hilton Jf] Peter "On the homotopy groups of the union of spheres", Journal of the London Mathematical Society, 30 (1955), p. 154-172. [Hu], Sze-Tsen Homotopy Theory, (New York: Academic Press 1959), i n particular Chapter V ; Obstruction Theory, and Exercise E . [James], loan "Reduced Product Spaces", Annals of Mathematics 62 (1955), pp. 170-192. [James 2], "Over Homotopy Type", Symposia Mathematica, Vol . IV, Rome: INDAM 1963/69. (London: Academic Press 1970), pp. 219-229. 46. [Kahn], Donald W. "Induced Maps for Postnikov Systems", Transactions of  the A.M.S. 197 (1963), pp. 432-450. [Meyer], J.P. "Principal Fibrations". Transactions of the A.M.S. 107, (1963), pp. 177-135. [Mimura-Toda] Mamoru Mimura and Uirosi Toda "On p-equivalences and p-universal spaces", Commentarii Mathematici Helvetici, 46:1 (1971), pp. 87-97. [Mimura-O'Neill-Toda] "On p-cquivalence in the sense of Serre", Japanese Journal of Mathematics, V. XL (1971), pp. 1-10. [Mosher-Tangora] Robert Mosher and Martin Tangora, Cohomology Operations and Applications in Homotopy Theory, (Net? York: Harper & Row, 1968). [Quillen] Dan, 'Rational Homotopy Type", Annals of Mathematics 90, (Sept., 1969), p. 205. [Serre] Jean-Paul "Groupes d'homotopie et classes de groupes abeliens", Annals of Mathematics 58 (1953), pp. 258-294, in particular Chapitre V, p. 288, "Applications d'un polyhedre dans une sphere'.'. Also p. 259, " I . La notion de classe". [Serre2] "Homologie Singuliere des espaces fibres", Annals of Mathematics 54 (1951), pp. 425-505. [Smith], Larry "On the Eilenberg-Moore Spectral Sequence", A.M.S. Lecture Notes, Summer Institute on Algebraic Topology 1970, Lecture #12. [Toda], Hirosi Composition Methods in Homotopy Groups of Spheres, (Princeton, J.J.: Princeton University Press 1962), Chapter II, i n particular Theorem 2,4, p. 19. [Whitehead] J.H.C. "Combinatorial Homotopy I-II", Bulletin of the A.M.S. 55 (1949), pp. 213-245 and pp. 453-496. [Hilton and Wylie] Homology Theory, An Introduction to Algebraic Topology (Cambridge: 1960), p. 151, Eg. 4.3.4. ' 

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