UBC Theses and Dissertations

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UBC Theses and Dissertations

Formality and finite ambiguity Verster, Jan Frans 1982

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FORMALITY AND FINITE AMBIGUITY by JAN FRANS VERSTER M.Sc, U n i v e r s i t y Of B r i t i s h Columbia , 1976 B.Math. U n i v e r s i t y Of Waterloo, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department Of Mathematics We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA September 1982 © Jan Frans V e r s t e r , 1982 In presenting t h i s thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his or her representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain' s h a l l not be allowed without my written permission. Department of Mathematics The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: October 19, 1982 i i A b s t r a c t S u p e r v i s o r : Dr. Roy R. Douglas The i n t e g r a l cohomology a l g e b r a f u n c t o r , H*( ;Z), was developed as an a i d i n d i s t i n g u i s h i n g homotopy types. We c o n s i d e r the problem of when there are only a f i n i t e number of homotopy equivalence c l a s s e s i n the c o l l e c t i o n of simply connected, f i n i t e CW complexes, f o r which the cohomology a l g e b r a i s isomorphic to a given a l g e b r a . We prove t h a t , i f we r e s t r i c t o u r s e l v e s to formal homotopy types, the set of such homotopy types i s always f i n i t e . T h i s i s shown by using the concept of d i s t a n c e between homotopy types. We b u i l d model spaces so that the d i s t a n c e from a CW complex, whose cohomology i s isomorphic to the given a l g e b r a , to one of the model spaces can be bounded. General r e s u l t s about d i s t a n c e then imply that the set of homotopy types i s f i n i t e . F o r m a l i t y i s a p r o p e r t y of r a t i o n a l homotopy type and we use i n f o r m a t i o n obtained from c a l c u l a t i o n s with the minimal models of S u l l i v a n as a guide i n the c o n s t r u c t i o n of spaces and maps. As a p a r t i a l converse, we show t h a t , f o r every nonformal space, X, there i s a homology s e c t i o n , X', of X such that there are an i n f i n i t e number of d i f f e r e n t homotopy types with cohomology a l g e b r a s isomorphic to H*(X';Z) and with the same r a t i o n a l homotopy type as X'. The dual problem, i n the sense of Eckmann-Hilton, i s a l s o shown to have s i m i l a r answers. For the dual problem one would r e p l a c e "cohomology a l g e b r a " with "Samuelson a l g e b r a " , "formal" with "coformal", and "homology s e c t i o n " with "Postnikov s e c t i o n " . The r e s u l t a p p l i e s to many n a t u r a l l y o c c u r i n g spaces, such as t o p o l o g i c a l groups, H-spaces, complex and q u a t e r n i o n i c p r o j e c t i v e spaces, Kahler manifolds and MoiSezon spaces. i v Table of Contents A b s t r a c t i i L i s t of Tables v Acknowledgement v i 1 I n t r o d u c t i o n 1 2 R a t i o n a l Homotopy Theory and Minimal Models 8 3 F o r m a l i t y and C o f o r m a l i t y 24 3.1 F o r m a l i t y 24 3.2 C o f o r m a l i t y 30 3.3 Examples and P r o p e r t i e s 37 4 D i s t a n c e Between Homotopy Types 45 5 F o r m a l i t y and F i n i t e Ambiguity 53 5.1 P r e l i m i n a r i e s 53 5.2 C o n s t r u c t i o n of Model Spaces 55 5.3 Main Theorem 61 6 I n f i n i t e Ambiguity 70 7 Co n c l u s i o n 82 BIBLIOGRAPHY 84 L i s t of Tables Eckmann-Hilton D u a l i t y Acknowledgement I would l i k e to thank Dr. Roy Douglas f o r h i s h e l p and encouragement i n the w r i t i n g of t h i s t h e s i s . I would a l s o l i k e to thank the N a t i o n a l Research C o u n c i l of Canada and the Isaac Walton K i l l a m Memorial S c h o l a r s h i p s f o r t h e i r generous f i n a n c i a l a s s i s t a n c e . 1 Chapter 1 INTRODUCTION The i n t e g r a l cohomology fu n c t o r was developed with the hope that i t would h e l p d i s t i n g u i s h homotopy types. For us, H*( ;Z) w i l l be a f u n c t o r from CW, the category whose o b j e c t s are p o i n t e d t o p o l o g i c a l spaces having the homotopy type of a simply connected, f i n i t e type CW complex, and whose morphisms are homotopy c l a s s e s of maps, to the category of a s s o c i a t i v e , graded commutative Z-algebras, which a s s i g n s to each space i t s s i n g u l a r cohomology a l g e b r a . One q u e s t i o n which a r i s e s immediately i s how w e l l t h i s f u n c t o r d i s t i n g u i s h e s homotopy types. In some cases i t works very w e l l . Example 1.1 A Moore space, K'(G,n), i s a simply connected space whose only n o n t r i v i a l , reduced homology group c o n s i s t s of the a b e l i a n group G i n p o s i t i v e dimension n. One way to r e a l i z e such a space i s to give a f r e e f i n i t e p r e s e n t i o n of G, R > F > G, and l e t K'(G,n) be the c o f i b r e of e K > L > K' (G, n ) where K and L are H*(L;Z) = F, and R > F. wedges of n-spheres with H*(K;Z) = R and e induces, on reduced homology, the map 2 Lemma 1.2 If X i s a simply connected space such that H*(X;Z) = H*(K'(G,n);Z), then X i s homotopic to K'(G,n). Proof: By the U n i v e r s a l C o e f f i c i e n t theorem, the homology of X i s isomorphic to the homology of K'(G,n). By the Hurewicz theorem, there i s a map f:L > X which induces the map F > G on reduced homology. Then fe i s homot o p i c a l l y t r i v i a l , and so there i s an extension g:K'(G,n) > X which i s a homotopy equivalence by the Whitehead theorem. For such examples we say that the s i n g u l a r cohomology f u n c t o r , H*( ;Z), uniquely determines homotopy type. T h i s i s not t r u e in g e n e r a l . Example 1.3 Let X be the c o f i b r e of the Hopf map n:S" > S 3. Lemma 1.4 i ) X i s not homotopic to S 3 v S 5 , but H*(X;Z) = H * ( S 3 v S 5 ; Z ) . i i ) Any other space with cohomology a l g e b r a H*(X;Z) i s homotopic to e i t h e r X or S 3 v S 5 . Proof: i ) The cohomology a l g e b r a s are isomorphic s i n c e m u l t i p l i c a t i o n by anything but the i d e n t i t y i s t r i v i a l f o r grading reasons. They are not homotopic s i n c e tr a(X) = 0 and i r„(S 3vS 5) = Z/2Z. i i ) Suppose that Y i s such that H*(Y;Z) =H*(X;Z). Let f : S 3 > Y generate i r 3(Y). The c o f i b r e of f i s homotopic to 3 S 5 . If we make a homology decomposition of f, then we o b t a i n a map g : S , ? A j e 5 > Y where the 5 c e l l i s attached by an element of t r < , ( S 3 ) . g i s a homotopy equivalence s i n c e the homology of Y i s t r i v i a l i n dimensions g r e a t e r than 5. Since »r a(S 3) = Z/2Z, S 3 u e 5 must be e i t h e r X or S 3 v S 5 . In t h i s example, although cohomology d i d not uniquely determine homotopy type, there were only a f i n i t e number of homotopy types with the same cohomology a l g e b r a . In such cases we s h a l l say that cohomology i s f i n i t e l y ambiguous. Example 1.5 For n > 0, l e t W(n) = ( S 3 v S 3 ) w e 8 , where the 8 c e l l i s a t t a c h e d by n [ i , , [ i , , i 2 ] ], ( i i and i 2 are the i n c l u s i o n s S 3 > S 3 v S 3 ) . A l l these spaces have isomorphic cohomology a l g e b r a s s i n c e the g r a d a t i o n f o r c e s a l l m u l t i p l i c a t i o n except by the i d e n t i t y to be t r i v i a l . [ i 1 , [ i 1 , i 2 ] ] i s of i n f i n i t e order i n r r 7 ( S 3 v S 3 ) and rr 7(W(n)) = i r 7 ( S 3 v S 3 )/<n[ i , , [ i , , i 2 ] ]>, where <n[i , , [ i , , i 2 ] ]> i s the subgroup generated by n f i , , [ i , , i 2 ] ]. T h e r e f o r e , i r 7(W(n)) i s not isomorphic to ir 7(W(m)) when n * m, and W(n) i s not homotopic to W(m). {W(n)} i s an i n f i n i t e f a m i l y of spaces with d i f f e r e n t homotopy types, but isomorphic cohomology a l g e b r a s . In t h i s case, H*( ;Z) i s not f i n i t e l y ambiguous. The above examples show that the a b i l i t y of cohomology to d i s t i n g u i s h homotopy types v a r i e s c o n s i d e r a b l y . Problem 5 of [Sta] asks the f o l l o w i n g q u e s t i o n . Problem: If A i s an a s s o c i a t i v e , graded, simply connected 4 Z-algebra, and T*(A) i s the set of homotopy types in CW with cohomology a l g e b r a A, when i s T*(A) f i n i t e ? C u r j e l and Douglas in [C&D] announced that i t i s f i n i t e i f AH© i s an e x t e r i o r a l g e b r a on odd dimensional g e n e r a t o r s . Body and Douglas in [B&D] showed that T*(A) i s f i n i t e i f A i s " r a t h e r n i c e " , and Body in [Bod] then showed that i t i s f i n i t e , i f A i s r e g u l a r . In t h i s t h e s i s , the above r e s u l t s are extended. The main theorem i s Theorem 5.8 Let B be a graded, simply connected, f i n i t e l y generated, f i n i t e dimensional Z-algebra. Then the set FT*(B) = {[X]|Xeobj(CW), H*(X;Z) = B, X i s formal} i s f i n i t e . ([X] denotes the homotopy type of X). F o r m a l i t y i s a p r o p e r t y of r a t i o n a l homotopy type and i s d i s c u s s e d i n chapter 3. T h i s theorem a p p l i e s to many n a t u r a l l y o c c u r i n g spaces. Some examples are spheres, complex and q u a t e r n i o n i c p r o j e c t i v e spaces, L i e groups, H-spaces, symmetric spaces, Kahler manifolds, MoiSezon spaces, and simply connected, compact manifolds with dimension l e s s than or equal to 6. Chapter 3 c o n t a i n s more examples and some g e n e r a l c o n d i t i o n s which imply that a space i s formal. Theorem 5.8 g i v e s the f o l l o w i n g p a r t i a l answer to the q u e s t i o n posed above. If B i s i n t r i n s i c a l l y formal ( d e f i n i t i o n 5 3.12), then T*(B) = FT*(B) i s f i n i t e . A l l of the examples above, with T*(B) f i n i t e , have B i n t r i n s i c a l l y formal. If B i s not i n t r i n s i c a l l y formal, the problem becomes more d i f f i c u l t . In chapter 6 the f o l l o w i n g p a r t i a l converse to the above theorem i s proved. P r o p o s i t i o n 6.1 Suppose that X i s a simply connected, non-formal, f i n i t e CW complex f o r which H*(X;Q) = 0, u n l e s s * = 0 o r r < * < m , and whose m-r-1 homology s e c t i o n i s formal. Then there are an i n f i n i t e number of d i f f e r e n t homotopy types with cohomology a l g e b r a s isomorphic to that of X, and which have the same r a t i o n a l homotopy type as X. In p a r t i c u l a r , any non-formal space has a homology s e c t i o n f o r which H*( ; Z) i s not f i n i t e l y ambiguous. T h i s p r o p o s i t i o n shows that T*(B) w i l l be i n f i n i t e i f i t c o n t a i n s one homotopy type which s a t i s f i e s the above c o n d i t i o n s . F i n d i n g such a space i s a d i f f i c u l t problem s i n c e i t i s s t i l l an open problem whether or not T*(B) i s nonempty. To prove theorem 5.8, we use the concept of d i s t a n c e between homotopy types ( d e f i n i t i o n 4.3) which was developed by Body and Douglas in [B&D]. A f i n i t e d i s t a n c e between homotopy types i s weaker than homotopy equ i v a l e n c e , but stronger than r a t i o n a l homotopy e q u i v a l e n c e . The main r e s u l t about d i s t a n c e i s that there are only a f i n i t e number of homotopy types w i t h i n a given d i s t a n c e N of a f i x e d space ( c o r o l l a r y 4.7). Since f o r m a l i t y i s a property of r a t i o n a l homotopy type, we can use 6 S u l l i v a n ' s theory of minimal models as an a i d i n c a l c u l a t i o n s . The r e s u l t s t h a t we use are o u t l i n e d i n chapter 2. There i s a dual v e r s i o n of the above problem, i n the sense of Eckmann-Hilton d u a l i t y [ H i l ] . Under t h i s d u a l i t y , the cohomology al g e b r a of X, H*(X;Z), corresponds to the Samuelson a l g e b r a of X, ir.(nx). Problem: Suppose n i s a connected, graded L i e a l g e b r a over Z. Under what c o n d i t i o n s are there only a f i n i t e number of homotopy types with Samuelson a l g e b r a IT? A l l of the r e s u l t s i n t h i s t h e s i s have dual v e r s i o n s . Using the dual notion to f o r m a l i t y ( i . e . c o f o r m a l i t y ) , we get a p o s i t i v e answer to the above q u e s t i o n , i f we r e s t r i c t o u r s e l v e s to coformal r a t i o n a l homotopy types. Eckmann-Hilton d u a l i t y i s not p r e c i s e and one cannot always d u a l i z e a proof to get a proof of the dual r e s u l t . There i s no dual v e r s i o n of the Serre s p e c t r a l sequence, and the dual v e r s i o n of any theorem using i t needs a new p r o o f . An example of t h i s occurs i n lemma 4.6. F o r t u n a t e l y , most p r o o f s d u a l i z e e a s i l y . T h i s is- t r u e f o r any a l g e b r a i c proof which doesn't i n v o l v e u n i t s or a s s o c i a t i v i t y . In t h i s t h e s i s , i f the dual proof i s s t r a i g h t f o r w a r d , i t w i l l e i t h e r be omitted, or j u s t an o u t l i n e w i l l be g i v e n . The f o l l o w i n g t a b l e g i v e s the d u a l v e r s i o n of some of the p r o p e r t i e s d i s c u s s e d i n t h i s t h e s i s . Table I - Eckmann-Hilton D u a l i t y Property Dual Property Cohomology al g e b r a Samuelson a l g e b r a Minimal a l g e b r a model Minimal L i e a l g e b r a model F o r m a l i t y C o f o r m a l i t y P o s i t i v e Weight P o s i t i v e Weight D i f f e r e n t i a l has degree +1 D i f f e r e n t i a l has degree -1 Postnikov tower CW homology s e c t i o n s 8 Chapter 2 RATIONAL HOMOTOPY THEORY AND MINIMAL MODELS The purpose of t h i s chapter i s to summarize the r a t i o n a l homotopy theory necessary to prove the r e s u l t s in t h i s t h e s i s . These ideas may be found i n [ S u l ] , [ Q u i ] , [ N e i ] , [B&L] and i n other papers. By the work of Q u i l l e n [ Q u i ] , there are s e v e r a l e q u i v a l e n t viewpoints from which one can study the r a t i o n a l homotopy type of a simply connected CW complex of f i n i t e type. One way i s to use d i f f e r e n t i a l graded algebras which look p r i m a r i l y at cohomology, and another way i s to use d i f f e r e n t i a l graded L i e al g e b r a s which look p r i m a r i l y at homotopy. Def i n i t i on 2.1; A graded a l g e b r a (g . a . ) , A, i s a r a t i o n a l v e c t o r space with an a s s o c i a t i v e m u l t i p l i c a t i o n M:AHA > A (write xy for » i (x8y) ) and a d i r e c t sum decomposition (a grading) n A = © A neZ n m n+m such that i ) i>(A 8 A ) c A and |x||y| n i i ) xy = (-1) yx where |x| = n, i f xeA . Denote by A + the c o k e r n e l of the i n c l u s i o n of the u n i t , <Q > A. Denote by A(Z) the f r e e g.a. on the graded v e c t o r space Z, and denote by A(x,y,...) the f r e e g.a. on the graded set 9 {x, y, ...}. (Of course, they c o i n c i d e i f the set i s a b a s i s of Z.) In g e n e r a l a f r e e g.a. w i l l be the tensor product of a polynomial a l g e b r a on the even dimensional generators and an e x t e r i o r a l g e b r a on the odd dimensional g e n e r a t o r s . D e f i n i t i o n 2.2: A d i f f e r e n t i a l graded a l g e b r a (d.g.a.) i s a graded a l g e b r a together with a degree 1 l i n e a r t r a n s f o r m a t i o n d:A > A ( c a l l e d a d i f f e r e n t i a l ) such that d 2 = 0 and d(xy) = (d( x ) ) y + (-1) xd(y) When the d i f f e r e n t i a l must be s p e c i f i e d , w r i t e (A,d). D e f i n i t i o n 2.3: A graded coalgebra ( g . c ) , C, i s a r a t i o n a l v e c t o r space with a c o m u l t i p l i c a t i o n A:C > CBC which i s c o a s s o c i a t i v e and cocommutative with a d i r e c t sum decomposition n C = ffi C neZ n p q such that A(C ) £ © C 8C . Denote by C + the k e r n e l of the p+q=n c o u n i t , C > (Q. A d i f f e r e n t i a l graded c o a l g e b r a ( d . g . c ) , C, i s a graded c o a l g e b r a together with a degree -1 map d:C > C (a d i f f e r e n t i a l ) which s a t i s i f i e s d 2 = 0 and, f o r ctC with A(c) = ExBy, 10 A(d(c)) = l(dx8y+(-1)' 'xBdy). D e f i n i t i o n 2.4: A graded L i e a l g e b r a ( g . l . a . ) , L, i s a r a t i o n a l v e c t o r space with a m u l t i p l i c a t i o n [,]:L8L > L and a d i r e c t sum decomposition (a grading) n L = © L ncZ n m n+m such that i ) [L ,L ] £ L MM i i ) [x,y] = -(-1) [y,x] |x||z| |y||x| i i i ) (-D [ x , [ y , z ] ] + (-1) [ y , [ z , x ] ] + I z | | y | (-1 ) [ z , [ x , y ] ] = 0 An a l t e r n a t e v e r s i o n of i i i ) i s l * l | y | i i i ' ) [ x , [ y , z ] ] = [ [ x , y ] , z ] + (-1) [ y , [ x , z ] ] Denote by L(Z) the f r e e L i e a l g e b r a on the graded v e c t o r space Z and denote by L(x,y,...) the f r e e L i e a l g e b r a on the graded set {x, y, ...}. (Again, these c o i n c i d e i f {x, y, ...} i s a b a s i s of Z.) If L and K are g . l . a . ' s , LvK i s d e f i n e d to be the graded L i e a l g e b r a generated f r e e l y by L and K. 11 D e f i n i t i o n 2.5: A d i f f e r e n t i a l graded L i e a l g e b r a (d.g.l.a.) i s a graded L i e algebra together with a degree -1 map d:L > L (a d i f f e r e n t i a l ) which s a t i s f i e s d 2 = 0 and M d[x,y] = [dx,y] + (-1) [x,dy] D e f i n i t i o n 2.6: For any d.g.a. ( d . g . c , d.g.l.a.) A, the homology of A i s d e f i n e d to be the graded a l g e b r a (g.c., g.l.a.) H*(A) = ker d/Im.d.. A d.g.a. ( d . g . c , d.g.l.a.) map f:A > B i s d e f i n e d to be a weak equivalence, i f the induced map on homology, f*:H*(A) > H*(B), i s an isomorphism. D e f i n i t i o n 2.7: a) Let dga be the category of connected (H°(A) = Q), simply connected (H 1(A) = 0) d i f f e r e n t i a l graded a l g e b r a s of f i n i t e type. b) For any g.a., A, we have the c a t e g o r i c a l l y dual graded coalgebra ( g . c ) , Hom(A,Q), which i s a d i f f e r e n t i a l graded coalgebra (d.g.c.) when A i s a d.g.a. Let dgc be the category whose o b j e c t s and morphisms are dual to those of dga. c) Let dgla be the category of connected (H°(L) = 0) d i f f e r e n t i a l graded L i e al g e b r a s of f i n i t e type. The c a t e g o r i e s dga, dgc, and dgla are Cl o s e d Model (CM) c a t e g o r i e s (see [ Q u i ] ) . A c y l i n d e r o b j e c t , f o r an o b j e c t , A, i n a c l o s e d model category, i s an o b j e c t A x l , together with a weak equivalence *:AxI > A, and two maps 6 0,6,:A > Axl such that tf60 and are the i d e n t i t y on A, and <60,6 y > : h v h -—> Axl 1 2 i s a c o f i b r a t i o n (AvA i s the c a t e g o r i c a l coproduct of A with i t s e l f ) . I A path o b j e c t f o r A i s an o b j e c t A , together with a weak I I equivalence *:A > A and two maps 6 0 , 6 i i A > A such that I 60c and 6,c are the i d e n t i t y on A, and { 6 0 » 6 i } : A > AxA i s a f i b r a t i o n (AxA i s the c a t e g o r i c a l product of A with i t s e l f ) . D e f i n i t i o n 2.8: a) Two maps f,g:A > B i n a c l o s e d model category are d e f i n e d to be l e f t homotopic i f and only i f there i s a c y l i n d e r o b j e c t Axl f o r A and a map h:AxI > B such that h6 0 = f and h6, = g. b) Two maps f,g:A > B i n a c l o s e d model category are d e f i n e d to be r i g h t homotopic i f and only i f there i s a path I I ob j e c t B f o r B and a map h:A > B such that 6 0h = f and 6 , h = g. Q u i l l e n proves that l e f t homotopy i s an equivalence r e l a t i o n i f A i s c o f i b r a n t (lemma 4, s e c t i o n 1 of [Qu2]). T h e r e f o r e , i n t h i s t h e s i s , homotopy w i l l mean l e f t homotopy. In any case, both n o t i o n s c o i n c i d e when A i s c o f i b r a n t and B i s f i b r a n t (lemma 5 ( i ) and i t s d u a l , s e c t i o n 1 of [Qu2]). Lemma 2.9 a) I f A = (A(Z),d) i s a f r e e d.g.a., a c y l i n d e r o b j e c t f o r A i s given by Axl = (A(Z'®Z"@sZ),D), where Z' and Z" are isomorphic c o p i e s of Z and sZ i s the suspension of Z, 1 3 i . e . obtained from Z by r a i s i n g degrees by one. Define the map S:A > Axl by i ) Sz = sz, f o r zeZ, |x| i i ) S(xy) = (Sx)y" + (-1) x'(S y ) , f o r x,y €A. Then the d i f f e r e n t i a l D i s d e f i n e d to agree with d on Z' and Z" and, f o r szcsZ, D(sz) = z"-z'-Sdz. The maps 6 0 and 6 , are d e f i n e d by i n c l u d i n g Z as Z' and Z" r e s p e c t i v e l y . The map <r:AxI > A i s d e f i n e d by tf(z') = z, <r:(z") = z, and <r(sz) = 0. b) If L = (L(Z),d) i s a f r e e d . g . l . a . , a c y l i n d e r o b j e c t f o r L i s given by L x l = (L( Z'©Z"©s" 1Z),D), where Z' and Z" are isomorphic c o p i e s of Z, and s" 1Z i s the desuspension of Z, i . e . obtained from Z by lowering degrees by 1. If the map S:L > L x l i s d e f i n e d by i ) Sz = s _ 1 z , f o r a l l zeZ i i ) S[x,y] = [Sx,y"] + (-1) [x',Sy ] , f o r x,y 6L, then D i s d e f i n e d to agree with d on Z' and Z", and D ( s " 1 z ) = z"-z'-Sdz f o r s ^ z t s - ' Z . Remark: The above c o n s t r u c t i o n s are from [B&L]. Let xi (dga) and TT (dgla) be the c a t e g o r i e s which have as o b j e c t s the f i b r a n t and c o f i b r a n t d.g.a.'s (resp. d.g.l.a.'s) and where the morphisms are homotopy c l a s s e s of maps. By Q u i l l e n [ Q u i ] , these c a t e g o r i e s are e q u i v a l e n t to each other and e q u i v a l e n t to the l a r g e r homotopy c a t e g o r i e s Ho(dga) and Ho(d g l a ) . To study these c a t e g o r i e s one uses the concept of minimal model. 1 4 D e f i n i t i o n 2.10: a) A minimal a l g e b r a model, M, i s a fr e e d.g.a. i n dga whose d i f f e r e n t i a l i s decomposable, i . e . d r e s t r i c t e d to Q(M) = M/V(M +HM +) i s zero.(Note that t h i s i m p l i e s that M° s Q and M1 = 0). b) A minimal coalgebra model, C, i s a f r e e d.g.c. i n dgc such that the d i f f e r e n t i a l r e s t r i c t e d to the p r i m i t i v e s of C, P(C) = ker(A:C + > C + B C + ) , i s zero . c) A minimal L i e a l g e b r a model, L, i s a f r e e d . g . l . a . i n dgla such that the d i f f e r e n t i a l i s decomposable, i . e . d r e s t r i c t e d t o Q(L) = L/[L,L] i s z e r o . Minimal algebra and L i e a l g e b r a models are both f i b r a n t and c o f i b r a n t o b j e c t s i n t h e i r r e s p e c t i v e c l o s e d model c a t e g o r i e s , dga and d g l a . For any d.g.a., A, i n dga there i s a minimal a l g e b r a model M such t h a t there i s a weak equivalence p:M > A. M i s r e f e r r e d to as a minimal model f o r A, and i t i s unique up to isomorphism. S i m i l a r r e s u l t s h o l d f o r dgc and d g l a . If N i s another minimal a l g e b r a model, and f:N > A i s a d.g.a. map, t h e r e i s a l i f t i n g #:N > M, unique up to homotopy, such that the f o l l o w i n g diagram homotopy commutes. N > A f T h i s map i s obtained as f o l l o w s . F i r s t by CM axiom 5, f a c t o r p i p as M > M' > A where i i s a c o f i b r a t i o n and a weak equ i v a l e n c e , and p i s a f i b r a t i o n ( i . e . a s u r j e c t i o n ) . Then 1 5 using CM axiom 4 twice we- get maps 0':N > M' and r:M' > M such that p*' = f and r i = 1 . Let <t> = r * ' . To show that the diagram homotopy commutes, we need only f i n d a homotopy from i r to 1. Since M' can be chosen to be M' = MBlA(Z©dZ) where A(Z©dZ) i s a c y c l i c , a homotopy from 1 to 0 on A(Z@dZ) can be extended to the r e q u i r e d homotopy. For any t o p o l o g i c a l space, X, there i s a d.g.a. A*(X), the ( s i n g u l a r ) de Rham complex of X, which i s d e f i n e d to be the de Rham a l g e b r a of the geometric r e a l i z a t i o n of the s i n g u l a r complex of X. The homology of A*(X) i s n a t u r a l l y isomorphic to the r a t i o n a l s i n g u l a r cohomology H*(X,Q>). D e f i n i t i o n 2.11: The minimal model of X i s d e f i n e d to be the minimal model of A*(X). Once minimal models have been chosen, there i s a fun c t o r M:CW > tr (dga) which a s s i g n s to a t o p o l o g i c a l space X i t s minimal model MX (see Chapter XIV of [G&M]). We can use the l i f t i n g p r o p e r t y above to get M(f) f o r f:X > Y. S p e c i f i c a l l y M(f) = [</>] where <t> i s such that the f o l l o w i n g diagram homotopy commutes. MX < MY v v A*(X) < A M Y ) f* There i s a functor £:dgc > dgla d e f i n e d by Q u i l l e n [ Q u i ] , f o r which f ( C ) = L ( s - 1 C + ) , the f r e e L i e a l g e b r a on the 16 desuspended module s ~ 1 C + obtained from C + by s h i f t i n g degrees down by 1. I f s ~ 1 c i s a generator such that MM A(c) = c»1 + 18c + I(x8y + (-1) y » x ) , then the d i f f e r e n t i a l d i s d e f i n e d to be |x| d ( s ' 1 c ) = - s ' 1 d c - E(-1) [ s - 1 x , s " 1 y ] . There i s (up to isomorphism) an unique minimal L i e al g e b r a model L such that there i s a weak equivalence L > £( C ) . I f C i s the minimal c o a l g e b r a model dual t o the minimal a l g e b r a model fo r X, then LX = L i s d e f i n e d to be the minimal L i e a l g e b r a model f o r X. L can be viewed as a f u n c t o r , CW > ir (dgla) , as f o l l o w s . I f f:X > Y i s a map, there i s where M(f) = [*], such that 0 MX < MY v v A*(X) < A*(Y) f * homotopy commutes. Then there i s * such that 17 LX > LY v v f((MX)*) > £((MY)*) homotopy commutes. Define L ( f ) to be [*]. There i s an isomorphism *:H*(LX) > jr*(nx)i85) which i s determined by the map LX > £((MX)*) ( p r o p o s i t i o n 8.3 of [ N e i ] ) . D e f i n i t i o n 2.12: A weight s p l i t t i n g f o r an o b j e c t i n one of the c a t e g o r i e s dga, dgc, or dgla i s d e f i n e d to be an i n t e g e r indexed, d i r e c t sum decomposition of the u n d e r l y i n g 3) v e c t o r space, which i s compatible with the a l g e b r a i c s t r u c t u r e (see [Dou]). For M, an o b j e c t i n dga, t h i s means M = @ nM neZ which s a t i s f i e s k k M = ffi nM (grading) neZ »/(ftM!SkM) < h +k)M ( m u l t i p l i c a t i o n ) d( hM) c hM ( d i f f e r e n t i a l ) The d e f i n i t i o n s f o r dgc and dgla are s i m i l a r . 18 D e f i n i t i o n 2.13: A p o s i t i v e weight s p l i t t i n g f o r M i n one of dga, dgc, or d g l a i s a weight s p l i t t i n g which i s t r i v i a l f o r a l l n o n - p o s i t i v e weights. If M i s a d.g.a., t h i s means (B, n = 0 0, n < 0 If M i s a d . g . l . a . , t h i s means ftM = 0 f o r n < 0. D e f i n i t i o n 2.14: An obj e c t i n CW has p o s i t i v e weight i f any type of minimal model f o r i t has p o s i t i v e weight. Theorem 2 of [Dou] shows that the p o s i t i v e weight concept i s independent of the type of minimal model used. T h e r e f o r e , p o s i t i v e weight i s s e l f d u a l , i n the sense of Eckmann-Hilton. The p r o p e r t y of p o s i t i v e weight i s e q u i v a l e n t to p - U n i v e r s a l i t y [M&T]. Formal and coformal spaces have p o s i t i v e weight by theorem 3.1 g) and theorem 3.3 g ) . The H i r s c h lemma t e l l s how to f i n d the minimal a l g e b r a model of the t o t a l space of a f i b r a t i o n which has f i b r e K(G,n), where G i s a f i n i t e l y generated a b e l i a n group, and a given minimal model f o r the base space. I f these are p r i n c i p a l f i b r a t i o n s , we can use the group a c t i o n to change l i f t i n g s . The f o l l o w i n g lemma g i v e s the corres p o n d i n g changes i n the maps between minimal models. 19 Lemma 2.15 f P a) Suppose K(Z,n) > X > Y i s a p r i n c i p a l f i b r a t i o n , and there i s a map g:W > X which l i f t s h = pg:W > Y. Suppose a l s o that the minimal a l g e b r a models f o r W, Y, and X are M, N, and NSA(w) (by the H i r s c h lemma). Then the set { [ * ' ( w ) - * ( w ) ] e H * ( M ) | U 3 = M(g), [ * ' ] = M ( g ' ) , g ' a l s o l i f t s h} forms an i n t e g r a l l a t t i c e i n H*(M) which i s independent of g. f p k b) Suppose that X > Y > S i s a p r i n c i p a l c o f i b r a t i o n , by which we w i l l mean a c o f i b r a t i o n induced, as a pushout, from the (cone-suspension) c o f i b r a t i o n , k-1 k-1 k-1 k-1 S > CS > IS , and a map e:S > X. Suppose a l s o that there i s a map g:Y > W which extends h = gf:X > W, and that the minimal L i e alg e b r a models f o r W, X, and Y are M, N, and NvL(w) by the dual form of the H i r s c h lemma ( p r o p o s i t i o n 8.11 of [ N e i ] ) . Then the set {[•'(w)-*(w) ] tH*(M)|[•] = L ( g ) , [•'] = L ( g ' ) , g' a l s o extends h} forms an i n t e g r a l l a t t i c e i n H*(M) which i s independent of g. Proof: a) g' can be obtained as $(g,c) i n the diagram 20 0 K ( Z , n ) x K ( Z , n ) > K(Z ,n) f v <t> v XxK(Z,n) > X (9,c) v -> Y Then i s homotopic to the top row of M <- ( N B A ( w ) ) B A ( w ) <-v A*(W) <-NBA(w) A * ( X x K ( Z , n ) ) < A* (X) and t * ' ( w ) ( w ) ] = (rr* )" 1 c * [ f *p (w) ] which i s independent of g and forms a l a t t i c e s ince c*+d* = (c+d)* . b) Dual p r o o f . In [H&S] another type of model i s g i v e n . For a s imply connected d . g . a . , A, a b igraded model (A(Z) ,d ) i s f i r s t c o n s t r u c t e d for H * ( A ) . The top g rada t ion of the b i g r a d i n g n (Z = © Z ) i s the t o p o l o g i c a l degree. The d i f f e r e n t i a l has n , m m degree +1 w i t h respect to the t o p o l o g i c a l degree, and degree -1 w i t h respect to the bottom grada t ion ( c a l l e d the f i l t r a t i o n ) . The model can be given a p o s i t i v e we ight ing by us ing the sum of the t o p o l o g i c a l degree and the f i l t r a t i o n . Th i s e x h i b i t s a homology d i a g o n a l , p o s i t i v e weight s p l i t t i n g , which i m p l i e s that 21 t h i s i s a formal minimal model (see theorem 3.1). To get a model f o r A the d i f f e r e n t i a l d i s "perturbed" to D, where D-d lowers f i l t r a t i o n s by at l e a s t two, i n such a way that there i s a weak equivalence (A(Z),D) > A. T h i s p e r t u r b a t i o n i s unique up to isomorphism, but the models are not n e c e s s a r i l y minimal. T h i s c o n s t r u c t i o n can be d u a l i z e d to pr o v i d e a bigraded model f o r L, an o b j e c t i n d g l a . An o u t l i n e i s given below. Let H(=H*(L)) be a connected g . l . a . We f i r s t c o n s t r u c t the bigraded L i e model f o r H, />:(L(Z),d) > (H,0), where Z = ZoffiZ,©... and Z i s the ve c t o r space spanned by the generators with f i l t r a t i o n i (the lower degree). The d i f f e r e n t i a l w i l l have degree -1 with respect to both the t o p o l o g i c a l degree and the f i l t r a t i o n . The d i f f e r e n c e between the t o p o l o g i c a l degree and the f i l t r a t i o n p r o v i d e s a homology d i a g o n a l p o s i t i v e weight s p l i t t i n g f o r the coformal minimal L i e model ( L ( Z ) , d ) . The bigraded model i s c o n s t r u c t e d i n d u c t i v e l y as f o l l o w s . n = 0 Set Z 0 = H/[H,H] with d = 0. Def i n e ./>:L( Z 0) > H so that i t s p l i t s H > Z 0. Then p i s s u r j e c t i v e with k e r n e l K. n = 1 Set Z, = K / [ K , ( L ( Z 0 ) ) ] , with a s h i f t upward by 1 of both degrees. Extend d to Z, by r e q u i r i n g t h a t i t be a l i n e a r map Z, ——> K s p l i t t i n g the p r o j e c t i o n . Then d:Z, > L ( Z 0 ) , and so d i s homogeneous of lower f i l t r a t i o n -1 i n L ( Z 0 © Z 1 ) . Extend p to be zero on Z,. 22 n = k+-1 I n d u c t i v e l y l e t P p-1 Z = (H ( L ( Z ( n ) ) , d ) / [ H ( L ( Z ( n ) ) , d ) , H ( L ( Z ( n ) ) , d ) ] ) n+1 n n 0 where Z(n) = Z 0@Z 1©...©Z . Extend d so that d:Z > ( L ( Z ( n ) ) ) f\ ker d s p l i t s the p r o j e c t i o n of n+1 n (L ( Z ( n ) ) ) A ker d onto Z . Extend p to be zero on Z n n+1 n+1 T h i s g i v e s a weak equivalence />:(L(Z),d) > (H,0) with Z = Z o©Z1 © . . . . P r o p o s i t i o n 2.16 i ) p*:H0(L(Z),d) > H i s an isomorphism. i i ) H + ( L ( Z ) , d ) = 0 i i i ) />:(L(Z),d) -> (H,0) i s the minimal L i e model. Proof: The proof i s dual to that of theorem 3.4 i n [H&S]. Theorem 2.17 Suppose that A i s a connected d . g . l . a . Let p:(L(Z),d) > (H*(A),0) be the bigraded L i e model f o r H*(A). Then there i s a g . d . l . a . (L(Z),D) and a homomorphism i r:(L(Z),D) > A such that i ) D-d lowers f i l t r a t i o n s by at l e a s t two. i i ) [ TTZ ] = pz , zeL ( Z 0 ) . 23 i i i ) rr* i s an isomorphism. Moreover, suppose n':(L(Z),D') > A a l s o s a t i s f i e s these c o n d i t i o n s . Then there i s an isomorphism * : ( L ( Z ) , D ) > ( L ( Z ) , D ' ) such that i ) 0-1 lowers f i l t r a t i o n s . i i ) n' <f> i s homotopic to i r : ( L ( Z ) , D ) > A. Proof: The proof i s dual to that of theorem 4.4 i n [H&S], 24 Chapter 3 FORMALITY AND COFORMALITY 3. 1 FORMALITY The s i m p l e s t r a t i o n a l homotopy types are those which are formal, indeed, they are uniquely determined by t h e i r r a t i o n a l cohomology a l g e b r a s . F o r m a l i t y has been s t u d i e d by many authors and many n a t u r a l l y a r i s i n g examples are formal. The d u a l concept, i n the sense of Eckmann and H i l t o n , i s c o f o r m a l i t y . Coformal spaces are those whose r a t i o n a l homotopy type i s determined s o l e l y by t h e i r homotopy groups and t h e i r (primary) Whitehead product s t r u c t u r e . T h i s s e c t i o n c o n t a i n s a theorem l i s t i n g many of the c o n d i t i o n s e q u i v a l e n t to f o r m a l i t y which occur i n v a r i o u s p l a c e s i n the l i t e r a t u r e . S e c t i o n 2 d u a l i z e s the theorem to get s i m i l a r c o n d i t i o n s f o r c o f o r m a l i t y . S e c t i o n 3 g i v e s examples and p r o p e r t i e s of formal and coformal spaces. Assume that M .is a minimal d.g.a., C i s the coalgebra dual to M, and L i s the corresponding minimal d . g . l . a . A l s o assume that K i s a simply connected CW-complex whose minimal model i s M. Theorem 3.1 The f o l l o w i n g are e q u i v a l e n t and any one of them can be taken as the d e f i n i t i o n of f o r m a l i t y (b i s the usual def i n i t i o n ) . a) The p r o j e c t i o n of Z(M), the c o c y c l e s of M, onto H*(M) 25 l i f t s to a g.a. map <t> i n the f o l l o w i n g diagram: Z(M) > H*(M) b) M i s the minimal model f o r H*(M), c o n s i d e r e d as a d.g.a. with zero d i f f e r e n t i a l . c) The c a n o n i c a l map Aut(M) > Aut(H*(M)) i s an epimorphism. d) Let ir:$* = ©.-{0} > Aut(H*(M)) be the map given by r r(t) = t*1, the grading homomorphism, which i s d e f i n e d by KI t * l ( x ) = t x, f o r x a homogeneous element of H*(M). Then, in the p u l l b a c k E > Aut(M) v v Q* > Aut(H*(M)) tr there i s a t * ±1 i n the image of e) In d) , l e t IT be given by n(t) = t*a where a i s some f i x e d a n i s o t r o p i c map (see d e f i n i t i o n 3.2). Then there i s a t * ±1 i n the image of *. f) There e x i s t s a homology d i a g o n a l weight decomposition P of M. That i s , H (nM) = 0 when p * n. g) There e x i s t s an upper d i a g o n a l weight decomposition of P P M. That i s nM =0 when p > n and H (^ M) = 0 when p * n. 26 h) Suppose M = A ( Z 2 © Z 3 © . . . ) . Each Z\ , the vec tor space spanned by the generators of dimension i , can be w r i t t e n as a d i r e c t sum Z\ = C\©Nj , where C\ c o n s i s t s of the c o c y c l e s , i n such a way that every c o c y c l e i n the i d e a l generated by ©Ni i s a coboundary. i ) The o b s t r u c t i o n s to f o r m a l i t y de f ined by H a l p e r i n and Stashef f i n [H&S] v a n i s h . j ) I f k i s a f i e l d of c h a r a c t e r i s t i c 0, then M55k i s formal ( i . e . s a t i s f i e s b ) . k) There i s a weak equiva lence H#(K;$) > c. 1) L i s isomorphic to f (H^. (K;Q)) . m) L i s isomorphic to a minimal L i e a lgebra whose d i f f e r e n t i a l has zero p e r t u r b a t i o n . In a minimal model, zero p e r t u r b a t i o n means that there i s a ba s i s for L for which the d i f f e r e n t i a l i s quadra t i c (see [ N e i ] ) . T h i s should not be confused w i t h the concept of p e r t u r b a t i o n i n a b igraded model . For the f o l l o w i n g two c o n d i t i o n s we need to assume that K i s a f i n i t e CW complex. n) I f r i s an i n t e g e r , r * 0 , ± 1 , then there e x i s t s s, a non-zero m u l t i p l e of r , and f : K > K such that f * = S * 1 : H * ( K ; Z ) > H * ( K ; Z ) o) There e x i s t s te@, t * 0 , ± 1 , and a map f : K ( 0 ) > K ( 0 ) such that 27 f* = t * l : H * ( K ( 0 ) ; Z ) > H * ( K ( 0 ) ; Z ) where K ( 0 ) denotes the l o c a l i z a t i o n of K at z e r o . D e f i n i t i o n 3 .2 : I f V i s a f i n i t e d imens iona l Q-vector space, we w i l l c a l l a e Aut(V) a n i s o t r o p i c i f and only i f a l l the Q - i r r e d u c i b l e f a c t o r s of the minimal po lynomia l of a are of the form n-1 n f ( x ) = 1 + b x + . . . + b x ± x , b e Z 1 n-1 i The above d e f i n i t i o n i s equ iva l en t to V having a ba s i s for which the mat r ix r e p r e s e n t a t i o n of a i s i n t e g r a l and has determinant ±1 . T h i s d e f i n i t i o n can be extended to i n f i n i t e d imens iona l vec tor spaces which can be w r i t t e n as a d i r e c t sum of f i n i t e d imens iona l vec tor spaces each of which i s i n v a r i a n t under a . P r o o f : a) => b) * i s a weak equiva lence of n e c e s s i t y . b) => c) Standard l i f t i n g p r o p e r t y . c) => d) * i s , i n f a c t , on to . d) => e) The i d e n t i t y map i s a n i s o t r o p i c . e) => f) Let ( t ,# )eE, where t * ± 1 . The weight s p l i t t i n g we get , M = @ rM, w i l l have the proper ty that <t> r e s t r i c t e d to r M i s r t a ( r ) w i t h a ( r ) a n i s o t r o p i c . Th i s w i l l be homology d iagona l s ince <t>* = t * a . Without l o s s of g e n e r a l i t y we may assume that <fi and a are semis imple . Suppose that i n d u c t i v e l y we have a weight s p l i t t i n g M, . , = @rM\ . , such that <t> r e s t r i c t e d to r M , . , i s 28 r t a ( r ) , where a ( r ) i s a n i s o t r o p i c . Suppose that M\ = M - l . 1 H A ( Z ) . Z can be p i c k e d so that <f>(l) = Z, because * i s semis imple . We can a l s o w r i t e Z = X@Y, where X = ker(d) and * (X) = X and * (Y) = Y . S ince X = H*(X) S H* (M) , <t> r e s t r i c t e d to X i s t' a , and we can l e t \My = \M-, . ,©X. * (dY) = d U ( Y ) ) = dY so dY i s i n v a r i a n t under <t>. We now c l a i m that dY = ©(dYnrM-, . , ) . Suppose that W i s an i r r e d u c i b l e subspace of dY ( i . e . ^(W) = W and no n o n - t r i v i a l proper subspace of W i s i n v a r i a n t under . We w i l l show that W s r M j . , for some r . Let. weW, where w = u + v , 0 t U€ r M - , and ve © SM-, . , = V . Let F be the minimal po lynomia l for a ( r ) s>r r e s t r i c t e d to the subspace of M . •, generated by the set {u, a ( r ) ( u ) , a ( r ) 2 ( u ) , . . . } . The i r r e d u c i b l e f a c t o r s of the r minimal po lynomia l for 0' = ( 1/t ) <t> r e s t r i c t e d to V are of the form k m-1 m f (x ) = t + c x + . . . + c x ± x 1 m-1 w i t h k > 1, and each c an in teger times a power of t . No such f can d i v i d e F s ince t * ±1 and and a l l the i r r e d u c i b l e f a c t o r s of F are as g iven i n d e f i n i t i o n 3 .2 . Thus F U ' M w ) = F U ' H u ) + F U ' M v ) = F ( a ( r ) ) ( u ) + F ( * ' ) ( v ) = F U ' ) (v) i s a non-zero element i n WnV. S ince W i s i r r e d u c i b l e i t has to 29 be a subset of V which c o n t r a d i c t s the fac t that u * 0. Thus W c r M ; . , , and the c l a i m h o l d s . To get the r e q u i r e d we ight ing we l e t i i M = M ® d - 1 ( M n dY) r i r i - 1 r i - 1 S ince M i s f r e e l y generated, t h i s can be extended to a weight r decompos i t ion . $ r e s t r i c t e d to M w i l l be t a ( r ) w i t h a ( r ) a n i s o t r o p i c because i t i s so when r e s t r i c t e d to Z and hence to A(Z) . P f) => g) Suppose * 0 w i t h p > n and (n,p) c l o s e s t to P ( 0 , 0 ) . S ince d i s decomposible d ( n M ) = 0. S ince (n,p) i s P P c l o s e s t to ( 0 , 0 ) , H (ftM) = n M * 0, which i s a c o n t r a d i c t i o n ; so the r e s u l t h o l d s . n g) => a) The compos i t ion M > @hM > H*(M) s a t i s f i e s the requirements . Remark: c) and d) are Theorem 12.7 of [ S u l ] . b) <=> h) i s Theorem 4.1 of [DGMS3. b) <=> i ) i s Chapter 6 of [H&S] b) <=> j ) <=> k) <=> 1) <=> m) i s p r o p o s i t i o n 3.2 of [N&M]. a) <=> b) <=> c) <=> f) <=> m) i s a l s o proved i n [Dou]. b) <=> j ) i s a l s o c o r o l l a r y 6.9 of [H&S] and Theorem 12.1 of [ S u l ] . b) <=> n) <=> o) i s g iven i n [ S h i ] . d) <=> o) f o l l o w s from the f ac t that the functor M:CW > TT (dga) g ives a b i j e c t i o n [B,C] > [ MC, MB ] for C -30 l o c a l . See, for example, theorem 14.7 of [G&M]. 3.2 COFORMALITY The concept of c o f o r m a l i t y i s dua l to f o r m a l i t y i n the sense of Eckmann-Hi l ton . The f o l l o w i n g theorem i s dua l to theorem 3.1 and g ives e q u i v a l e n t c o n d i t i o n s for c o f o r m a l i t y . Assume that M i s a minimal d . g . a . , C i s the coa lgebra dual to M, and L i s the corresponding minimal d . g . l . a . A l s o assume that K i s a s imply connected Pos tn ikov s e c t i o n whose minimal model i s M. Theorem 3.3 The f o l l o w i n g are e q u i v a l e n t and any one of them can be used as the d e f i n i t i o n of c o f o r m a l i t y (b i s the usua l def i n i t i o n ) . a) The p r o j e c t i o n of Z ( L ) , the c y c l e s of L , onto H*(L) l i f t s to a g . l . a . map 0 : Z(L) > H*(L) b) L i s the minimal model for H * ( L ) , cons idered as a d . g . l . a . w i t h zero d i f f e r e n t i a l . c) The c a n o n i c a l map Aut (L) > A u t ( H * ( L ) ) i s an epimorphism. 31 d) Let ir:g>* = $-{0} > A u t ( H * ( L ) ) be the map g iven by i r ( t ) = t * 1 , the grading homomorphism. Then, i n the p u l l b a c k E > Aut (L) * v v (B* > Aut (H*(L) ) jr there i s a t * ±1 i n the image of e) In d ) , l e t rr be g iven by rr ( t ) = t * a where a i s a f i x e d a n i s o t r o p i c map. Then there i s a t * ±1 i n the image of f) There e x i s t s a homology d i agona l weight decomposi t ion P of L . That i s , H ( n L) = 0 when p * n . g) There e x i s t s an upper d i agona l weight decomposi t ion of P P M. That i s n L =0 when p > n and H (nL) = 0 when p * n . h) Suppose L = L ( Z , © Z 2 © . . . ) . Each Z; , the vec tor space spanned by the generators of dimension i , can be w r i t t e n as a d i r e c t sum Zj = Ci©Nj , where Ci c o n s i s t s of the c y c l e s , i n such a way that every c y c l e i n the i d e a l generated by ©Ni i s a boundary. i ) The o b s t r u c t i o n s to c o f o r m a l i t y v a n i s h . These are de f ined d u a l l y to the o b s t r u c t i o n s to f o r m a l i t y d e f i n e d i n [H&S]. j ) I f k i s a f i e l d of c h a r a c t e r i s t i c 0, then LHk i s c o f o r m a l . k) C i s isomorphic w i t h £ ( H * ( L ) ) , where C i s the a d j o i n t functor to £ g iven in [Qui ] . 1) M i s isomorphic w i t h C * ( H * ( L ) ) . 32 m) M i s isomorphic to a minimal a lgebra whose d i f f e r e n t i a l has zero p e r t u r b a t i o n . (Aga in , t h i s means that there i s a ba s i s for M for which the d i f f e r e n t i a l i s q u a d r a t i c . ) For the f o l l o w i n g two c o n d i t i o n s we need to assume that L i s the minimal L i e model for a CW complex K which i s a f i n i t e Pos tn ikov sect i o n . n) I f r i s an i n t e g e r , r * 0 , ± 1 , then there e x i s t s s, a non-zero m u l t i p l e of r , and f : K > K such that f* = S*' 1 1 :tr*(K) > rr* (K) o) There e x i s t s teQ, t * 0 , ± 1 , and a map f : K ( 0 ) > K ( 0 ) such that f* = t * " 1 1 : r r x ( K ( 0 ) ) > T r * . ( K ( 0 ) ) where K ( 0 ) denotes the l o c a l i z a t i o n of K at z e r o . P r o o f : These proofs are g e n e r a l l y d u a l to those of theorem 3 . 1 . a) <=> b) <=> c) <=> d) <=> e) <=> f) <=> g) are e n t i r e l y dual to those i n theorem 3 . 1 . b) <=> j ) <=> k <=> 1) <=> m) i s p r o p o s i t i o n 3.3 of [N&M]. a) <=> b) <=>c) <=> f) <=> m) i s a l s o proved i n [Dou]. g) => h) Let C; be the space spanned by the generators w i t h dimension equal to t h e i r weight and l e t N\ be the space spanned by those generators w i t h dimension l e s s than t h e i r we ight . Then 33 every homogeneous c y c l e i n the i d e a l generated by ©N; has weight grea ter than d imens ion, so i t has to be a boundary. h) => a) Def ine <p:L > H*(L) on generators by [z] i f zeC; * ( z ) =< 0 i f zcN b) => i ) Let />:(L(Z),d) > H * ( D be the b igraded model for H * ( L ) , and rr : ( L ( Z ) , D ) > L be the per turbed model for L . F i x n : H * ( L ) > L ( Z 0 ) l i n e a r so that pr\ = 1 . We de f ine the o b s t r u c t i o n s to c o f o r m a l i t y as f o l l o w s . Suppose there i s an i-1 r e a l i z e r 0 : ( L ( Z ( i ) ) , D ) > ( L ( Z ( i ) ) , d ) , i . e . an isomorphism such that 0-1 lowers f i l t r a t i o n s . The dua l concept a r i s e s i n [H&S]. Def ine 0(0)cHom" 1 (Z j + 1 ,H* (L) ) by 0(z) = [rr ^ D z ] . For <|; eHom0 ( Z; , H * (L) ) , de f ine 0(*), a d e r i v a t i o n of L ( Z ( i ) ) , by i f zeZ(i-1 ) e U ) ( z ) = { n* (z ) i f zeZi Then by de f ine X. :Hom° (Z\ , H * (L)) > Horn" 1 ( Z j + , , H * (L) ) X ( * ) ( z ) = [ y B e ( * ) d z ] . The i t h o b s t r u c t i o n , 0 ( i ) , i s de f ined to be the image of 0(0) i n Horn" 1 (Z,• + , , H * ( L ) ) / lm X. Assume that we have a weak equiva lence 0:L -> H*(L) . 34 Without l o s s of g e n e r a l i t y we may assume that <t>* = 1 : H * ( L ) > H * ( L ) . By s tandard l i f t i n g p r o p e r t i e s there i s an isomorphism p ' : ( L ( Z ) , d ) > L which i s such that = p. By uniqueness of b igraded models, there must be an isomorphism * : ( L ( Z ) , D ) > (L (Z) ,d ) such that *-1 lowers f i l t r a t i o n s and />'$ i s homotopic to rr. o ( * | ( L ( Z ( i ) , D ) ) ) i s always z e r o , so the o b s t r u c t i o n s v a n i s h . i ) => b) Suppose that the i t h o b s t r u c t i o n van i shed . Let <t> and * be such that 0(0) = Then o ( e x p ( -e )$ ) = 0, so e x p ( - e ) $ extends to an i r e a l i z e r ( L ( Z ( i+1 ) ) , D ) > ( L ( Z ( i+1 ) ) , d ) . exp i s the e x p o n e n t i a l map. C o n t i n u i n g i n d u c t i v e l y and t a k i n g l i m i t s we get an isomorphism (L (Z) ,D) > ( L ( Z ) , d ) . S ince L i s minimal i t i s a l s o isomorphic to (L (Z) ,d ) and hence c o f o r m a l . For d e t a i l s of the above see [H&S]-» m) => n) We c o n s t r u c t the map by i n d u c t i o n on the Pos tn ikov s e c t i o n of K. P ick t , so that t a n n i h i l a t e s the t o r s i o n n+2 n subgroup of H (K ;Z ) ) for a l l n between 0 and the homotopy dimension of K. The number s tha t r e s u l t s w i l l be a power of r t . n = 0 K° i s a p o i n t . The i d e n t i t y map t r i v i a l l y s a t i s f i e s the theorem. 35 n = i + 1 Suppose that we have c o n s t r u c t e d the f o l l o w i n g i n d u c t i v e l y . . There i s a power s of r t and a map f : K * > K' w i t h the p r o p e r t i e s 1) f * = s * " 11 : T T * ( K ' ) > FF»(KL) 2) The diagram M; A * ( K ' ) < A * ( K l ) f * 1*1-1 homotopy commutes, where * ( z ) = s z , for z a genera tor . 3) l m ( f * ) c t H * ( K ' ; Z ) for * l e s s than or equal to the homotopy dimension of K. Suppose M l + 1 = M i B A ( Z ) , where the d i f f e r e n t i a l i s quadra t i c "on Z. We get K l + 1 as the f i b r e of k K i + i -> K l > K ( i r i + 1 ( K ) , i + 2) I f a e H 1 + 2 ( K ( r f , + 1 ( K ) , i + 2 ) ) , there i s zeZ w i t h * * ( [ d z ] ) = q ( k ( a ) ) , where q i s the n a t u r a l t r a n s f o r m a t i o n q :H* ( ,Z) > H*( . S ince d i s q u a d r a t i c q ( f * k ( a ) ) = f * t f * [ d z ] = <y***[dz] = s ' « y * [ d z ] = s ' q ( k ( a ) ) 36 By c o n d i t i o n 3, f * ( k ( a ) ) i s d i v i s i b l e by t , so f * ( k ( a ) ) = s 'k (a ) and the diagram K v. K1 -> K ( i r i + i ( K ) , i + 2) S 1 -> K(ff i + 1 ( K ) , i + 2) homotopy commutes and we get a l i f t i n g g ^ K 1 * 1 > K l + 1 . Looking at the long exact homotopy sequences, we see that c o n d i t i o n 1 i s s a t i s f i e d for any such l i f t i n g . The corresponding map * ' i n the diagram M . 1 + 1 <— A * ( K , + 1 ) <-M i t i a J A*(K' ' + 1 ) i s such that * ' r e s t r i c t e d to Mi i s * and * ' ( z ) = s1 z + c ( z ) for zeZ and c ( z ) some c y c l e i n M j . [ c ( z ) ] i s an o b s t r u c t i o n to g e t t i n g the l i f t i n g * tha t we want. I f [ c ( z ) ] i s i n the l a t t i c e of lemma 2 .15 , we. can change the l i f t i n g g, to make [ c ( z ) ] z e r o . W r i t e c ( z ) = c , + c 2 + . . . +ck where Cj i s a sum of products of j genera tor s . S ince d i s q u a d r a t i c Cj i s a c y c l e for a l l j . F i n d 1 so that 1 [ CJ ] i s i n the above l a t t i c e for a l l j . Wr i te 1 = pq where (q ,s ) = 1, and p d i v i d e s some power of s. By i n d u c t i o n one can show that 37 N Ni ( N -1 ) i * ' (z) = s z + Ns c 1 k ( N - l ) ( i - j + l ) N ( j -1 ) j-1 + I s (s - 1 ) / ( s -1 )c j = 2 j Choosing N to be the l e a s t common m u l t i p l e of q , of m, where m r+1 r+1 j p | s , and of </>{q ) , where q )(s - 1 , for j = 1, k - 1 , and <t> i s E u l e r ' s <t> f u n c t i o n , makes 1 devide the c o e f f i c i e n t of c j for a l l j = 1, k. Thus the o b s t r u c t i o n for to the N i s zero and we can modify g , to the power N to get g 2 : K * 1 > K + 1 which s a t i s f i e s c o n d i t i o n 2. To get c o n d i t i o n 3 we note that powers of g 2 w i l l l o c a l i z e the homotopy groups at primes r e l a t i v e l y prime to s and there fo re w i l l do the same to the cohomology groups. Thus some power of g 2 , say g , s a t i s f i e s a l l three c o n d i t i o n s and the i n d u c t i o n c o n t i n u e s . n) => o) L o c a l i z e the map f . o) => d) L i f t f to a map <t>:L > L . Then ( t , * ) e E i s such that * ( t , $ ) = teQ* i s non t r i v i a l . 3.3 EXAMPLES AND PROPERTIES Formal and coformal spaces have been s t u d i e d by s e v e r a l a u t h o r s . Some of the f o l l o w i n g examples and theorems are found i n [DGMS], [H&S], [N&M], and [ S u l ] . When p o s s i b l e the duals to the theorems are g i v e n , as are proofs where these are s i m p l e . 38 P r o p o s i t i o n 3.4 Products and coproducts of formal ( re sp . coformal) CW-complexes are formal ( re sp . c o f o r m a l ) . P r o o f : Suppose M and N are the minimal d . g . a . models for the CW-complexes X and Y. Then MSN i s the minimal model for XxY by the Runneth theorem. I f X and Y are both f o r m a l , then MSN i s a l s o the minimal model for H*(XxY;Q) = H*(X;(£)BH*(Y;®) so XxY i s f o r m a l . I f X and Y are both c o f o r m a l , the d i f f e r e n t i a l i n MG5N s t i l l has zero p e r t u r b a t i o n so XxY i s c o f o r m a l . Coproducts can be proved d u a l l y us ing the minimal d . g . l . a . models. Remark: T h i s i s lemma 4.1 of [N&M]. P r o p o s i t i o n 3.5 a) A l l ske le tons and homology s e c t i o n s of formal CW-complexes are f o r m a l . b) A l l Pos tn ikov s e c t i o n s of coformal spaces are c o f o r m a l . c) Pos tn ikov s e c t i o n s of formal spaces need not be f o r m a l . d) Homology s e c t i o n s of coformal spaces need not be c o f o r m a l . P r o o f : a) Th i s i s lemma 3.1 of [ S h i ] . b) The minimal model of the i t h Pos tn ikov s e c t i o n of X i s M j , 39 which has a quadrat ic , d i f f e r e n t i a l , when M does. Hence i t i s c o f o r m a l . c) The s i x t h Pos tn ikov s e c t i o n of S 3 v S 3 has the minimal model M 6 = A ( x , y , z ) , where the dimensions of x and y are 3 and the dimension of z i s 5. The d i f f e r e n t i a l s of x and y are zero and dz = x y . I f t h i s were formal we would have to l e t the weights of x and y be 3, and thus the weight of z would be 6. Then [xz] i s a non-zero cohomology c l a s s w i t h weight 9 and dimension 8. Thi s v i o l a t e s theorem 3.1 f ) . d) The s i x t h homology s e c t i o n of K(Z©Z,3) i s a dual example to c) . P r o p o s i t i o n 3.6 a) I f X i s a s imply connected CW complex of f i n i t e type such that H*(X;q>) = 0, unless * = 0, or k < * < 3k + 2 for some k, then X i s f o r m a l . b) I f X i s a s imply connected Pos tn ikov s e c t i o n of f i n i t e type such that rr*(X)Kg) = 0, unless * = 0 or k < * < 3k + 2 for some k, then X i s c o f o r m a l . P r o o f : a) T h i s i s C o r o l l a r y 5.16 of [H&S]. I t a l s o f o l l o w s e a s i l y by the dual of proof b) below. b) The sma l l e s t dimension that a generator of the minimal model of X can have i s k + 1, and the l a r g e s t dimension i t can have i s 3k + 1. Thus the d i f f e r e n t i a l must be quadra t i c and have no p e r t u r b a t i o n . Hence X i s coformal by theorem 3.3 m). 40 Example 3.7 A l l spheres , Ei lenberg-MacLane spaces, and Moore spaces are both formal and c o f o r m a l . Example 3.8 A l l complex and q u a t e r n i o n i c p r o j e c t i v e spaces are f o r m a l . Only (CP(1), (CP(oo), ffip(l) and IP(oo) are c o f o r m a l . The others are not c o f o r m a l , s i n c e t h e i r minimal models have non-quadrat ic d i f f e r e n t i a l s . P r o p o s i t i o n 3.9 a) Suppose X i s a s imply connected CW complex whose r a t i o n a l cohomology has f i n i t e t y p e . I f a l l n o n - t r i v i a l Q-cohomology l i e s i n odd degrees, then i ) X i s formal and c o f o r m a l . i i ) X has the r a t i o n a l , homotopy type of a bouquet of odd spheres , and i i i ) A l l n o n - t r i v i a l Q-homotopy l i e s i n odd degrees . b) Suppose X i s a s imply connected Pos tn ikov s e c t i o n whose r a t i o n a l homotopy has f i n i t e t y p e . I f a l l n o n - t r i v i a l Q-homotopy l i e s i n even degrees , then i ) X i s formal and c o f o r m a l , i i ) X has the r a t i o n a l homotopy type of a product of even d imens iona l Ei lenberg-MacLane spaces , and i i i ) A l l n o n t r i v i a l Q-cohomology l i e s i n even degrees . P r o o f : a) E v e r y t h i n g except coformal i s theorem 1.5 and 7.10 of [H&S]. C o f o r m a l i t y i s by example 3 . 7 , together w i t h p r o p o s i t i o n 3 .5 . i i ) i s a l s o C o r o l l a r y 1.2 of [Bau] , The r e s u l t a l s o 41 f o l l o w s e a s i l y by the dua l of proof b) below. b) A l l the generators for the minimal d . g . a . model for X have even d imens ion . Hence the d i f f e r e n t i a l has to be z e r o , and so has no p e r t u r b a t i o n .and X i s coformal by theorem 3.2 m). The re s t f o l l o w s s ince a product of even d imens iona l Ei lenberg-MacLane spaces has the same minimal model. P r o p o s i t i o n 3.10 (Lemmas 4.2 and 4.3 of [N&M]) a) I f X i s a n i l p o t e n t space such that H*(X;Q) = S / I , where S i s a graded symmetric a l g e b r a , and I i s a B o r e l i d e a l , then X i s f o r m a l . I f the generators for I can be chosen to be q u a d r a t i c , then X i s a l s o c o f o r m a l . b) I f X i s a s imply connected space such that rr*(nX)Hg> = F / J , where F i s a f ree graded L i e a lgebra and J i s a B o r e l i d e a l , then X i s c o f o r m a l . I f the generators of J can be chosen to be q u a d r a t i c , then X i s a l s o f o r m a l . Example 3.11 The " r a t h e r n ice spaces" of [B&D] and the " r e g u l a r r a t i o n a l homotopy types " of [Bod] are formal by the above p r o p o s i t i o n . D e f i n i t i o n 3 .12 : a) A g . a . , H, i s i n t r i n s i c a l l y formal i f and only i f a l l d . g . a . ' s w i t h cohomology a lgebras i somorphic to H are f o r m a l . b) A g . l . a . , L , i s i n t r i n s i c a l l y c o f o r m a l , i f and only i f a l l d . g . l . a . ' s w i t h homology L i e a lgebra s i somorphic to L are c o f o r m a l . 42 Remark: P r o p o s i t i o n s 3 .5 , 3 .6 , 3.9 and 3.10 a l l g ive c o n d i t i o n s under which H ( resp . L) i s i n t r i n s i c a l l y formal ( resp . c o f o r m a l ) . Example 3.13 L i e groups and, more g e n e r a l l y , H-spaces are f o r m a l . Example 3.14 C l a s s i f y i n g spaces of L i e groups are formal [ S u l ] . Example 3.15 Symmetric spaces are formal [H&S]. Example 3.16 A wide c l a s s of (but not a l l ) homogeneous spaces are formal ( c f . Theorem I V , Chapter X I , of [GHV]). Example 3.17 The unstable Thorn spaces MU(n), and MSO(n) are formal [ S u l ] . Lemma 3.18 The spaces W(n) of example 1.5 are not f o r m a l . P r o o f : The minimal d . g . l . a . model for W(n) i s L [ x , y , z ] where the dimensions of x and y are 2 and the dimension of z i s 7. The d i f f e r e n t i a l s of x and y are zero and dz = n [ x , [ x , y ] ] which i s not q u a d r a t i c , so ( S 3 v S 3 ) u e 8 i s not formal by theorem 3.1 m). 43 P r o p o s i t i o n 3.19 ( P r o p o s i t i o n 4.4 of [N&M]) Every n-connected, compact, m-dimensional mani fo ld w i t h rank (PH(M;(B)) ^ 2 and m < 3n + 1 i s both formal and c o f o r m a l . P r o p o s i t i o n 3.20 ( P r o p o s i t i o n 4.6 of [N&M]) Every s imply connected compact mani fo ld of dimension l e s s than or equal 6 i s f o r m a l . P r o p o s i t i o n 3.21 (Main Theorem of [DGMS]) A l l Kahler mani fo lds and MoiSezon spaces are f o r m a l . P r o p o s i t i o n 3.22 a) I f F and X are s imply connected spaces of f i n i t e type , F i s formal and H*(X;Q) = H*(F;Q) as g . a . , then d i m U i (F)B(B) > d i m U i (X)H(B) for a l l i . I n t u i t i v e l y , a formal space has as much homotopy as i t s cohomology p e r m i t s . b) I f C and X are s imply connected spaces of f i n i t e type , C i s c o f o r m a l , and ir*(nC)iS<B = i r f i X j i S Q as g . l . a . , then d im(H ; (C;Q) ) > dim(H; (X;<B) ) for a l l i . P r o o f : a) Let (A(Z) ,d ) be the minimal b igraded model for H*(F) as i n [H&S], and l e t M be the minimal d . g . a . model for X . By 44 the r e s u l t s i n [H&S], we may per turb d i n t o D and o b t a i n a weak equiva lence pi(AZ,D) > M By the s tandard l i f t i n g p r o p e r t i e s of minimal models we may f i n d elM > (AZ,D) such that pe i s a weak e q u i v a l e n c e . But such a p<s has to be an isomorphism, so we may assume that pe = 1 . Hence, e induces a monomorphism on indecomposibles and the r e s u l t f o l l o w s , b) dual to a) . 45 Chapter 4 DISTANCE BETWEEN HOMOTOPY TYPES R. Body and R. Douglas i n [B&D] developed the concept of d i s t a n c e between homotopy types (see d e f i n i t i o n 4.3 below) which was used to prove that there are only a f i n i t e number of homotopy types w i t h i n any f i x e d (cohomology) d i s t a n c e of a f i x e d homotopy type X ( C o r o l l a r y 4.7 b ) ) . Th i s g ives a technique which can be used to show that c e r t a i n sets of homotopy types are f i n i t e . In t h i s chapter we d u a l i z e the concept of d i s t ance and t a l k about homotopy d i s t a n c e which i s g iven i n terms of the f i b r e of a map. Suppose tha t X and Y are s imply connected CW complexes, and that c :X > Y i s a map which has f i b r e F and c o f i b r e C. D e f i n i t i o n 4 . 1 : The cohomology l e n g t h of a i s i i l c ( a ) = n (order H (C;Z) ) i^1 D e f i n i t i o n 4 . 2 : The homotopy l e n g t h of a i s i l f ( o ) = n (order IT (F) ) i>1 i We note that these lengths are de f ined ( i . e . i s a r a t i o n a l homotopy e q u i v a l e n c e , d e f i n i t i o n s W has to be s imply connected . f i n i t e ) only when c In the f o l l o w i n g 46 D e f i n i t i o n 4 , 3 : The cohomology d i s t a n c e between X and Y i s a fi d i s t c ( X , Y ) = m i n { l o g 2 ( l c ( a ) l c ( < » ) ) | X > W < Y} W D e f i n i t i o n 4 . 4 : The homotopy d i s t a n c e between X and Y i s o fi d i s t f ( X , Y ) = m i n { l o g 2 ( l f U ) l f ( f i ) ) | X < W > Y} W I f these d i s t a n c e s are f i n i t e , then X and Y are r a t i o n a l l y homotopy e q u i v a l e n t , but the converse does not h o l d . In fac t i t i s p o s s i b l e for one d i s t a n c e to be f i n i t e whi l e the other i s i n f i n i t e . For example, l e t X be K(Z/pZ ,2 ) and Y be a p o i n t . Then d i s t f ( X , Y ) = l o g 2 p , whi l e d i s t c ( X , Y ) i s i n f i n i t e . Of course , r e s t r i c t i n g to f i n i t e d imens iona l CW-complexes ( resp . f i n i t e Pos tn ikov s e c t i o n s ) , d i s t c ( X , Y ) ( resp . d i s t f ( X , Y ) ) i s f i n i t e i f and only i f X and Y are r a t i o n a l l y homotopy e q u i v a l e n t . Lemma 4.5 a) l f ( * c ) < l f U ) l f ( o ) b) l c U a ) < l c ( * ) l c ( o ) c) d i s t f i s a m e t r i c , when r e s t r i c t e d to spaces which are w i t h i n some f i x e d r a t i o n a l homotopy type and which are homotopy equ iva l en t to a f i n i t e Pos tn ikov s e c t i o n . d) d i s t c i s a m e t r i c , when r e s t r i c t e d to spaces which are w i t h i n some f i x e d r a t i o n a l homotopy type and which are homotopy 4 7 e q u i v a l e n t to a f i n i t e CW-complex. P r o o f : a) The induced maps on the f i b r e s of o, 00 and 0 , F(a) > F ( 0 a ) > F(^) i s i t s e l f homotopy e q u i v a l e n t to a f i b r a t i o n . The long exact homotopy sequence then g ives o r d e r U ( F ( 0 o ) ) ) < o r d e r U (F ( a))) o r d e r (rr (F ( / » ) ) ) i i i and the r e s u l t f o l l o w s . b) Dual to a.). c) Only the t r i a n g l e i n e q u a l i t y needs to be proved. Let X , Y , V , W and W' be s imply connected CW complexes such that d i s t f ( X , Y ) = l o g 2 ( I f ( c ) l f ( 0 ) ) and d i s t f ( Y , V ) = l o g 2 ( l f ( r ) l ' f ( 6 ) ) . a and e map W to X and Y, and 7 and 6 map W' to Y and V . Let Y' be the p u l l b a c k of 0 and 7 . 0 ' Y ' > W' v W v -> Y S ince they have the same f i b r e s , l f ( 0 ' ) = l f ( 0 ) . Then (by a) above) I f ( 7 ' ) = I f ( 7 ) and d i s t f ( X , V ) < l o g 2 ( l f ( 0 7 ' ) l f ( 6 0 ' ) ) < d i s t f ( X , Y ) + d i s t f ( Y , V ) d) Dual to c ) . In [B&D], the d i s t a n c e between two f i n i t e CW-complexes 48 w i t h i n the same r a t i o n a l homotopy type i s g iven as i n d e f i n i t i o n 4 . 3 , but the minimum i s taken over sequences x — > x, < — x 2 — > . . . <— x n = y By the t r i a n g l e i n e q u a l i t y , t h i s w i l l be the same as d e f i n i t i o n 4.3 Lemma 4.6 Let M be a s imply connected CW complex of f i n i t e type and N be a p o s i t i v e i n t e g e r . Then the sets a) Ef(M) = { [X] jXeobj (CW), there e x i s t s c : X > M w i t h l f ( c ) = N} and b) Sc(M) = { [ X ] | X € o b j ( C W ) , t h e r e e x i s t s c :M ^—> X w i t h l c ( o ) = N} are both f i n i t e s e t s . c) I f i n a d d i t i o n M has f i n i t e homotopy d imens ion , the set Sf(M) = { [X3 |Xeobj (CW), there e x i s t s a:M > X w i t h I f ( a ) = N} i s a f i n i t e s e t . d) I f i s a d d i t i o n M i s a f i n i t e CW complex, the set Ec(M) = { [ X ] | X t o b j ( C W ) , t h e r e e x i s t s c :X > M w i t h l c ( o ) = N] i s f i n i t e . The f o l l o w i n g c o r o l l a r y i s the main r e s u l t of t h i s chap te r . 49 C o r o l l a r y 4.7 a) Let X have f i n i t e homotopy d imens ion, and N be a f i x e d p o s i t i v e i n t e g e r . Then { [ Y ] | Y e o b j ( C W ) , d i s t f ( X , Y ) < N} i s a f i n i t e s e t . b) Let X be a f i n i t e CW complex and N a f i x e d p o s i t i v e i n t e g e r . Then { [ Y ] | Y e o b j ( C W ) , d i s t c ( X , Y ) < N} i s a f i n i t e s e t . P r o o f : a) There i s a W such that there e x i s t a:W > X and 0:W > Y w i t h the homotopy lengths of a and p both l e s s than or equal to 2 to the power N . By Lemma 4.6 a) there are only a f i n i t e number of cho ice s for [W], and then by Lemma 4.6 c) there are then only a f i n i t e number of p o s s i b i l i t i e s for [ Y ] , b) Dual to a ) . C o r o l l a r y 4.7 b) i s Theorem 3.2 of [B&D]. P r o o f : ( o f Lemma 4.6) Sc(M) and Ec(M) were proved f i n i t e i n [B&D]. The proof s for the other 2 set s are not d u a l , so they are g iven below. a) Let XeEf (M) . Make a homotopy decomposit ion of o. X > Mi > K(ff£t |(F) , i + 2) a; k,-Here F i s the f i b r e of o, M, = M, a , = o and a; induces an isomorphism on homotopy for dimensions l e s s than i and an epimorphism i n dimension i . S ince the homotopy groups of F are 50 f i n i t e and bounded- there are only a f i n i t e , bounded number of cho ices for the k - i n v a r i a n t s at each l e v e l . S ince the homotopy groups of F van i sh for dimension greater than l o g 2 N , a-, i s a homotopy equiva lence for i s u f f i c i e n t l y l a r g e . Any other space X 1 , w i t h the same k - i n v a r i a n t s up to t h i s p o i n t , must be homotopy e q u i v a l e n t to X . Thus Ef(M) i s f i n i t e . c) Let XeSf (M) . Make a homology decomposit ion of a. K' (H-, + , (C;Z) , i ) M i + 1 Here C i s the c o f i b r e of a , M-, = M, a , = c , and a', induces an isomorphism on homology for dimensions l e s s than i and an epimorphism i n dimension i . In order to c o n t r o l the k ' - i n v a r i a n t s , we need to know about the order of H \ ( C ; Z ) . Th i s can be bounded by a f u n c t i o n of I f ( a ) = N, and the isomorphism type of H * ( M ; Z ) , *• < i as f o l l o w s . Assume that we have i n d u c t i v e l y bounded the order of H#.(C;Z) for *< i . F i r s t , because I f ( a ) = N , there are o n l y a f i n i t e number of cho ice s for the homotopy type of F, and so we may assume that i t i s f i x e d . The short exact sequence 0 > c o k e r U j ) > H (C;Z) > ker (o- , > 0 where <j' (:Hj(M;Z) > H j ( X ; Z ) i s the map induced by c , shows that we need only bound the orders of k e r ( a j . i ) and c o k e r ( c \ ) . 51 Since o i s a r a t i o n a l homotopy equiva lence we can bound the order of ker (c j.i) by the order of the t o r s i o n subgroup of H i . , ( M ; Z ) . The order of coker (a-,) i s bounded by l o o k i n g at the a p p r o p r i a t e edge homomorphism i n the Serre s p e c t r a l sequence for F > M > X . We f a c t o r a, as oo i + 1 1 3 2 H i ( M ; Z ) > E > E > . . . > E = H \ ( X ; Z ) . i , 0 i , 0 i , 0 Since the f i r s t map i s a s u r j e c t i o n and the second i s an isomorphism, we need only look at the re s t of the maps. Because they are i n c l u s i o n s i + 1 order(coker(a\ ) ) = n o r d e r ( c o k e r ( 1 ) k=3 But k k order (coker (1 ) = order(Im(d |E )) i , 0 k < order (E ) i-k,k-1 2 < order (E ) i-k,k-1 = order(H (X;H ( F ; Z ) ) ) i - k k-1 This depends on F, which i s f i x e d , r a n k ( H ( X ; Z ) ) = rank(Hj .^ (M;Z)) and the order of the t o r s i o n subgroup of H j . ^ (X;Z) and HJ.R-I (X; Z ) . But these can be bounded, us ing the long exact homology sequence, by the orders of the t o r s i o n subgroups of H*(M;Z) and H * ( C ; Z ) , for * < i . These have 52 been p r e v i o u s l y bounded by i n d u c t i o n . A l l t h i s shows that the order of H ; ( C ; Z ) i s bounded and thus there are only f i n i t e number of cho ices for the k ' - i n v a r i a n t s at each l e v e l . The homotopy dimension of X i s l e s s than or equal to the maximum of the homotopy dimension of M and l o g 2 N . S ince two s imply connected spaces w i t h the same k ' - i n v a r i a n t s up to t h e i r homotopy dimension p lus one are homotopy e q u i v a l e n t , and s ince there are only a f i n i t e number of cho ices for the k ' - i n v a r i a n t s up to t h i s d imens ion , Sf(M) i s f i n i t e . 53 Chapter 5 FORMALITY AND FINITE AMBIGUITY 5.1 PRELIMINARIES The purpose of t h i s chapter i s to prove the f o l l o w i n g r e s u l t and i t s d u a l . Theorem 5.8 Let B be a graded, s imply connected, f i n i t e l y generated, f i n i t e d imens iona l Z - a l g e b r a . Then the set FT*(B) = { [X] |Xtob j (CW) , H * ( X ; Z ) = B, X i s formal} i s f i n i t e . To prove t h i s r e s u l t , we w i l l c o n s t r u c t model spaces for any Y e F T * ( B ) . Then the cohomology d i s t ance between any XeFT*(B) and some model space i n a f i n i t e subset of the model spaces i s bounded. The r e s u l t then f o l l o w s from the prev ious chap te r . S ince X«FT*(B) i s formal we may assume the minimal d . g . a . model, M, for X and for B i s the b igraded model M = (A(Z) ,d ) g iven i n [H&S]. Let M(k) = ( A ( Z 0 © Z , © . . . © Z k ) , d ) be the subalgebra of M generated by a l l generators w i t h f i l t r a t i o n l e s s than or equal to k. The model spaces w i l l be cons t ruc ted as a Pos tn ikov tower b u i l t by i n d u c t i o n on the f i l t r a t i o n ra ther than the dimension of the generators of M. The advantage of doing the c o n s t r u c t i o n i n t h i s order i s t w o f o l d . F i r s t , the i n c l u s i o n M(0) > M i s onto i n cohomology, so the order of the 54 c o k e r n e l of H*(WY;Z) > H * ( X ; Z ) , where WY i s one of the model spaces, can be bounded at s tep 0. Second, H*(M(k)) i s zero i n f i l t r a t i o n s n , 1 ^ n < k. Thus i f c i s a c y c l e of f i l t r a t i o n k, and * i s a (k-1) r e a l i z e r ( i . e . $-1 lowers f i l t r a t i o n s ) , [ ( * - l ) ( c ) ] = [ c 0 ] , c o e M ( 0 ) . What happens here i s again determined i n s tep 0. The proof of the dual r e s u l t w i l l be e n t i r e l y analogous . For a coformal space we may assume that the minimal L i e model i s the b igraded model ( L ( Z ) , d ) that i s c o n s t r u c t e d in Chapter 2. For any YeCT*(n) a model space i s b u i l t as a CW complex w i t h c e l l s a t tached accord ing to f i l t r a t i o n , ra ther than d imens ion . Then the homotopy d i s t ance between any XcCTMn) and some model space i n a f i n i t e subset of a l l the model spaces w i l l be bounded. From the prev ious chap te r , the theorem f o l l o w s . Th i s proof i s e n t i r e l y d u a l , so d e t a i l s w i l l not be g i v e n . For t h i s chapter we need to f i x the f o l l o w i n g n o t a t i o n . a) For every X e F T * ( B ) , f i x an isomorphism $ : H * ( X ; Z ) > B. i w i l l a l s o be used for the corresponding isomorphism p:H*(X;Q) > BBQ. b) Let p:M > B8Q be a f i x e d weak equiva lence and n: BBQ > A ( Z 0 ) a l i n e a r map such that PD = 1 . c) For any X e F T * ( B ) , l e t ir:M > A* (X) be a weak equiva lence which s a t i s f i e s p([irz]) = pz for z e A ( Z 0 ) . Such a ir can be obta ined as f o l l o w s . F i r s t use theorem 4.4 of [H&S] to p e r t u r b d i n t o D and f i n d i r ' : ( A Z , D ) > A* (X) which has />([ir'z]) = pz. S ince the spaces are f o r m a l , a l l o b s t r u c t i o n s to f o r m a l i t y v a n i s h , and theorem 5.3 of [H&S] g ives an isomorphism 55 * : ( A ( Z ) , d ) —-> (A(Z),D) such that *-1 lowers f i l t r a t i o n s . Then TT = TT ' * has the r e q u i r e d p r o p e r t i e s . d) Let {b} be a f i x e d set of generators of B. S p e c i f i c a l l y we want <{q(b)}> = p ( Z 0 ) , where q i s the n a t u r a l i n c l u s i o n q:B > BHQ and <{q(b)}> denotes the vec tor space generated by {q(b)} . We a l s o use q for the n a t u r a l t r a n s f o r m a t i o n q : H * ( X ; Z ) > H*(X;5>) which i s such that q * = * q . For the dual theorem the f o l l o w i n g n o t a t i o n w i l l be used, a ' ) For any XeCT*(n) , l e t p:rr*.(nx) > n be a f i x e d isomorphism. p w i l l a l s o be used for the corresponding isomorphism obta ined by t e n s o r i n g p w i t h Q. b ' ) Let />:(LZ,d) > IIS5Q be a f i x e d weak equiva lence c ' ) Let {e} be a f i x e d set of generators of n such that <{q(e)}> = p ( Z 0 ) , where q i s again the n a t u r a l i n c l u s i o n q:n > nBQ. 5.2 CONSTRUCTION OF MODEL SPACES For each i , l e t K ( i ) = n K (Z ,d im z) be a g e n e r a l i z e d Ei lenberg-MacLane space, where the product i s taken over {z}, a set of homogeneous ba s i s elements for Zj . Any map X > K ( i ) can be given by a corresponding set of i n t e g r a l cohomology c l a s s e s . Suppose that Y c F T * ( B ) . We w i l l use Y to c o n s t r u c t a model space WY that w i l l be a homology s e c t i o n of a s u i t a b l e space in the Pos tn ikov tower c o n s t r u c t e d below. 5 6 n = 0 Let W Y ( O ) = K(0) and h 0 : Y > WY(O) be determined by the cohomology c l a s s e s ifi'Hb)}. The minimal model for WY(0) i s j u s t M(0) and we can f i n d <ro:M(0) > A*(WY(0)) so that the f o l l o w i n g diagram homotopy commutes i n M M(0) v v A* (Y) < A*(WY(0)) h 0 * where " i n " i s the i n c l u s i o n . n = k+1 Suppose that we have i n d u c t i v e l y c o n s t r u c t e d WY(k) w i t h minimal d . g . a . model M(k) and maps h and a such that i n M M(k) v v A * ( Y ) < A*(WY(k)) h * homotopy commutes. Suppose M(k+1) = M ( k ) H A ( Z k + , ) . We can f i n d {z}, a homogeneous set of generators of Z k + 1 , and a set {c} of cohomology c l a s s e s i n H*(WY(k) ;Z) such that for each oe{c} there i s zc{z} w i t h i ) q ( o ) = tf*([dz])€H*(WY(k)?Q) i i ) c = t a ' , where a' i s i n d i v i s i b l e , and t a n n i h i l a t e s the t o r s i o n subgroup of B. Let WY ( k+1) be the f i b r e of f 57 P f WY(k+l) > WY(k) > K(k+1) where f i s determined by the cohomology c l a s s e s {c}. By the H i r s c h lemma, the minimal d . g . a . model of WY(k+1) i s M(k+1), together w i t h a weak equiva lence which extends p*c. For any a, q(h*(a)) = h * U * ' d z ] j = i r * ( [dz ] ) = 0, so 0 ( h * ( c ) ) e t o r ( B ) . But by c o n d i t i o n i i ) , a can be d i v i d e d by t so h* (a) = 0. Thus fh i s h o m o t o p i c a l l y t r i v i a l and there i s a l i f t i n g h ' of h WY(k+1 ) P v Y —> WY(k) h Let <r":M(k+1) > A*(WY(k+1)) be a weak equiva lence such that tf"|M(k) = p*<y, which e x i s t s by the H i r s c h lemma. By standard l i f t i n g p r o p e r t i e s , we may f i n d r such that M <- M(k+1) rr v v A* (Y) < A*(WY(k+1)) h ' * homotopy commutes. S ince r r e s t r i c t e d to M(k) i s the i n c l u s i o n and r * : H * ( M ( k+1 ) ) > H*(M) i s a s u r j e c t i o n , we may assume that r ' i n r f ap tor s as M(k+1) > M(k+1) > M. Then i f tf' = ff"(r')"1 58 i n M < M(k+1) v v A * ( Y ) < A*(WY(k+1)) h ' * homotopy commutes, and we can cont inue the c o n s t r u c t i o n i n d u c t i v e l y . The model space for Y e C T * (n ) i s c o n s t r u c t e d as f o l l o w s . Let W'Y(O) = v S , where the wedge product i s taken over the set {sp'Me)} (s i s the suspension map). Let h o : W ' Y ( 0 ) > Y be g iven by the homotopy c l a s s e s { s p ~ 1 ( © ) } . I n d u c t i v e l y , suppose that we have c o n s t r u c t e d W'Y(k) w i t h minimal L i e model L ( k ) , and a map h :W'Y(k) > Y w i t h L(k) = [ i n ] ( " i n " i s the i n c l u s i o n ) . I f L(k+1) = L ( k ) v L ( Z k + ! ) , we can f i n d {z}, a homogeneous set of generators of Z k + 1 , and a set {y} of homotopy elements i n ffft(W'Y(k)) such that for each re { r ] there i s ze{z} w i t h i ) q ( r ) = s * ( [ d z ] ) e»* (W ' Y ( k ) ) B ® i i ) r = t r ' , where t a n n i h i l a t e s the t o r s i o n subgroup of n . We use ( r ) to a t t a c h c e l l s to W'Y(k) to get W'Y(k+l) w i t h minimal L i e model L(k+1) ( p r o p o s i t i o n 8.11 of [ N e i ] ) . As i n the above c o n s t r u c t i o n we can f i n d a map h:W'Y(k+1) > Y w i t h the r e q u i r e d p r o p e r t i e s . Def i n i t i o n 5 . 1 : a) Suppose that B i s N d imens iona l and BBQ i s n connected. The model space corresponding to Y, WY, w i l l be the (N+2) homology s e c t i o n of WY(k), where k = [ (N+2 ) /n ] -1 . b) Suppose that, the dimension of n i s N and i s n connected. The model space corresponding to Y, W'Y, w i l l be the 59 N+1 Pos tn ikov s e c t i o n of W ' Y ( k ) , where k = [(N+3)/(n+2)]-1. Remark: D e f i n i t i o n a) ensures that H*(WY;Q) = BSSQ and b) ensures that n*.(nW'Y)8$ = 118$. The spaces WY(k) are geometric models for M(k) and we get the f o l l o w i n g r e s u l t about (k-1) r e a l i z e r s . Lemma 5.2 a) I f * i s a k r e a l i z e r , there e x i s t s a homotopy equiva lence g:WY(k+l) > WY(k+l) , and a p o s i t i v e in teger n , n such that M(g) = ], i . e . n M(k+1) < M(k+1) v v A*(WY(k+l)) < A*(WY(k+D) g* homotopy commutes. b) I f <t> i s a k r e a l i z e r , . there e x i s t s a homotopy equiva lence g:W*Y(k+1) > W'Y(k+1) and a p o s i t i v e in teger n n such that L(g) = ] . P r o o f : a) g i s cons t ruc ted i n d u c t i v e l y . n = 0 Let g = 1:WY(0) > WY(0) 60 n = k+1 Suppose that we have c o n s t r u c t e d f :WY(k) > WY(k) w i t h j M(f) = [ 0 | M ( k ) ] . Without l o s s of g e n e r a l i t y we may assume that K(k+1) = K(Z,m) for some m, and that c i s the cohomology c l a s s " k i l l e d " to get WY(k+1). Then q ( p * f * ( c ) ) = p * f * U * ( [ d z ] ) ) j = a*<t> * ( [ d z ] ) = 0cH*(WY(k+1);Q) because 0 i s de f ined on M(k+1). Thus p * f * ( a ) i s a t o r s i o n element i n H*(WY(k+1) ;Z) . S ince the k e r n e l of p * i s the subgroup generated by a, and s ince f i s a homotopy e q u i v a l e n c e , f * ( o ) = ± 0 + t o r s i o n . Then, for some r and s w i t h r > s, V s f* (a) = f* ( a ) , because there i s only a f i n i t e amount of t o r s i o n i n dimension m. Then r-s f* (o) = o r-s and p * f * (a) = p * U ) = 0 Hence we get a l i f t i n g g ' :WY(k+l) > WY(k+l) w i t h M(g' ) = [ 0 ' ] ( r - s ) j ( r - s ) j where 0 ' | M ( k ) = 0 |M(k) and 0 ' ( z ) = 0 (z) + c . I f [ c ] t H * ( M ( k ) ) were i n the l a t t i c e of p o s s i b l e changes (see lemma 2.15) we c o u l d change the l i f t i n g g ' and be f i n i s h e d . I f not 61 w r i t e c = + . . . + c , + c 0 , where each c-, has f i l t r a t i o n i . By i n d u c t i o n one can show that N N ( r - s ) j 0' (z) =0 (z) + N c t + + c , ' + c 0 ' I f N i s such that N[c^] i s i n the l a t t i c e of p o s s i b l e changes we N N(r - s ) can change the l i f t i n g g ' of f to reduce the number of terms i n c . C o n t i n u i n g i n d u c t i v e l y we can f i n d n g :WY(k+l) > WY(k+l) w i t h M(g) = [0 ] as r e q u i r e d . b) Dual P r o o f . 5.3 MAIN THEOREM D e f i n i t i o n 5 . 3 : a) Let F T * ( B , Y , k ) be the subset of FT*(B) d e f i n e d by X t F T * ( B , Y , k ) i f and only i f i ) There i s a l i f t i n g fj :X ——> W Y ( i ) , i < k. i i ) f 0 i s determined by { p " 1 ( b ) } . i i i ) There i s *«Aut'(M(k)) such that M ( f ; ) = [ i n (0 |M( i )) ] , i < k, and 0-1 lowers f i l t r a t i o n s . ( i . e . 0 i s a k-1 r e a l i z e r ) . b) Let C T*(n , Y , k ) be the subset of 'CT*(n) de f ined by XeCT*(n ,Y ,k ) i f and only i f i ) There i s an extens ion f , : W ' Y ( i ) > X , i < k. i i ) f 0 i s determined by { s p " 1 ( © ) } . i i i ) There i s 0 *Aut(L(k)) such that L ( f j ) = [ i n ( 0 | L ( i ) ) ] and 0-1 lowers f i l t r a t i o n s . ( i . e . 0 i s a k-1 r e a l i z e r ) . 62 Lemma 5.4 a) For every k > 0, there i s a f i n i t e subset I (k ) of FT*(B) such that FT*(B) = U F T * ( B , Y , k ) Yf i l (k ) b) For every k > 0, there i s a f i n i t e subset I (k ) of CT*(n) such that CT*(n) = U C T*(n , Y , k ) Y e l ( k ) P r o o f : a) n = 0 F i x Y c F T * ( B ) . Then WY(0) = K(0) and for any X € F T * ( B ) , l e t f : X > WY(0) be the map determined by the cohomology c l a s s e s {*?- 1(b)}. Then i n 1 M < M(0) < M(0) v A* (X) <- f* A*(WY(0)) homotopy commutes, s ince p([vz]) = pz f o r Z € A ( Z 0 ) . Thus FT*(B) = F T * ( B , Y , 0 ) . 63 n = k+1 Let X € F T * ( B ) . Assume that X e F T * ( B , Y , k ) for some Y e l ( k ) . By c o n s t r u c t i o n , WY(k+l) i s the f i b r e WY(k+1) > WY(k) > K(k+1) Since any Pos tn ikov tower can be b u i l t by " k i l l i n g " one element at a t i m e , assume that K(k+1) = K(Z,m) for some m, and a i s the c l a s s k i l l e d . We want to l i f t f : X > WY(k) to f ' : X > WY(k+l) . The element f * ( o ) e H * ( X ; Z ) i s the o b s t r u c t i o n to l i f t i n g f . I f M(k+1) = M ( k ) B A ( z ) , where z i s such that <f * ( [dz]) = q(a) , then f*(q(<*)) = [rr0 (dz) ] i s an element used i n [H&S] for the ( k - l ) t h o b s t r u c t i o n to f o r m a l i t y . S ince e v e r y t h i n g i s formal the o b s t r u c t i o n can be r a t i o n a l l y overcome. However, we need to accompl i sh t h i s i n t e g r a l l y . D e f i n i t i o n 5 . 5 : a) Let C(WY(k)) be the subset {g} of [WY(k),WY(k)] w i t h the p r o p e r t i e s : i ) g i s a homotopy e q u i v a l e n c e . i i ) g i s a l i f t i n g of g ' e C ( W Y ( k -1)). i i i ) M(g) = [ r ] , where y-\ lowers f i l t r a t i o n s . b) Let C(W'Y(k)) be the subset {g} of [W*Y(k) ,W*Y(k) ] w i t h the p r o p e r t i e s : i ) g i s a homotopy e q u i v a l e n c e . i i ) g i s an ex tens ion of g' eC (W' Y ( k- 1 )) . i i i ) L(g) = [ r ] , where r _1 lowers f i l t r a t i o n s . I f h :Y > WY(k) i s the map c o n s t r u c t e d i n s e c t i o n 2, l e t 64 H:C(WY(k)) > B be g iven by H(g) = ph*(g*(o)-c). Note that q(H(g)) = * h * ( [ ( r - 1 ) U d z ) ] = *h$[ *o(a 0 . ) ] = q ( * f * ( g * ( e ) - a ) ) where a o e M ( 0 ) , s ince 7 - 1 lowers f i l t r a t i o n s . Thus H(g) = * f * ( g * ( c ) - a ) ) . D e f i n i t i o n 5 . 6 : a) The o b s t r u c t i o n to X being i n F T * ( B , Y , k + l ) i s de f ined to be 0 ( X , k ) = [fif*(a)]eB/lm(H) b) The o b s t r u c t i o n to X being i n C T*(n , Y , k + l ) i s de f ined to be 0 ' ( X , k ) = [ p s - 1 f * ( r )]€n / I m ( H ' ) where H' i s de f ined ana logous ly to H. 0 ( X , k ) , i f d e f i n e d , w i l l be a f i n i t e element i n B/Im(H). The f o l l o w i n g diagram homotopy commutes i n <t> M < M(k) < M(k) v A* (X) <- A*(WY(k)) f* w i t h 0 a k-1 r e a l i z e r . We can f i n d gcC(WY(k)) w i t h M(g) = [* ] for some p o s i t i v e n by lemma 5 . 2 . Then 65 n q(H(g)) = *h *<r * (U - l ) * ( [ d z ] ) ) = n$h*<y*[a0] = npf*<r*[a 0] = n ^ n * [ a 0 ] = np j r * * * [dz ] = n * f * ( q ( e ) ) In the above c a l c u l a t i o n [ a 0 ] = [ ( * - l ) d z ] w i t h a o e M ( 0 ) . <t>-\ lowers f i l t r a t i o n s so i t i s the i d e n t i t y on M(0 ) . Thus H(g) = npf * (o ) + t o r s i o n . Because a = t a ' , H(g) = n p f * ( c ) and 0 ( X , k ) i s a f i n i t e element. We need Lemma 5.7 a) I f 0 ( X ' , k ) = 0 ( X , k ) , then X ' c F T * ( B , X , k + 1 ) . b) I f 0 ' ( X ' , k ) = 0 * ( X , k ) , then X ' e C T *(n , X , k + 1 ) . The proof of lemma 5.4 i s now completed us ing lemma 5 . 7 , s ince there are only a f i n i t e number of o b s t r u c t i o n c l a s s e s p o s s i b l e . Thus FT*(B) = U F T * ( B , Y , k ) , for some f i n i t e set I ( k ) . Y t l ( k ) P r o o f : ( o f above lemma) a) S ince the o b s t r u c t i o n s both e x i s t , we have f ' : X ' > WY(k) and f : X > WY(k) w i t h the r e q u i r e d p r o p e r t i e s . Let geC(WY(k)) be such that H(g) = 0 f * ( c ) - pf'*(c). Then fit'*(g*(c) - a) = <?f*(a) - 0 f ' * ( c ) 66 0 f ' * g * ( a ) = fit*(a) Thus, wi thout l o s s of g e n e r a l i t y , we may assume that 0 f ' * ( o ) = 0 f * ( c ) . To b u i l d the model space for X , l e t WX(i) = WY(i) for i < k. Because the o b s t r u c t i o n e x i s t s i n <t> M < M(k) < M(k) Tf V A*(X) <- A*(WX(k)) f * homotopy commutes. 0 i s an isomorphism, so i n M <- M(k) v v A* (X) < A*(WX(k)) f * homotopy commutes, w i t h r = e<t>~ 1 and we can cont inue the c o n s t r u c t i o n i n d u c t i v e l y . Suppose that {6} i s " k i l l e d " to get WX(k+l), where i ) q(6) = T * ( [ r d z ] ) , where * * [ d z ] = q ( o ) , and reQ). i i ) 6 = t 6 ' , where t a n n i h i l a t e s the t o r s i o n subgroup of B. Then q 0 f ' * ( 6 ) = fit ' * r * ( [ r d z ] ) = fit1)*[rdz] = 0 f ' * * * ( [ r d z ] + [ a 0 ] ) = r 0 f ' * ( q ( c ) ) + r 0 f ' * f f * [ a o ] 67 = r * f * ( q ( c ) ) + r p f * t f * [ a 0 ] = q*f . * (6) = 0 S ince 6 i s d i v i s i b l e by t,. * (6 ) = 0, and X ' £ F T * ( B , X , k + 1 ) . b) Dual p r o o f . We now can prove Theorem 5.8 a) Let B be a graded, s imply connected, f i n i t e l y generated, f i n i t e d imens iona l Z - a l g e b r a . Then the set FT*(B) = { [X] |Xeobj (CW), H * ( X ; Z ) = B, X i s formal} i s f i n i t e . b) Let n be a graded, connected, f i n i t e l y generated, f i n i t e d imens iona l Z L i e a l g e b r a . Then the set CT*(n) = {[X] |Xeobj (CW) , jr*(nx) = n, X i s coformal} i s f i n i t e . P r o o f : a) Suppose that the dimension of B i s N and BBQ i s n-connected . Let k = [ (N+2)/n]-1 , and WY be the N+2 homology s e c t i o n of WY(k). S ince FT*(B) = U F T * ( B , Y , k ) , we need only Y e l ( k ) show that F T * ( B , Y , k ) i s f i n i t e . Suppose tha t X e F T * ( B , Y , k ) . S ince we can assume that X and Y are N + 1 d imens iona l 68 CW-complexes, by c e l l u l a r approximat ion we can l i f t f and h to f and h ' X > WY(k) < Y f h such that f and h ' are isomorphisms on r a t i o n a l cohomology. The orders of the cokerne l s of the homomorphisms induced by f and h ' (on i n t e g r a l cohomology) are bounded by the order of B/<{b}> and the orders of t h e i r k e r n e l s are-bounded by the order of the t o r s i o n subgroup of WY. Each of these orders i s f i n i t e and independent of X . Thus we have bounded d i s t c (X ,WY) and d i s t c ( Y , W Y ) and by c o r o l l a r y 4.7 b) the set F T * ( B , Y , k ) i s f i n i t e . b) Dual proof to a ) . The f o l l o w i n g example i l l u s t r a t e s the o b s t r u c t i o n s de f ined i n t h i s c h a p t e r . Example 5.9 For p a f i x e d pr ime , and for k = 0, p - 1 , l e t X(k) = ( S 2 v S 2 v S 3 ) u e 5 , where the 5 c e l l i s a t tached by the map p [ i 2 r j ] + k [ i 2 , [ i 2 i i 1 ] ] ( i i and i 2 are the i n c l u s i o n s S 2 > S 2 v S 2 v S 3 and j i s the i n c l u s i o n S 3 > S 2 v S 2 v S 3 ) . I f a , b , c and x are the cohomology c l a s s e s corresponding to the c e l l s of X ( k ) , then each space has the cohomology a lgebra H * ( X ( k ) ; Z ) = Z [ a , b , c , x ] / < a 2 , a b , a c , a x , b 2 , p b c - x > 69 where Z [ a , b , c , x ] i s the graded commutative Z-algebra generated by the elements a , b, c , and x . They are a l l f o r m a l , because the d i f f e r e n t i a l i n the minimal L i e a lgebra model i s quadra t i c for the set of genera tor s , { s " ' a , s " 1 b , s " 1 c + ( k / p ) [ s - 1 b , s " ' a ] , s ~ 1 x } , even though the Massey product <a,b,b> i s n o n - t r i v i a l i n t e g r a l l y i n X(k) for k > 1 ( i t i s t r i v i a l r a t i o n a l l y ) . I f WX i s the model space cons t ruc ted for X ( 0 ) , then WX(0) = K ( Z , 2 ) x K ( Z , 2 ) x K ( Z , 3 ) . The f i r s t map f : X ( 0 ) > WX(0) w i l l be given by the cohomology c l a s s e s a , b, and c . Suppose that we have i n d u c t i v e l y cons t ruc ted f : X ( 0 ) > WX(1) and h :X(k ) > WX(1), so t h a t , by d e f i n i t i o n 5 . 3 , X ( k ) e F T * ( H * ( X ( 0 ) ; Z ) , X ( 0 ) ,1 ) . Then f*(<a,b,b>) i s t r i v i a l but h*(<a,b,b>) i s not t r i v i a l . S ince the Massey product i s k i l l e d to o b t a i n WX(2), the o b s t r u c t i o n i s non-zero and we need to b u i l d new model spaces. 70 Chapter 6 INFINITE AMBIGUITY The prev ious chapter showed t h a t , for formal spaces, the cohomology f u n c t o r , H * , i s f i n i t e l y ambiguous. Of course , H* i s not always f i n i t e l y ambiguous. The spaces W(n) of example 1.5 comprise an i n f i n i t e set of CW complexes w i t h isomorphic cohomology a lgebras and d i f f e r e n t homotopy t y p e s . Thus H * i s i n f i n i t e l y ambiguous for the a lgebra H * ( W ( n ) ; Z ) . By Lemma 3.18, the spaces W(n) are non- forma l . The f o l l o w i n g p r o p o s i t i o n prov ides a p a r t i a l converse for theorem 5 .8 . P r o p o s i t i o n 6.1 a) Suppose that X i s a s imply connected, non- formal , f i n i t e CW complex such that H*(X;<B) = 0, unless * = 0 or r < * < m, and that the m-r-1 homology s e c t i o n of X i s f o r m a l . Then there are an i n f i n i t e number of spaces X(n) w i t h H * ( X ( n ) ; Z ) = H * ( X ; Z ) as graded a l gebra s , the same r a t i o n a l homotopy type as X , and a l l d i f f e r e n t homotopy t y p e s . b) Suppose that X i s a s imply connected, non-coformal space whose Pos tn ikov tower has f i n i t e type . Then there i s a Pos tn ikov s e c t i o n X ' of X , such that there are an i n f i n i t e number of spaces X ' ( n ) w i t h Samuelson a lgebras isomorphic to the Samuelson a lgebra of X ' , the same r a t i o n a l homotopy type of X ' and a l l d i f f e r e n t homotopy types . To prove par t a) of t h i s p r o p o s i t i o n , the spaces X(n) w i l l be c o n s t r u c t e d i n such a way that we can appeal to the f o l l o w i n g lemma to get the r e s u l t . 71 Lemma 6.2 a) Let X and Y be s imply connected, non-formal CW complexes such that H * ( X ; Z ) = H*(Y;Z) as graded groups and that there i s a map f : X > Y w i t h f* = t *1 :H*(Y ;Z) > H* (X ;Z) where t * 0 ,±1 i s an in teger r e l a t i v e l y prime to the order of any f i n i t e t o r s i o n subgroup of H * ( X ; Z ) . Then i ) H * ( X ; Z ) = .H*(Y;Z) as graded a l g e b r a s . i i ) X and Y have the same r a t i o n a l homotopy t y p e . i i i ) X and Y have d i f f e r e n t homotopy types . b) Let X and Y be s imply connected non-coformal CW complexes such that fr*(X) = rr*-(Y) as graded groups and that there i s a map f : X > Y w i t h f * = t * - 1 1 : » r * ( X ) > ir*(Y) where t 4- 0 ,±1 i s an in teger r e l a t i v e l y prime to the order of any f i n i t e t o r s i o n subgroup of tr * (X) . Then i ) tr*(nx) = ir*(nY) as graded L i e a l g e b r a s . i i ) X and Y have the same r a t i o n a l homotopy t y p e . i i i ) X and Y have d i f f e r e n t homotopy types . P r o o f : a) i ) f* = t *1 means that there i s an isomorphism 1:H*(Y;Z) > H* (X ;Z) of cohomology groups, and that |y| f * ( y ) = t 1<y) for y tH*(Y ;Z ) . Now 72 I xy I t l ( x y ) = f * ( x y ) = f * ( x ) f * ( y ) | x | | y | = t 1(x) t 1(y) xy = t 1(X)1(y) and, s ince t i s r e l a t i v e l y prime to the order of any t o r s i o n subgroup of H * ( X ; Z ) , l ( x y ) = l ( x ) l ( y ) . Thus the map 1 i s an isomorphism of a l g e b r a s . i i ) f * : H * ( Y ; @ ) > H*(X;Q) i s an isomorphism and thus f i s a r a t i o n a l homotopy e q u i v a l e n c e , s ince X and Y are s imply connected . i i i ) Suppose that X were homotopic to Y, i . e . there i s a homotopy equiva lence h :Y > X . Then the composite h f : Y > Y i s such that ( h f ) * = ( t *1 ) h * : H * ( Y ; Q ) > H*(Y;f£) . h * i s a n i s o t r o p i c , because i t can be i n v e r t e d i n t e g r a l l y . Then the corresponding map * :MY > MY i s such that <t>* = ( t * 1 ) (a* )' 1 h*c* and s i n c e (a*)'xh*a* i s a l s o a n i s o t r o p i c , Y i s formal by Theorem 3.1 e ) . Thi s i s a c o n t r a d i c t i o n , so X and Y must have d i f f e r e n t homotopy types , b) Dual proof to a ) . P r o o f : (of p r o p o s i t i o n 6.1) Let the minimal L i e model for X be L which can be assumed to be such that d r e s t r i c t e d to L m . r . ! has no p e r t u r b a t i o n , i . e . i s q u a d r a t i c . Let * be the isomorphism H*(L) > Tr^nXjBfl). The proof w i l l c o n s i s t of three p a r t s . 73 F i r s t , as i n chapter 5, we i n d u c t i v e l y c o n s t r u c t a model space, W(1,m), for X which has c e l l s i n one-to-one correspondence w i t h the generators of L and which i s r connected. S ince X i s not f o r m a l , i n d u c t i o n must be done on the t o p o l o g i c a l degree. Next we prove the p r o p o s i t i o n for W(1,m) and then use that r e s u l t to prove the p r o p o s i t i o n for X. n = 0 Let W(1,0) be a p o i n t , and h : W ( l , 0 ) > X be the basepoint map. n = k Suppose that we have c o n s t r u c t e d W(1 , k-1) w i t h minimal L i e a lgebra model L k . , , and a map h : W ( 1 , k -1 ) > X such that the corresponding map L k . •, > L i s the i n c l u s i o n . Then the f o l l o w i n g diagram commutes. i n * H * ( L k . 1 ) > H*(L) »r*(nW( 1 ,k'-1 ) ) m > Jr*-(nX)B(e h * Without l o s s of g e n e r a l i t y , assume that L k = L k . , v L ( z ) . I f dz = 0, l e t W ( l , k ) = W( 1 ,k-1 ) v S w + 1 . Let ecir*.(X) be such that q ( s " 1 © ) = * ( [ r d z ] ) for some r a t i o n a l number r . Def ine h ' : W d , k ) > X by l e t t i n g h ' r e s t r i c t e d to W ( l , k - 1 ) be h and h ' r e s t r i c t e d to S * + 1 be 9. The i n d u c t i o n c o n t i n u e s . I f dz * 0, the [dz] represents a c l a s s i n L K . , . We can f i n d eeir*(W( 1 ,k-1 )) such that 74 i ) q ( s" 16) = * ( [ r d z ] ) , for some re®. i i ) 9 = i e ' , where i a n n i h i l a t e s the t o r s i o n subgroups of both IT k ( W ( 1 , k-1 ) ) and r r k ( X ) . Let W ( l , k ) be the c o f i b r e of 9 : k e S > W ( 1 , k - 1 ) > W ( l , k ) By c o n d i t i o n i i ) , he = 0eir ( X ) , and so there i s an extens ion h ' : W ( 1 , k ) > X . The corresponding map * : L > L w i l l be such that c r e s t r i c t e d to L ^ . , i s the i n c l u s i o n , and e(z) = z+c, for some c y c l e c . I f we were c a r e f u l enough to f i r s t add the generators w i t h zero d i f f e r e n t i a l , so that ff*:H*(L>.) > H*(L) i s onto i n dimension k, we can assume that c e L k , and there i s a r i g h t inverse T : L ^ > L k to c. I f one rep laces />:Lk > £ ( ( M W ( 1 , k ) ) * ) w i t h pr , the map corresponding to h ' w i l l be the i n c l u s i o n and the i n d u c t i o n c o n t i n u e s , u n t i l we get W( 1,m). In L the generators w i t h dimension between m-r-1 and m have d i f f e r e n t i a l s which can be w r i t t e n as dz = c 2 + . . . + C j , where c; e L m . r . , i s a sum of products of i genera tor s . S ince the d i f f e r e n t i a l i s quadra t i c i n L m . r . , , each c ; i s a c y c l e , and there i s a non-zero in teger N such that N s * [ c ; ] i s i n the image of q for a l l i and a l l z . Let p be a prime which does not d i v i d e N and the orders of any of the t o r s i o n subgroups of i ) r r * ( W ( 1 ,k) ) , k < m i i ) r r * ( X ) 75 i i i ) cokerne l (h * :H* (W(1 ,m) ) > H*(X) ) i n a l l dimensions up to and i n c l u d i n g m. There are an i n f i n i t e number of such cho ices for p . We c l a i m that there i s a map f (p ) :W(1 ,m-r-1 ) > W ( l , m - r - l ) which induces , on cohomology, the grad ing homomorphism t * 1 , where t i s a power of p . S ince the space i s formal t h i s map i s very s i m i l a r to the map of theorem 3.1 n) which i s c o n s t r u c t e d i n [ S h i ] . S ince he i s not c a r e f u l enough i n choosing h i s maps (see example 7 . 1 ) , the c o n s t r u c t i o n i s g iven below. n = 0 There i s only one map from a p o i n t to i t s e l f and i t t r i v i a l l y s a t i s i f i e s the c o n d i t i o n s . n = k Suppose that we have i n d u c t i v e l y cons t ruc ted f (p,k-1 ) :W( 1 ,k-1 ) > W d , k - 1 ) which s a t i s f i e s i ) f ( p , k - 1 ) * = t * 1 : H * ( W ( 1 , k - 1 ) ; Z ) > H* (W(1 ,k-1 ) ;Z ) where t i s some power of p . i i ) The map corresponding to f ( p , k - l ) i s $ : L k . , > L k . , | x ] +1 where, for x a generator of Ly,. , , * ( x ) = t x . Without l o s s of g e n e r a l i t y , assume that L k = L k . , v L ( z ) . I f dz = 0, de f ine f ( p , k ) : W ( 1 , k ) > W ( l , k ) to be f ( p , k - 1 ) k+1 k+1 when r e s t r i c t e d to W ( l , k - 1 ) and t on S I f dz * 0, then W ( l , k - 1 ) i s the c o f i b r e of © where q ( s ~ 1 e ) = 4>([dz]). S ince there i s no p e r t u r b a t i o n , dz = Eab. 76 The f o l l o w i n g diagram commutes 0* H * ( L k . •, ) > H * ( L k . , ) •IT*(nw( 1 , k-1 ) )HQ > rr*(nw( 1 , k-1 ) ) 8 $ f ( p , k - 1 > » k+1 and so f ( p , k - 1 ) * (9 ) = t e, s ince © i s d i v i s i b l e by enough to e l i m i n a t e t o r s i o n . Hence there i s a l i f t i n g f ' : W ( 1 , k ) > W ( l , k ) . The corresponding map 0' :L j , > L k i s 0 k+1 when r e s t r i c t e d to L k . , , and 0'(z) = t z+c where c i s some c y c l e . As i n the c o n s t r u c t i o n of W ( l , k ) we may assume that ceL fc . i . Us ing the technique i n the proof of m) => n) of theorem 3 .3 , there i s a power of the map f which can be changed to a l i f t i n g f ( p , k ) : W ( 1 , k ) > W ( l , k ) of the same power of f ( p , k - 1 ) which s a t i s f i e s the i n d u c t i o n hypotheses . Thus for an i n f i n i t e set of pr imes , {p}, there are maps f (p ,m-r -1 ) :W(1 ,m-r -1 ) > W(l ,m~r-1) which induce the grading homomorphism t * 1 , w i t h t a power of the g iven prime on cohomology. W(1,k) for k > m-r-1 can not be assumed to be f o r m a l , so new CW complexes W(p,k) need to be c o n s t r u c t e d i n order to have a map f ( p , k ) : W ( 1 , k ) > W(p,k) which induces a grading homomorphism t * 1 , w i t h t a power of p , on cohomology. Let W(p,m-r-1) = W(1,m-r-1) and suppose that we have i n d u c t i v e l y cons t ruc ted f ( p , k - 1 ) : W ( 1 , k - 1 ) > W(p,k-1) which induces the grad ing homomorphism t *1 on cohomology. Assume that L k - L k - 1 v L ( z ) • k+1 I f dz = 0, de f ine W(p,k) to be W ( p , k - l ) v S and f ( p , k ) to 77 k+1 be f(p,k-1> when r e s t r i c t e d to W(p,k-1) and t when r e s t r i c t e d k+1 to S I f dz 0, then dz = c 2 + . . . + C j , where c ; i s a sum of products of i genera tor s . W ( l , k ) i s the c o f i b r e of ©, where q ( s " 1 e) = * ( [ d z ] ) : k e S > W ( l , k-1) > W ( l , k ) k + 1 To get W ( p , k ) , we need to f i n d ctir k (W(p, k-1 )) so that the f o l l o w i n g diagram homotopy commutes. © k -> w( 1 , k - l ) f ( p , k - 1 ) f S > W(p,k-1) a Then, i f W(p,k) i s de f ined to be the c o f i b r e of a, there w i l l be an ex tens ion f ( p , k ) : W (1 , k ) > W(p,k) as r e q u i r e d . S ince W(1,k-1) and W(p,k-1) are r connected, there are isomorphisms nk(W( 1 ,m-r- 1 ) ) = irk(W( 1 , k-1 )) and ffk (W( 1 ,m-r-1 )) = ir k (W(p, k-1 )) induced by the i n c l u s i o n s . Let ©' correspond to © under t h i s isomorphism. S ince d z e L m . r . 1 f q ( s " 1 © ' ) = * ( [ d z ] ) . N was de f ined to be such that N s * ( [ c ; ] ) € l m ( q ) for a l l i , so 78 q ( s " ' f ( p , m - r - 1 ) * ( N e ' ) ) = f (p ,m-r-1 ) .# (N[dz] ) = • ( N [ * d z ] ) k+1 j - 2 = t * (N[c +tc +...+t c ]) 2 3 j k+1 = t q ( s - ' o " ) for some o " en k ( W ( p , m - r - 1 ) ) . k+1 Thus f ( p , m - r - 1 ) * ( N © ' ) = t a " + t o r s i o n . S ince t i s a power of p, where p was chosen so that i t does not d i v i d e the order of the t o r s i o n subgroup of ir k(W( 1 ,m-r-1 )) , we may assume that o " k+1 was chosen so that f ( p , m - r - 1 ) * ( N e ) = t o " . S ince p does not k+1 d i v i d e N , There are i n t e g e r s a and b such that aN+bt = 1. k+1 Then f ( p , m - r - 1 ) * ( e ) = t a ' , where c ' = a a " + b f ( p , m - r - 1 ) . ( © ) . The corresponding ceiry- (W(p, k-1 ) ) i s the map that we want. Continue i n d u c t i v e l y u n t i l the map f(p,m):W(1,m) > W(p,m) i s c o n s t r u c t e d . On cohomology, f ( p , m ) * i s the grad ing homomorphism t * 1. By lemma 6 .2 , W(1,m) and W(p,m) have i somorphic cohomology a lgebras and d i f f e r e n t homotopy t y p e s . T h i s c o n s t r u c t i o n can be repeated , s t a r t i n g w i t h W(p,m), to produce W(p'p,m) where p' i s a prime d i f f e r e n t from p . C o n t i n u i n g i n d u c t i v e l y , the proceedure generates an i n f i n i t e set of spaces (W(n,m)} w i t h maps W(n,m) > W(n ' ,m) , when n < n ' , which induce a grading homomorphism on cohomology and hence, by lemma 6 . 2 , have isomorphic cohomology a lgebras and d i f f e r e n t homotopy t y p e s . 79 Let X(p) be the pushout W(1,m) h v X -> W(p,m) h ' i -> X(p) By the M a y e r - V i e t o r i s theorem, H * ( X ; Z ) = H * ( X ( p ) ; Z ) s ince p does not d i v i d e the order of the c o k e r n e l of h . By the U n i v e r s a l C o e f f i c i e n t theorem and lemma 6 .2 , X and X(p) have isomorphic cohomology a lgebras and d i f f e r e n t homotopy types . Let X (p 'p ) be the pushout W(p,m) > W(p'p,m) X(p) > X(p 'p ) and cont inue i n d u c t i v e l y to generate on i n f i n i t e set of CW complexes w i t h isomorphic cohomology a lgebras and d i f f e r e n t homotopy types . Remark: I f the minimal model for W(1,m) i s ( L ( d ) , then the minimal model for W(p,m) i s ( L , $ d 0 ~ 1 ) , where * : L > L i s g iven | z | + 1 on generators by * ( z ) = t z , w i t h t a f i x e d power of p . Th i s i s an example where the change i n d i f f e r e n t i a l does not change the r a t i o n a l homotopy type ( t h i s i s d i scus sed i n [L&S]) but i t does change the homotopy t y p e . To prove p r o p o s i t i o n 6.1 b ) , we t r y to mimic the map 0 : ( M , 0 " ' d 0 ) > (M,d) , where (M,d) i s the minimal a lgebra model 80 | z | -1 for X , and <t> i s g iven on generators by <t>(z) = t z , w i t h t some power of a prime p . U n f o r t u n a t e l y , we cannot d u a l i z e the technique used i n the proof of 6.1 a ) . Thi s would i n v o l v e r e p l a c i n g X w i t h a s impler space whose homotopy i s t o r s i o n free and then us ing a p u l l b a c k argument, but the map <t> i s i n the wrong d i r e c t i o n for t h i s to work. Th i s i s another example where one can not s imply d u a l i z e a p r o o f . T h i s i s the reason why, i n the statement of p r o p o s i t i o n 6.1 b ) , we only have t h a t , for every non-formal X , there i s a Pos tn ikov s e c t i o n of X for which the Samuelson a lgebra i s not f i n i t e l y ambiguous. Let X ' be the kth Pos tn ikov s e c t i o n of X , where k i s chosen so that X ' i n non-coformal and the ( k - 1 ) t h Pos tn ikov s e c t i o n , X " , i s c o f o r m a l . Suppose that the generators of M have been chosen so that the d i f f e r e n t i a l r e s t r i c t e d to M ^ . , has no p e r t u r b a t i o n . Then, for an i n f i n i t e set of primes {p}, us ing a s i m i l a r c o n s t r u c t i o n to that g iven i n the proof of theorem 3.3 m) => n ) , there i s a map f : X " > X" such that i ) f induces the grading homomorphism, t * " 1 1 , on homotopy, where t i s a power of p, and i i ) The diagram 81 M k . , < M k . , v v A * ( X " ) < A * ( X " ) f * homotopy commutes. Suppose that X ' i s the f i b r e of © © X' > X" > K(n,k+1) To complete the c o n s t r u c t i o n , we need to f i n d ©' :X" ——> K(n,k+1) so that the f o l l o w i n g diagram homotopy commutes. X " v X" © -> K (n ,k+1 ) —> K(n,k+1) © ' Then, l e t X ' ( p ) be the f i b r e of ©' , and the induced map g : X ' > X ' ( p ) s a t i s f i e s the requirements . Assuming s u i t a b l e r e s t r i c t i o n s on p, t h i s can be done by the dua l to the method of choos ing o ' i n the proof of p r o p o s i t i o n 6.1 a ) . 82 Chapter 7 CONCLUSION T h i s t h e s i s leaves open the ques t ion of the l o c a t i o n of the exact d i v i d i n g l i n e between f i n i t e and i n f i n i t e a m b i g u i t y . S ince cohomology i s f i n i t e l y ambiguous for formal spaces , and for every non-formal CW complex, there i s a s k e l e t o n for which cohomology i s i n f i n i t e l y ambiguous, i t seems reasonable to c o n j e c t u r e that cohomology i s f i n i t e l y ambiguous for formal spaces, and i n f i n i t e l y ambiguous for non-formal spaces. One way to extend the r e s u l t s . o f chapter 6 would be to show t h a t , u s ing s u f f i c i e n t c a r e , the i n d u c t i v e c o n s t r u c t i o n of the spaces W(p,m) c o u l d be c o n t i n u e d . Consider the f o l l o w i n g example. Example 7.1 Suppose that W(1,5) was the space S 2 v ( S 2 x S 3 ) and that we had i n d u c t i v e l y de f ined g : S 2 v S 2 v S 3 > S 2 v S 2 v S 3 to be the map < p 2 i , , p 2 i 2 , p 3 j + [ i , , i , ] > , where i , and i 2 are the two i n c l u s i o n s S 2 > S 2 v S 2 v S 3 and j i s the i n c l u s i o n S 3 > S 2 v S 2 v S 3 and p i s some pr ime. g induces the grading homomorphism, p * 1 , on cohomology, but , s ince 9*(C i 2 r j 1 ) = P 5 [ i 2 , j ] + p 2 [ i 2 , [ i 1 , i , ] ] there i s no way to a t t a c h a 5 c e l l to S 2 v S 2 v S 3 i n order to be able to extend g to a map h : S 2 v ( S 2 x S 3 ) > S 2 v S 2 v S 3 w e 5 which induces a grading homomorphism on cohomology. Thus, before we can get W(p,m), g has to be r e p l a c e d . Th i s example i s one that 83 S h i g a ' s method [Sh i ] does not h a n d l e . The example shows t h a t , i n a d d i t i o n to knowing that the map induces a grad ing homomorphism on cohomology, we a l s o need to know the induced map on homotopy. For formal spaces the maps are from a space to i t s e l f , and we can get enough i n f o r m a t i o n from the corresponding automorphism of the minimal L i e model . Once we are no longer d e a l i n g w i t h f o r m a l i t y , the spaces are d i f f e r e n t , and there i s no longer a s t rong enough c o n n e c t i o n . I t might be u s e f u l to get an a c t u a l bound on the number of formal homotopy types which have cohomology a lgebras isomorphic to a g iven a l g e b r a . By l o o k i n g c a r e f u l l y at the c o n s t r u c t i o n s g iven i n t h i s t h e s i s , one c o u l d come up w i t h some s o r t of bound, but the methods used are too c rude , and the r e s u l t i n g bound would be too poor to j u s t i f y the amount of work i n v o l v e d . Examples 1.3 and 5.9 show t o r s i o n as one reason for g e t t i n g d i f f e r e n t homotopy types . In g e n e r a l , such t o r s i o n i s d i f f i c u l t to determine , and that would h inder any e f f o r t s to o b t a i n a p r e c i s e bound. 84 BIBLIOGRAPHY [B&L] - H . J . Baues and J . M . Lemaire , "Min ima l Models i n Homotopy T h e o r y " , M a t h . Ann. 225 (1977), pp 219-242. [Bau] - H . J . Baues, " R a t i o n a l Homotopietypen", Manuscr ipta Math. 20 (1977) no. 2, pp 119-131. [B&D] - R. Body and R.R. Douglas , "Homotopy Types W i t h i n a R a t i o n a l Homotopy Type" , Topology 13 (1974), pp 209-214. [Bod] - R. Body, "Regular R a t i o n a l Homotopy Types" , Comment. Math. H e l v . 50 (1975), pp 89-92. [C&D] - C. C u r j e l and R.R. Douglas , "On S t a s h e f f ' s F i f t h Prob lem" , N o t i c e s of the AMS 18:5 (Aug.1971), pg 787. [DGMS] - P . D e l i g n e , P. G r i f f i t h s , J . Morgan, and D. S u l l i v a n , "Rea l Homotopy Theory of Kahler M a n i f o l d s " , Invent iones Math. 29 (1975), pp 245-274. [Dou] - R .R. Douglas , " P o s i t i v e Weight Homotopy Types" , I l l i n o i s J o u r n a l on M a t h . , to appear. [G&M] - P . A . G r i f f i t h s and J .W. Morgan, " R a t i o n a l Homotopy Theory and D i f f e r e n t i a l Forms" , B i rkhauser 1981. [GHV] - W.H. Greub, S. H a l p e r i n , and J . R . Vanstone, "Connect ions , Curvature and Cohomology", V o l I I I , Academic Pre s s , N . Y . , 1 9 7 6 [H&S] - S. H a l p e r i n and J . D . 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Q u i l l e n , "Homotopical A l g e b r a " , Lec ture Notes i n Math . no. 43, Spr inger V e r l a g , 1967. [ Sh i ] - H . S h i g a , " R a t i o n a l Homotopy Type and S e l f Maps", Jour of the Math. Soc. of Japan 31 (1979) no. 3, pp 427-434. [Sta] - J . D . S t a she f f , "H-space Problems" , H-spaces, Neuchatel ( S u i s s e ) , Lec ture Notes i n Math. no. 196, Spr inger V e r l a g , pp 122-135. [Sul ] - D. S u l l i v a n , " I n f i n i t e s i m a l Computations in Topology" , P u b l . I . H . E . S . , v o l . 47 (1978), pp 269-331. 

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