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Sullivan’s theory of minimal models Deschner, Alan Joseph 1976

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SULLIVAN'S THEORY OF MINIMAL MODELS by ALAN JOSEPH DESCHNER B.Sc, University of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of MATHEMATICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1976 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of this thesis for f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of Mathematics The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1WS-Date May 4 , 1 9 7 6 Supervisor: Dr. Roy R. Douglas Abstract: For a s i m p l i c i a l complex K, the de Rham algebra E (K) i s the d i f f e r e n t i a l graded algebra (DGA) of Q-coefficient polynomial forms i n the barycentric coordinates of the simplices of K which agree as d i f f e r e n t i a l forms on common faces. The associated de Rham cohomology algebra i s isomorphic to the s i m p l i c i a l cohomology of K with Q- c o e f f i c i e n t s by in t e g r a t i o n of forms over simplices. Given a 1-connected DGA, A, the minimal model of A i s a DGA, M, which i s free as an algebra, has a d i f f e r e n t i a l which decomposes the generators, and which computes the cohomology of A . Such minimal models e x i s t and are unique up to isomorphism. The minimal model M(X) of a 1-connected s i m p l i c i a l complex * X i s the minimal model of E (X) . I t depends only on the' r a t i o n a l homotopy type of X . For a f i b r a t i o n K(ir,n) -— > E — > Y, with E and Y 1-connected, we have (under mild hypothesis) M(E) = M(Y) 8 H*(K(TT,n) ;Q) with a s u i t a b l y defined d i f f e r e n t i a l . This i s applied i n d u c t i v e l y to the Postnikov decomposition of X to show that the free generators of M(X) correspond to the generators of ^ C X ) 8 Q . The number of these generators which are cocycles i s the rank of the r a t i o n a l Hurewicz homomorphism. - i i -Table of Contents Page Introduction i v Chapter 0 : Algebraic Preliminaries 1 Chapter 1 : De Rham's Theorem for S i m p l i c i a l Complexes 6 Chapter 2 : The Minimal Model 34 Chapter 3 : The Minimal Model i n Rational Homotopy Theory 54 Bibliography 84 - i i i -Acknowledgments The author thanks h i s supervisor, Dr. Roy R. Douglas, whose • help and encouragement has made th i s thesis possible. The author also thanks the Uni v e r s i t y of B r i t i s h Columbia and the National Research Council of Canada for t h e i r f i n a n c i a l support (1973-1975). - i v -Introduction This thesis i s a presentation of Dennis Sullivan's theory of minimal models of r a t i o n a l homotopy type. Our purpose i s to give a comprehensive development of the basic theory i n the simply connected case as f i r s t outlined by S u l l i v a n i n [13], and to prove Theorems A and B of that paper. Further r e s u l t s i n the theory and applications to manifolds can be found i n [1]. Chapter 1 develops rational'de Rham theory for s i m p l i c i a l complexes. The c l a s s i c a l theory considers the d i f f e r e n t i a l graded algebra (DGA) of smooth d i f f e r e n t i a l forms on a manifold, and shows that the r e s u l t i n g cohomology agrees with s i m p l i c i a l theory. We achieve the gene r a l i z a t i o n to s i m p l i c i a l complexes and r a t i o n a l c o e f f i c i e n t s by using r a t i o n a l c o e f f i c i e n t polynomial forms defined on the various simplices i n the space, and requ i r i n g that they "patch together" along common faces. The c o l l e c t i o n of such forms i s a DGA whose cohomology algebra i s na t u r a l l y isomorphic by int e g r a t i o n to s i m p l i c i a l cohomology with r a t i o n a l c o e f f i c i e n t s . The proof of t h i s equivalence, which occupies the major part of the chapter, i s e s s e n t i a l l y the same as that of the c l a s s i c a l de Rham theorem as presented by H. Whitney [15]. One advantage of using d i f f e r e n t i a l forms rather than s i m p l i c i a l cochains i s that the wedge product of forms i s graded commutative, whereas the cup product of cochains i s not. This property i s needed i n Chapter 3. In Chapter 2 we discuss Sullivan's minimal models from a purely algebraic point of view. For a given simply connected DGA we construct i t s minimal model, a DGA which i s free as a graded algebra, has a - V -decomposable d i f f e r e n t i a l , and which computes the cohomology of the o r i g i n a l DGA . The purpose of th i s chapter i s to prove the existence and uniqueness of the minimal model. To th i s end we must study induced maps of minimal models, and t h i s leads to a discussion of homotopy i n the category of DGA's . The uniqueness of the minimal model i s e s s e n t i a l i n Chapter 3. Chapter 3, the main chapter i n the th e s i s , r e l a t e s the r a t i o n a l homotopy theory of a space to the algebraic construction of Chapter 2. The minimal model of a simply connected, s i m p l i c i a l complex i s defined to be the minimal model of the r a t i o n a l de Rham algebra of the complex; i t depends only on the r a t i o n a l homotopy type of the space (Theorem 3.4). The construction of the model p a r a l l e l s the Postnikov decomposition of the space i n such a way that the algebra generators of degree n correspond to the generators of ^(X) ® >^ a n d the d i f f e r e n t i a l s of these generators correspond to the r a t i o n a l k-invariants i n the decomposition. The proof i s based on the Guy Hirsch method for computing the cohomology of a p r i n c i p a l " K ( i r , n ) - f i b r a t i o n , which i s e s s e n t i a l l y our Theorem 3.9 . This method of attack was suggested by Rend Thorn i n the Cartan Seminar, 1954, and our proof i s a corrected version of that given by Dennis S u l l i v a n and Roy Douglas i n the summer of 1975. Our treatment stops short of the complete r e s u l t [13: Theorem C], which states that the r a t i o n a l homotopy type of a simply connected space i s uniquely determined by i t s minimal model. The main technique used for t h i s r e s u l t i s the Hirsch method, which we present i n d e t a i l . Chapter 0 A l g e b r a i c P r e l i m i n a r i e s In t h i s chapter, we define the a l g e b r a i c objects which w i l l be used i n the remainder of the t h e s i s . 0.1 D i f f e r e n t i a l Graded Algebras and Cohomology: Let Q denote the r a t i o n a l numbers. By a graded algebra, A , over Q we mean a graded Q-vector space A = © A n together w i t h an a s s o c i a t i v e n>0 m u l t i p l i c a t i o n u : A ® A •+ A which i s graded (u(A n8A m) A n + m ) and graded commutative (a*b = (-l)™ b*a when a e A H and b e A™) We a l s o assume, unless otherwise s t a t e d , that A has an i d e n t i t y 0 n element 1 £ A . The elements of A are s a i d to be homogeneous of degree n (or dimension n) . A d i f f e r e n t i a l graded algebra (or DGA) i s a graded algebra A , together w i t h a d i f f e r e n t i a l , d , of degree +1 , which i s a d e r i v a t i o n . This means that f o r each n there i s a vector space homomorphism d = d n : A n -* A n + 1 s a t i s f y i n g d°d = 0 ( d i f f e r e n t i a l ) and d(a-b) = d(a)-b + ( - l ) n a-d(b) f o r a £ A n ( d e r i v a t i o n ) . I f A i s a DGA , l e t Z n(A) = Ker{d : A n -* An+"'"} = subspace of cocycles of A U , B (A) = Im{d : A -> A } = sub space of coboundaries of A Z*(A) = © Z n(A) , B*(A) = « B n(A) . n>0 n>_0 As d = 0 , we have B (A) «= Z (A) . Def ine the n cohomology - 2 -space of A to be the quotient v e c t o r space Z n ( A ) / B n ( A ) , which we n * denote by H (A) As d i s a d e r i v a t i o n , we see that Z (A) i s a subalgebra of A , and B (A) i s an i d e a l i n Z (A) . Hence H*(A) = ©' H n(A) = Z*(A)/B*(A) i s a graded algebra (with i d e n t i t y n>0 i f A i s ) , c a l l e d the cohomology algebra of A . We say that the DGA , A , i s connected i f H^(A) = Q , and that A i s simply connected i f i t i s connected and H^(A) = 0 . We w i l l be mainly concerned w i t h simply connected DGA's . I f A and B are graded algebras, a f u n c t i o n f : A •+ B i s a homomorphism i f i t preserves a l l the a l g e b r a i c s t r u c t u r e ; that i s , f ( A N ) < = B N , f(a+b) = f(a) + f(b) , and f(a-b) = f ( a ) - f ( b ) . We al s o assume that f ( l ) = 1 . I f A and B are DGA's, we r e q u i r e a l s o that f commute w i t h the d i f f e r e n t i a l s ; f°d. = d °f . A B I f f : A -> B i s a DGA homomorphism, then f induces a map f'' : H*(A) -> H'{(B) by the r u l e f ( [ z ] ) = [ f ( z ) ] , where [z] denotes the cohomology c l a s s of the element z e Z (A) . C l e a r l y f i s a homomorphism of graded algebras. So we have categ o r i e s GA and VGA of graded and d i f f e r e n t i a l graded algebras, r e s p e c t i v e l y , and the cohomology funct o r H* : VGA + GA . 0.2 Tensor Products and Free Algebras: I f A and B are objects i n GA , we may form t h e i r tensor product A 8 B which, as a graded n vec t o r space, i s (A8B) n = ® (A 1 8 B N ^) , the tensor product on the i=0 - 3 -r i g h t being the usual one f o r vect o r spaces. We define the m u l t i p l i c a t i o n i n A 8 B by (a8b) • (a'8b') = (-l)™ (a-a') 8 (b-b') when b e B n and a' e A™ . One v e r i f i e s that A 8 B i s a graded algebra. Note that A 8 B i s the coproduct of A and B i n the category GA . To see t h i s , note that there are ca n o n i c a l " i n c l u s i o n s " A > A 8 B by a \ > a 8 1 , and B > A 8 B by b I > 1 8 b . Now given any diagram w i t h C , f , g a r b i t r a r y i n GA A A 8 B h => C there i s a unique h : A 8 B C i n G A given by h( £ a. 8 b.) = £ f ( a . ) * g ( b . ) which makes the t r i a n g l e s commute, i i I f A and B are DGA's , we can define a d i f f e r e n t i a l i n the graded algebra A S B as f o l l o w s : f o r a e A n and b e B> , set d(a8b) = d(a) 8 b + ( - l ) n a 8 d(b) , and extend by l i n e a r i t y . One checks that t h i s makes A 8 B a DGA, which i s the coproduct i n the category VGA . However, we w i l l u s u a l l y consider A 8 B as a graded al g e b r a , and define d i f f e r e n t i a l s d i f f e r e n t from the one above. - 4 -The f r e e graded algebra ^ ( x ) o n a generator, x , of degree n , i s the polynomial algebra on x i f n i s even, and the e x t e r i o r algebra on x i f n i s odd. That i s , i f n i s even, k (A (x)) = 0 i f k i 0 (mod n) and i s the one dimensional v e c t o r n OL space spanned by x i f k = an . I f n i s odd, we add the 2 k r e l a t i o n x = 0 (graded-commutativity) so (A^(x)) = 0 i f k ^ 0 , n A graded algebra A i s f r e e on a set of generators x^,...} i f A i s a tensor product of f r e e algebras on each of the generators. We w r i t e A = A (x,,x_,...) i f a l l the generators are of degree n . n 1 Z I f A i s a graded algebra or a DGA , we w i l l w r i t e A = Q i f A° = Q and A n = 0 f o r n >_ 1 ; A i s c a l l e d the t r i v i a l DGA (with i d e n t i t y ) . 0.3 Examples: (a) I f K i s a s i m p l i c i a l complex, the s i m p l i c i a l cohomology algebra H (K;Q) of K w i t h c o e f f i c i e n t s i n Q i s a graded algebra. (b) I f M i s a smooth manifold, the c o l l e c t i o n E (M) of smooth d i f f e r e n t i a l forms on M i s a d i f f e r e n t i a l gra'ded algebra (over the r e a l s ) w i t h the wedge product as- m u l t i p l i c a t i o n and the e x t e r i o r d e r i v a t i v e as d i f f f e r e n t i a l . De Rham's theorem s t a t e s that the •k cohomology of E (M) i s isomorphic to the ordinary s i m p l i c i a l * cohomology H (M;R) of a smooth t r i a n g u l a t i o n of M [see 15]. We g e n e r a l i z e t h i s example i n Chapter 1. (c) The s i m p l i c i a l cochain complex C (K;Q) of a s i m p l i c i a l complex - 5 -K i s not a DGA ; the m u l t i p l i c a t i o n i s not graded commutative, even though i t i s on the cohomology l e v e l . This i s why we use d i f f e r e n t i a l forms i n s t e a d of cochains. - 6 -Chapter 1. De Rham's Theorem f o r S i m p l i c i a l Complexes In t h i s chapter, we extend the concepts of de Rham cohomology from a r e a l theory on smooth manifolds to a r a t i o n a l theory on s i m p l i c i a l complexes. We do t h i s by c o n s i d e r i n g r a t i o n a l - c o e f f i c i e n t polynomial forms i n each simplex that "patch together" on common faces. We w i l l see that de Rham's theorem holds i n t h i s g e n e r a l i z e d s e t t i n g . We assume only a b a s i c f a m i l i a r i t y w i t h d i f f e r e n t i a l forms, eg. [12]. Our treatment f o l l o w s that of H. Whitney [15]. 1.1 The de Rham Algebra: We w i l l work on an o r i e n t e d s i m p l i c i a l complex K . R e c a l l that K i s the union of o r i e n t e d n - s i m p l i c e s (n=0,l,...) , each of which i s homeomorphic to the standard n-simplex A - R n + 1 : n n+1 n " A = {(x.,...,x ) e R I x. > 0 and 7 x. = 1} . n 0 n ' x — , n x 1=0 The x.'s are the b a r y c e n t r i c coordinates of A . Note that the x n boundary of the standard n-simplex i s a union of ( n - 1 ) - s i m p l i c e s , and i s a s i m p l i c i a l complex homeomorphic to the (n-1)-sphere S n "*"<^  R n . In A we consider d i f f e r e n t i a l k-forms n co = I co . (x ,.. . ,x ) dx A • • • A dx 0<i <...<! <n X V ' \ 0 n \ \ — 1 k— where x„,...,x are the b a r y c e n t r i c coordinates i n A and the O n n co. . are polynomials i n x , . . . ,x w i t h r a t i o n a l c o e f f i c i e n t s . x 1 , . . . , x k O n 7 -In f a c t , these forms are defined on the n-dimensional hyperplane i n determined by . On t h i s hyperplane, we have the r e l a t i o n s X . + . . - . + x = 1 , 0 n dx„ + ... + dx =0 0 n I f K i s an (oriented) s i m p l i c i a l complex, a d i f f e r e n t i a l k-form co on K i s a c o l l e c t i o n {u^} , one f o r each simplex a of K , of Q-polynomial k-forms i n the b a r y c e n t r i c coordinates of a (as above) which s a t i s f y the f o l l o w i n g coherence c o n d i t i o n : I f a i s a simplex of K and x i s a face of a , then CO X = CO , a 1 x where the l e f t hand s i d e i s the r e s t r i c t i o n of co to x i n the a sense of d i f f e r e n t i a l forms. Let E (K) denote the c o l l e c t i o n of a l l such k-forms on K , and set E*(K) = k © E k(K) We d e f i n e a product A . and a d i f f e r e n t i a l d i n E (K) as f o l l o w s : k £ k+£ I f co e E (K) and n e E (K) , then co A n e E (K) and k+1 dco e E (K) are given on a simplex a of K by ("An)a = V A na (dco) = d(co ) , a a where A and d on the r i g h t are the usual wedge product and e x t e r i o r d e r i v a t i v e on a ~ (as a subset of the n-dimensional hyperplane i n Rn+"'") . C l e a r l y co A 1 and dco s a t i s f y the coherence c o n d i t i o n of the l a s t paragraph. A l l the usu a l p r o p e r t i e s of forms hold i n E (K) : (co +co2) A H = (to A n) + (" 2An) , (co A n) A 8 = co A (n A 6) , (qco) A n = q(co A n) f o r q e <Q , kJl k £ UA n = (-1) riAw when co e E (K) , n e E (K) , d(d6) = 0 , k k d(coAl) = dco A n + (-1) to A dn when co e E (K) . Note that E^(K) c o n s i s t s of equivalence c l a s s e s of Q-polynomials i n the b a r y c e n t r i c coordinates of the va r i o u s s i m p l i c e s of K ; the equivalence r e l a t i o n i s the r e s u l t of the f a c t that the sum of the b a r y c e n t r i c coordinates i s i d e n t i c a l l y 1 at every p o i n t of K . The constant polynomial 1 e E^(K) acts as the i d e n t i t y f o r the wedge product i n E (K) . A Hence E (K) i s a d i f f e r e n t i a l graded algebra over Q , c a l l e d the r a t i o n a l de Rham algebra of K . The cohomology algebra A A H ( E (K)) i s c a l l e d the de Rham cohomology algebra of K , and w i l l X be denoted H^_.(K) . DK I f K and K' are s i m p l i c i a l complexes and f : K -> K' i s a s i m p l i c i a l map ( i . e . , takes s i m p l i c e s to s i m p l i c e s l i n e a r l y ) , then f induces a map of DGA's E ( f ) = f E (K') + E (K) k by s u b s t i t u t i o n of b a r y c e n t r i c coordinates. That i s , i f co e E (K 1) , f (co) e E (K) i s given on a simplex a of K by x (f co) (x) = (w) . . (f (x)) , x e a . o t (.cr; & %k ft One v e r i f i e s that (g°f) = f °g f o r f : K -> K' and g : K' -> K" , X X X and that ( i d e n t i t y on K) i s the i d e n t i t y on E (K) So E i s a co n t r a v a r i a n t f u n c t o r from the category of s i m p l i c i a l complexes to the category VGA . k Note that i f co e E (K) and d i s an Jl-simplex of K , H < k , then (co)^ = 0 . Hence i f f : K -> K 1 co e E k(K') and a i s a simplex of K f o r which f ( a ) has dimension < k , then X (f co) = 0 , even i f a has dimension > k . a — I f i : L —> K i s the i n c l u s i o n of a subcomplex L i n t o the s i m p l i c i a l complex K , then the induced map x x x i : E (K) -> E (L) corresponds to r e s t r i c t i o n of the forms on K to L . We w i l l show x ', ( P r o p o s i t i o n 1.9) that i i s always an epimorphism. In other words, - 10 -any form on the subcomplex L can be "extended" to a form on a l l of K • We use t h i s f a c t to de f i n e the r e l a t i v e de Rham algebra of K modulo L as the k e r n e l of i 1.2 S i m p l i c i a l Cohomology: We wish to e s t a b l i s h a r e l a t i o n s h i p between de Rham cohomology and s i m p l i c i a l cohomology. In t h i s s e c t i o n we set the n o t a t i o n of the l a t t e r . I f K i s an o r i e n t e d s i m p l i c i a l complex, l e t C^CKjQ) denote the chain complex of s i m p l i c i a l chains i n K , w i t h Q - c o e f f i c i e n t s . So C (K;Q) i s the vect o r space over Q w i t h b a s i s c o n s i s t i n g of the n o r i e n t e d n - s i m p l i c e s of K , -a being i d e n t i f i e d w i t h the opposite o r i e n t a t i o n of a . There are boundary operators 8 : C (K;Q) C ,(K;(D) , and the homology of C.(K;Q) w i t h respect n n - i * to 3 i s the s i m p l i c i a l homology of K , denoted Hx(K;(Q) ; i t i s a graded ve c t o r space over Q . S i m i l a r l y , the s i m p l i c i a l cochain complex C (K;Q) of K i s given by Cn(K;(Q) = Hom^(Cn(K;Q) ,Q) , w i t h coboundary operator 6 : C n(K;Q) •> C n + 1(K;Q) dual to 3 . I f c e C n(K;Q) , we denote the value of t h i s homomorphism on a chain z e Cn(K;(Q) by <[c,z^> e Q ; ( , y i s a b i l i n e a r p a i r i n g of cochains and chains. To each o r i e n t e d n-simplex o of K , there corresponds a cochain c^ e C n(K;Q) whose value on a b a s i s element (that i s , an o r i e n t e d n-simplex) x E C n(K;Q) i s given by 1 i f x = a 0 i f x ^ a - 11 -Every element of C n(K;Q) can be w r i t t e n uniquely as a ( p o s s i b l y i n f i n i t e ) l i n e a r combination of the cochains c . When there i s no confusion, we use a e C n(K;Q) to denote the cochain C q , as w e l l as the chain a £ C n(K;Q) and the n-simplex a <=- K . •k There i s a l s o a m u l t i p l i c a t i o n i n C (K;Q) given by the cup product u : C n(K;Q) 8 C m(K;Q) -> C n 4™(K;Q) ; i t i s a s s o c i a t i v e , has an i d e n t i t y 1 e C^(K;Q) given by ^ 1 , v ^ = 1 E Q f o r every v e r t e x v e K , and w i t h respect to t h i s * product, 6 i s a d e r i v a t i o n . > However C (K;Q) i s not a DGA i n our sense, as the cup product f a i l s to be graded commutative. On passage to cohomology, we o b t a i n the s i m p l i c i a l cohomology of K , denoted * H (K;Q) ; i t i s a graded algebra, as the cup product is_ graded commutative on the cohomology l e v e l . 1.3 D e f i n i t i o n : We de f i n e a map of graded ve c t o r spaces i|/ : E*(K) —> C*(K;Q) , c a l l e d i n t e g r a t i o n , as f o l l o w s : ' i f to e E (K) , ^(to) i s the k-cochain whose value on a k-simplex, a , of K i s <( ^ (to), a / = to . J a This i s a r a t i o n a l number as we are using polynomial forms w i t h r a t i o n a l c o e f f i c i e n t s . In f a c t , an easy computation shows that - 12 -where x n,...,x are the b a r y c e n t r i c coordinates i n the k-simplex a (with the c o r r e c t o r i e n t a t i o n ) . As the i n t e g r a l i s a d d i t i v e , ty i s a morphism of graded vec t o r spaces. In f a c t , more i s t r u e : 1.4 P r o p o s i t i o n : ty i s a n a t u r a l homomorphism of cochain complexes. Proof: To show that ty i s a cochain map, we v e r i f y the commutativity of the f o l l o w i n g diagram: E k(K) > C K(K;Q) E k + 1 ( K ) > C k + 1(K;Q) For to e E (K) and a a (k+1)-simplex of K , we have <^6Kto),a> = do} = da dco = ^(dco) , where the t h i r d e q u a l i t y i s j u s t Stokes' Theorem. As t h i s holds f o r a l l such a , <StJj(co) = ^(doo) , and the diagram commutes. For n a t u r a l i t y , i f f : K > K' i s a s i m p l i c i a l map, we - 13 -must e s t a b l i s h the commutativity of the diagram E U(K) E k(K') 1 > C k(K;Q) ^ > Ck(K';(Q) Suppose co e E (K') and a i s a k-simplex of K . I f f(o) i s a simplex of K' of dimension k , then ( f % t » , a ) = < % 0 ) , f^(a>) = < % ( » , ± f ( a ) ) = ( * / * \ co= fco = < i j ; ( f c o ) , a > , j f ( a ) J a the + depending on whether f ( a ) has the c o r r e c t or opposite o r i e n t a t i o n i n K' . I f f ( a ) has dimension < k , then f,(cr) =,0 e C.(K';Q) and (f*to) = 0 , so x K. 0 < ( f % ( c o ) , a ) = 0 = <(<J>(f'Vco),a^ Hence f ^(co) = i|»(f u) , and the diagram commutes. Q.E.D As i|> i s a cochain map, i t induces a map of graded ve c t o r spaces, * : H D R ( K ) -> H (K;Q) We w i l l see that 4* i s , i n f a c t , a map of graded algebras, and, as 0 0 i s the case w i t h C -forms on a manifold, we have - 14 -1.5 de Rham's Theorem: ty : H (—) H (—;Q) i s a n a t u r a l equivalence of func t o r s DR from the category of s i m p l i c i a l complexes to the category of graded algebras. Before proving de Rham's Theorem, we need some p r e l i m i n a r y r e s u l t s . k R e c a l l that co e E (K) i s closed i f dco = 0 , and exact k-1 'k i f co = dn f o r some n e E (K) ; R^CK) i s j u s t the closed forms modulo the exact forms on K . 1.6 P r o p o s i t i o n (Poincare Lemma) For k _^ 1 , every close d k-form on A^ i s exact. k Proof: For k > n + l , E ( A ) = 0 and there i s nothing to prove. — n k So suppose l < k < n , c o e E ( A ) , and dco = 0 . We can assume — — n that XQ and dx^ do not appear i n the expression f o r . co ( i f they do, r e w r i t e x,. = 1 - x, -...-x , dx r t = - dx., - . . . - dx ) . Let U 1 n (J 1 n H be the l a r g e s t i n t e g e r (k <_ £ <_ n) f o r which dx appears i n co Then we can w r i t e co = E, + 0 A d x ^ , where £ and 6 do not i n v o l v e dx ,...,dx (or x , dx ) . Write 0 = J 0 dx , where the sum i s J6 n \j u J over a l l J e {(j^ .. . .Jk_x) I1 1 i± < • • • < J k - 1 £ £ - 1} and dx T = dx. A . . . A dx. . Now 0 = dco = d£ + d6 Adx„ , and J J-l J k - 1 1 n 36 d e = ^ \ a T 1 d x i * d x i J i = l I - 15 -As any term i n dZ can have only one f a c t o r dx f o r ot >_ £ , and the dQj a . ' terms dx. A dx A dx. have two such f a c t o r s when i > £ + 1 , 3x. 1 J £ — ' l 80 36 we have 3x £+1 By i n t e g r a t i n g Qj w i t h respect to x^ , we can f i n d r a t i o n a l polynomials A^ s a t i s f y i n g 9x XJ k-1 9 X J = (-1) 8 . 7-^=0 f o r a > £ + 1 . Set £ J ' 9x. X = I X dx e E k 1 ( A ) . Then dA = I I —- d x . A d x T + T J J n _ . - ox. x J £-1 3A. J 3A J i = l " i I r — dx A dx . The second terms i s j u s t I (-1) 9 T dx„ A d x T J £ J I Qj d X j A d x ^ = 6 A d x ^ . So J £-1 3A co - dA = ? - V T dx. A dx, , which does not i n v o l v e . 3x. l J J 1=1 l dx , dx , ... , dx . Repeating t h i s process £ - k + 1 times X/ Xji A. n k - l y i e l d s n e E (A ) f o r which co - dn does not i n v o l v e dx, ,...,dx n k n But t h i s leaves only dx , ...,dx 1 , and co - dn i s a k-form, so co = dn • Q.E.D. 1.7 C o r o l l a r y : Q , k = 0 0 , k > 1 - 16 -Proof: The Poincare Lemma gives the r e s u l t when k >_ 1 . E^(A^) c o n s i s t s of a l l r a t i o n a l polynomials i n x^,...,x n . I f co e E^(A^) and dco = 0 , we have n 0 = do, = % | 2 - dx. 1=1 3x_. 1 1 we have 3 0) 0 r - * So = 0 , 1 <_ i £ n , and hence to i s constant. As B (E (A )) = 0 H° R(A n) = Z°(E*(An)) = Q . Q.E.D We now look at the problem of extending forms. Denote by F.A the (n-1)-face of A given by x. = 0 , and by 3A the t o p o l o g i c a l boundary of A n as an (n-1)-dimensional s i m p l i c i a l complex (with the induced o r i e n t a t i o n s ) k — k 1.8 Lemma: I f co e E (F.A ) , there i s a form to e E (A ) such that: - J n n i ) to IF . A = to 3 n i i ) to I F.A = 0 whenever to I F.A nF.A = 0 . i n 1 i n j n Proof: Without l o s s of g e n e r a l i t y , assume to e E (F A ) ( i . e . , that n n j=n) . Let C = {(x.,...,x ) e A Ix ^ 1} be the complement of 0 n n n th n v e r t e x . Let p : C > F^A^ be the p r o j e c t i o n defined by x x p(x.,...,x ) = (-2— ^SZ±. 0) e F A 0 n 1-x 1-x n n n n the - 17 -* n+1 Then p (to) i s a k-form on C as a subset of R , but i s not a polynomial form. In f a c t , f o r 0 <_ i <_ n - 1 , we have p*(x ) = x /(1-x ) , and p*(dx ) = d(p*(x )) = d(x / ( l - x n ) ) = n „ x. x. I -r-*- ( T f - ^ - ) dx. = (dx. + dx ) . . n dx. 1-x j 1-x x 1-x n 3=0 j n n n So f o r s u f f i c i e n t l y l a r g e N , (1-x )• p (to) i s a Q-polynomial form on a l l of A , and we set n — N * k to = (1-x ) p (to) e E (A ) . n n On F A , dx = 0 and (1-x ) = 1 , and n n n n p|FA : F A -> F A i s the i d e n t i t y map. Hence 1 n n n n n n to F A = to . I f to F. A n F A = 0 , w e have p (to) C n F. A = 0 , ' n n ' x n n n . 1 x n and hence tolF.A = 0 . Q.E.D. ' x n k — k 1.9 Lemma: I f to e E (3A ) , there i s an to e E (A ) such that n n co I 9A = to . .' n Proof: Let co. = COIF.A e E K ( F . A ) . The coherence c o n d i t i o n on co J J n J n says t h a t , f o r a l l 0 <_ i , j <_ n , to. F.A o F.A = to. IF.A n F A . i i n j n j ' x n j n — k — We must f i n d to e E (A ) so that to | F A = to i = 0 n n ' j n j u,...,n. By Lemma 1.8, we can f i n d to e E^(A ) so that 0 n - 18 -— i — k to„ F„A = to . Suppose we have constructed to. e E (A ) , f o r some 0' 0 n 0 j n i < n - 1 , so that to. I F.A = to. f o r a l l i < i . Set J - J i n l - J w Li = " - J - I " "-lF-o.iA e Ek(F-a.iA > • T h e n u Li l F-4.i A ^ F - A = 0 3+1 3+I j ' j+1 n j+1 n 3+1 J+l n i n f o r 1 <_ j by the coherence c o n d i t i o n . By Lemma 1.8, we can f i n d to.,, e E K ( A ) so that I O ' I F . ^ A = co!,. j + l n 3+1 1 3+1 n j+1 and co! IF. A = 0 f o r i < j . 3+1' i n — J Set to . ,. = to. + to ! ,. e E k(A ) . Then co . ,.. IF. A = co. f o r a l l 3+1 3 3+1 n 3+1' i n l i <_ j + 1 •, and the i n d u c t i o n continues. We e v e n t u a l l y get — — k to = co E E (A ) w i t h the de s i r e d property. Q.E.D n n 1.10 P r o p o s i t i o n : I f i : L ^ K i s the i n c l u s i o n of a subcomplex L i n a s i m p l i c i a l complex K , then i : E (K) E (L) i s an epimorphisfn. — k * — — i Proof: R e c a l l t h a t , f o r to e E (K) , i (co) = co|L . So, given to E E (L) , we must f i n d to e E (K) f o r which co|L = to . Let K k k-1 denote the n-skeleton of K . Note that E (K ) = 0 . We define to i n d u c t i v e l y over the r e l a t i v e s k e l e t a . Set to|L = to , — i k-1 — i co|K = 0 , and . to | a = 0 f o r any k-simplex o of K - L . This — k — defines co coherently on L u K . Suppose we have defined to on L u K n f o r some n >_ k . I f a i s an (n+1)-simplex of K - L , then 3o •= K n , and Lemma 1.9 allows us to extend to from 3o to 19 -a . Doing t h i s f o r every simplex of K n + ^ - L defines to on L V Kn+^~ . So, by i n d u c t i o n , we can define to on a l l of K so that WIL^ = •<*).. = Q.E.D. 1.11 The R e l a t i v e de Rham Algebra: I f i : L c->-K i s the i n c l u s i o n of a subcomplex, we def i n e E k ( K , L ) = K e r { i * : E k ( K ) - » E k ( L ) } = {co e E k ( K ) | co|L = 0} Then E*(K,L) = ® E k ( K , L ) i s a DGA (without i d e n t i t y , unless L = 0) c a l l e d the r e l a t i v e de Rham algebra of K modulo L . I f f : (K,L) — > (K',L') i s a s i m p l i c i a l map of p a i r s ( i . e . , L c K' , L' <= K'- , and f ( L ) L ' ) . , then f induces a map of DGA's * A A f : E (K 1,L') — > E (K,L) i n the obvious way. Thus E i s a c o n t r a v a r i a n t functor from the * * category of s i m p l i c i a l p a i r s to the category VGA, and E (K,0) = E (K) . By P r o p o s i t i o n 1.10, there i s a short exact sequence of DGA's * J A i * 0 — > E (K,L) > E (K) » E (L) > 0 where L —^—> K —^—> (K,L) are the s i m p l i c i a l i n c l u s i o n s . So passing to cohomology, we have a long exact sequence - 20 -... ~ > H* R(K,L) X H£ R(K) X H ^ L ) — > H^OC.L) — > A * where H (K,L) denotes the cohomology algebra of E (K,L) . DR Another consequence of P r o p o s i t i o n 1.10 i s : 1.12 P r o p o s i t i o n : Let K be a s i m p l i c i a l complex and I a d i r e c t e d s e t . I f { L a : a e 1} i s a d i r e c t system of subcomplexes of K (under i n c l u s i o n ) whose union i s K , then there are n a t u r a l isomor-phisms E*(K)> = l i m E*(L ) of DGA's, < — I A „ A and H (K) = l i m H (L ) of graded algebras. DK ^ DK Ot I Proof: I t i s c l e a r that {E (L ) : a e 1} i s an i n v e r s e system of a DGA's i n which a l l the maps are epimorphisms, and that the f i r s t isomorphism holds. D. W. Kahn [4; Theorem 1.1] has shown that f o r such systems of DGA's, cohomology commutes w i t h i n v e r s e l i m i t s , so the second isomorphism holds . Q.E.D. 1.13 A Right Inverse f o r I n t e g r a t i o n : We now begin the proof of A A de Rham s Theorem by c o n s t r u c t i n g a r i g h t i n v e r s e (j) : C (K;Q) —> E (K) to the i n t e g r a t i o n map y . I f T and T' are s i m p l i c e s of K , w r i t e x < T' i f x i s a face of x' . Define the s t a r of the simplex x to be 21 -S t ( T ) = (J { < T ' > | K T ' J , where denotes the open simple L e x Then St(x) i s an open subset of K . A l s o , St(x) = f] { S t ( v ) | v i s a v e r t e x of x} Let a = [p ,...jp^] be an o r i e n t e d n-simplex of K , where p , ;..,p are the v e r t i c e s , and l e t x.,...,x be the U n O n b a r y c e n t r i c coordinates w i t h respect to p ,...,p considered as 0 n fu n c t i o n s (O-forms) on a l l of K . So x. = 1 at p. , and x. = 0 l 1 1 on K - St(p^) . Consider a as a cochain i n C n(K;Q) as i n 1.2, and d e f i n e an n- form <Ka) e E (K) by r i / \ (1) cj)(a) = n! I (-1) x d x Q A . . . dx_^ . . . A dx^ , i=0 where the symbol over dx. i n d i c a t e s that i t i s omitted. l Sub lemma: (j>(a) = 0 on K - St(a) . Proof: I f x i s a simplex of K - St(a) , there i s an i f o r which p. i s not a ver t e x of x . So x. = 0 and dx. = 0 on x . But I i i every term i n (1) has e i t h e r x_^  or dx^ as a f a c t o r , and hence <KCT)|T = 0 . Q.E.D. We wish to extend the d e f i n i t i o n of cb l i n e a r l y over C*(K;Q) . Every element of C n(K;Q) has the form I c a °a ' a - 22 -where e Q , a i s an n-simplex, and the sum i s over a l l r\-simplices of K . The problem i s that an i n f i n i t e number of the c a may be non-zero, and we cannot add an i n f i n i t e number of d i f f e r e n t i a l forms. However i f T i s any simplex of K , V'T) d s t ( a ) i f f a < x , and hence d)(a ) I T ^ 0 i f f a < x . ct a As x has only a f i n i t e number of fa c e s , a l l but a f i n i t e number of the forms ^ ( a a ) a r e zero on x . So we can define (2) <K I C a ) = V c <Ka ) , a a keeping i n mind that the sum on the r i g h t i s f i n i t e on any simplex of K . So we have a l i n e a r map of graded ve c t o r spaces * : C*(K;Q) > E*(K) . 1.14 P r o p o s i t i o n : (a) d) i s a homomorphism of cochain complexes. (b) ty ° cj) i s the i d e n t i t y on C (K;Q) (c) cb preserves i d e n t i t y elements. Proof: For part ( a ) , we must show 23 -dcp = q><5 : Cn(K;Q) > E n + 1 ( K ) I t s u f f i c e s to show t h i s on an a r b i t r a r y o r i e n t e d n-simplex a of K Let {p } be the v e r t i c e s of K , and a = [p_....,p 1 • a r0 n n . d<K°) = n! T (-1) dx.Adx^A... dx. . . . A dx . r, x 0 x n x=0 n = n! I dx Q A•••A d x n = ( n + 1 ) ! d x 0 A•••A d x n • i=0 Now oa = 2, [p > Pn»-->»P J where 2. i n d i c a t e s that the sum i s a a over a l l a f o r which [p^, p ,...,p ] i s an (n+l)-simplex of K ( 3 ) (n+1) I = ^ { x a d X 0 A A d X n 1 x i d x a A d x 0 A ' " ' ' A d x n } a i=0 I f p , p ,...,p are not the v e r t i c e s of a simplex and x ct (J n i s any simplex, there i s a j e {a, 0,...,n} so that x.. = 0 and dx. = 0 on x . Hence J x dx.. A . . . A dx = 0 and a 0 / x A n x. dx A dx y\ • • -dx; . . . A d x = 0 (i=0, . . . ,n) on x . x ct (j X n • v* As t h i s i s true f o r every x i n K , 2. i n (3) can be replaced by a n 1 . Als o 2\ dx = 0 on K , so £ dx. = - \ dx . a^O,...,n- a a j=0 3 a^0,...,n a So (3) becomes - 24 -, = I x a dx 0 A .. . A d x n + I I dx A dx Q A .. .dx... . A d x n K l v r j- J ct^0,...,n i=0 3-0 I x a d x Q A . . . A d x n + I (-1) 1 x. dx. A d x Q A ...dx.. a^0,...,n i=0 dx a - 0 — d X n + \ *< d X a^0,...,n i=0 I x , d x n A ' - - A d x n + % x ± d X ( ) A . . . A d x ( ^ V d x 0 A•••A d x n = dxQ A »••A d x n ' as y x = 1 . Hence d<j>(a) = <K<5cf) For part (b) , we must show that # ( a ) = a f o r every n-simplex a of K . This means t h a t , f o r any o r i e n t e d n-simplex Uo) = 1 i f T = o 0 i f T 4 a I f x $ o, then T n St(cr) = $ , so that <j>(a)|x = 0 , and the r e s u l t f o l l o w s . I f o = [p ,...,p n] , then the r e l a t i o n s n n I x. = 1 and I dx. = 0 1=0 1=0 are v a l i d on o . So we have Uo) L! 0 x„ dx n . n A . . . A d x + I . ( - l ) V (- I dx ) A dx A ...dx . 1=1 1 j = l 3 A d x l '» n - 25 -1+1 = X 0d x 1 A . . . A d x n + I (-1) x ± d x ± A d X ; L A . . .dx ±. . . A d x n i = l ^Qdx1 A . . . A d x n + I x ± d X ; L A . . . A d x n i = l n = ( I x ± ) d x 1 A . . . A d x n = dx^ A . . . A d x n i=0 So ty(a) = n! I dx^ /\ . .. A dx = n! • = 1 . This proves (b) a ' a For part ( c ) , r e c a l l that the i d e n t i t y element of C (K;Q) i s 1 = 1 p a e C°(K;Q) . But rp(P a) = x & , so a H i ) = <K I p a ) = I <Kp a) = I x a = l , a a a and ty preserves i d e n t i t i e s . Q.E.D 1.15 C o r o l l a r y : ty : E (K) > C (K;Q) and / : H* R(K) > H*(K;Q) are both epimorphisms. Proof: ty ° ty = i d , so ty i s epimorphic, and * * * * • * „ ty o ty = (jf) o ty) = (i d ) = i d , so ty i s epimorphic. y.E.D. - 26 -1.16 Proof of de Rham's Theorem: We must show that ty" : H" ( K ) > H"(K;Q) i s a n a t u r a l DR A isomorphism of graded algebras. By C o r o l l a r y 1.15, ty i s an epimorphisrrij and n a t u r a l i t y f o l l o w s from P r o p o s i t i o n 1.4. We leave the proof that ty i s an algebra homomorphism t i l l l a s t , and prove the isomorphism part by i n d u c t i o n on the dimension of the complex K . I f K i s O-dimensional, we have H D R ( K ) = E * ( K ) " C * < K 5 ^ = H * ( K ; Q ) , A and c l e a r l y ty i s an isomorphism. Suppose i n d u c t i v e l y that ty i s an isomorphism (of graded v e c t o r spaces) f o r a l l complexes of dimension l e s s than n , and suppose K has dimension n . As ty i s an epimorphism, we have a short exact sequence of cochain complexes 0 — > Ker(»J0 «=->E*(K) — — » C*(K;Q) —> 0 . This induces a long exact sequence i n cohomology A ... — > H k ( K e r ty) — > H K (K) ^ — » H k ( K ; Q ) — > H k + 1 ( K e r ty) —> ... DK A A So ty i s an isomorphism i f and only i f H (Ker ty) = 0 . The proof r e l i e s on the f o l l o w i n g : Lemma: Suppose c o e E ( A ) , l < _ k < _ n , d c o = 0 , and ty(u) = 0 , and k-1 | n e E (3A ) , dn = co 3A , and ty(r)) = 0 . Then there i s an n n n e E k \ A ) f o r which n|3A = n , dn = co , and ty(r\) = 0 . n n - 27 -k-1 Proof: By Lemma 1.9, we can f i n d n' e E (A ) so that ; n n' | 9A = n . Then d(co-dri') = 0 , so the Poincare Lemma gives k-1 6 e E (A ) so that d6 = to - d n ' . We have n d(8 I 9A ) = J d A - d(n '|3A ) = dn - dn = 0 . n n n I f k = 1 , 9 3A i s a closed 0-form, and hence i s 1 n constant: 6I3A = c , c e Q . Set n = n ' + 0 - c e E°(A ) . Then n n dn = d n ' + d6 = to , "nl3A = n'|3A + e|3A - c = n'|3A = n , 1 n 1 n 1 n 1 n and ijj(n) = ipCni 3A ) = ty(r\) = 0 , as d e s i r e d . So suppose k >_ 2 . By the i n d u c t i o n hypothesis: / : H ^ O A ) ~^-> H k _ 1 ( 3 A ;<Q) , and H k - 1 ( 3A n;Q) = 0 i f k + n . So i f k ± n , 613A^ e E k - 1 ( 3 A _ ) n n i s exact. I f k = n , we have HDR 1 ( 3 V = H n" 1OA n;Q) = Ho» Q(H n_ 1OA n;Q), Q) by the u n i v e r s a l c o e f f i c i e n t theorem, and t h i s composite isomorphism takes [8|3A n] to the homomorphism given by [z] — > ^ ^ ( 6 | 3A )» f f o r [z] e H ..(3A ;<Q) . Now H _(3A ;Q) i s a 1-dimensional v e c t o r n — i n n-1 n space generated by . [3A^] (here, 3 i s the boundary operator i n - 28 -CK(An;<Q)). But <^ (e|8A n ) , 3 A n ) = <^6^(0), A n ^ = <(^(de), A n y = ^ *(<*>), A ^ - <^ (dn'), A ^ = - (<Kh') , 3 A ^ = - ^ K n ) , 3 A n ) = 0 as I J J ( C O ) = 0 and ijj(n) = 0 . So [8|3A ] maps to the zero homomorphism, and hence [8|3A^] = 0 e ^ ^ ( S A ) . i k-2 So, f o r k > 2 , 6 8A i s exact, and we have X £ E. (3A ) — n n such that dX = 013A . Again by Lemma 1.9, we can f i n d n X' e E k _ 2 ( A ) so that X*|3A = X . Set n = n' + 0 - dX* e E k _ 1 ( A ) n n n Then dn = dn.' + d0 = co , .n|3A = n*ISA +-0|3A - d(X'!3A ) 1 n ' -n ' n 1 n = n + 0 I 3A - dX •= n , 1 n and <Kn) = <Kn7|3A ) = <Kn) = 0 . n This completes the proof of the Lemma. We now show that H*(Ker(i|> : E*(K) — » C*(K;Q))) = 0 f o r K an n-dimensional complex. Suppose co E E (K) i s such that k-1 dco = 0 and IJJ(CO) = 0 . We must f i n d r) e E ( K ) f o r which dn = w and ty(r\) = 0 . I f k = 0 , dco = 0 i m p l i e s that co i s constant on each path - 29 -component of K ; and ^(co) = 0 gives co = 0 on K . So H°(Ker ty) = 0 . So suppose k >_ 1 . Then co|Kn ^ e £k(K.n ^) i s such that d(co|Kn "*") = 0 and ^ (co | K n 1 ) = 0 , where K n _ 1 i s the (n-1)-skeleton of K . By the i n d u c t i o n hypothesis, ty* : H* ( K n _ 1 ) = H*(K n - 1;Q) , DR and hence H*(Ker(iJj : E * ( K n - 1 ) — » c' C(K n _ 1;Q) )) = 0 . So there i s an n' e E k - 1 ( K n - 1 ) f o r which iKn') = 0 and dn' = co IK n-1 Then, working one n-simplex at a time, the Lemma allows us to extend k-1 n' to n £ E (K) i n such a way that ty(r\) = 0 and dn = to . Hence, H*(Ker ty) = 0 , and ** : H D R ( K ) " H * ( K ^ ) fo r n-dimensional K . This concludes the i n d u c t i o n step, and shows that ty i s an isomorphism of graded ve c t o r spaces f o r a l l f i n i t e dimensional complexes. I f K i s i n f i n i t e dimensional, we have a map of in v e r s e systems of graded ve c t o r spaces: * , 0 * 1 H D R ( K > < - H D R ( K > < -* ;„0 * 1 H (K ;Q) < H (K ;Q) < — . . . < — H (K ;Q) < — . . . - 30 -By P r o p o s i t i o n 1.12, l i m H*_(K n) = H* (K) . Also ' DK DR < H (K;Q) = H (K ;Q) when m >_n + 1 , so l i m H (K ;Q) = H (K;Q) . As each K n i s a f i n i t e dimensional complex, the v e r t i c a l maps i n the diagram are isomorphisms, and hence the induced map on the inv e r s e l i m i t s i s an isomorphism. So ** : H* R(K) = H*(K;(Q) . X X X Note that t h i s a l s o shxws ty : H (K;Q) — > R^CK) i s an isomorphism, and that -ty •= (ty ) We conclude the proof of de Rham's theorem by showing that X X ty preserves products. We de f i n e a product, A . i n C (K;Q) as X f o l l o w s : f o r c^, c^ e C (K;Q) , c x A, c 2 = K K c . ^ A <KC 2)) » x where A o n the r i g h t i s the product i n E (K) . This product i s graded and graded-commutative, but i s not a s s o c i a t i v e (as ty ° ty ^  i d ) Whitney [14] has shown that a graded product, A » on C (K;Q) X induces the cup product on H (K;Q) provided i t s a t i s f i e s the f o l l o w i n g p r o p e r t i e s : ( i ) i f 0^ and are k- and A-s i m p l i c e s , resp., considered as cochains, and i f T i s a (k+£)-simplex, A ' T ^ ^ ^ - 31 -im p l i e s 0 ^ < T and 0 ^ < T , ( i i ) i f C ; L e C k(K;Q) and c 2 e C*(K;Q) , < S^ C1 A C2) = <S°1 A °2 + Cl A 6 c 2 ( i i i ) 1 A C = C = C / \ 1 f o r a l l c e C (K;Q) . We v e r i f y these p r o p e r t i e s f o r the product, A > defined above. C l e a r l y , <|)(a )A ty(.°0) = 0 on K - ( S t ( o 1 ) H S t ( a 0 ) ) i f 0 T4 ° 2» 0 =So ty(a-,) A ty(a?) > we must have T <( T )> d S t ( a 1 ) A S t ( ° 2 ) » a n d hence a < T and < T . This proves ( i ) . Property ( i i ) f o l l o w s immediately from the f a c t s that * d i s a d e r i v a t i o n on E (K) , and ty and <j> are chain maps. Property ( i i i ) f o l l o w s from P r o p o s i t i o n 1.14(c): l A c = K<KD A <Kc)) = IK1A<K C ) . ) = <K<Kc)) = c . So A induces the cup product i n H (K;Q) . That i s , i f c^ and c 2 are co c y c l e s , [c±] [ c 2 ] = [ c 1 A c 23 . To see that ty i s m u l t i p l i c a t i v e , take any closed forms to , co2 e E (K) , and set c ± = ^(to ±) , i = 1, 2 . Then $ ( [ c ^ ] ) = [w ] , i = 1, 2 , and we have * * ( [ " 1 l A . [ w 2 ] ) [ C X ^ A <I>*([C2])) - 32 * ( [ K ^ ) A K c 2 ) ] ) lUHCj) A +(c 2))] [ c 1 A c 2 ] = [ c ^ K / t ^ ] This completes the proof of de Rham's Theorem. I f L i s a subcomplex of K , then c l e a r l y i K E * ( K , L ) ) «= C*(K,L;Q) , and we have 1.17 C o r o l l a r y : ty : H (K,L) = H (K,L;Q) i s a n a t u r a l equivalence of functors from the category of s i m p l i c i a l p a i r s to the category of graded algebras. Proof: N a t u r a l i t y i s c l e a r . We have a map of short exact sequences 0 — > E*(K,L) > E * ( K ) ' — — E * ( L ) > 0 0 — > C (K,L;Q) ^ C (K;Q) —•» C (L;Q) —> 0 which induces a map of long exact cohomology sequences - 33 --> H J R ( K ) — > l £ R ( L ) — > C(K,L) -> H ^ C K ) — > H ^ C L ) -> H N(K;Q) — > H N(L;Q) — > H N + 1(K,L;Q) — > H N + 1(K;Q) — > H N + 1 ( L ; Q ) — > The r e s u l t f o l l o w s by the 5-lemma. Q.E.D - 34 -Chapter 2 The Minimal Model In t h i s chapter, we describe an a l g e b r a i c c o n s t r u c t i o n on simply-connected DGA's which i s , i n some sense, minimal. This w i l l be a p p l i e d i n Chapter 3 to the de Rham algebra E (K) of a simply-connected s i m p l i c i a l complex K , and we s h a l l see that the c o n s t r u c t i o n p a r a l l e l s the Postnikov decomposition of K . We f i r s t need some d e f i n i t i o n s . 0 0 oo 2.1 D e f i n i t i o n : I f A = ® ^ A™ i s a graded a l g e b r a , set A = ® ^ A™ + + Define D(A) to be the image of A 0 A under m u l t i p l i c a t i o n . D(A) i s c l e a r l y an i d e a l of A , c a l l e d the i d e a l of decomposables; i t c o n s i s t s of a l l sums of n o n - t r i v i a l products i n A . I f A i s a DGA, we say that A has a decomposable d i f f e r e n t i a l i f the image of the d i f f e r e n t i a l i s contained i n the i< i d e a l of decomposables; B (A) cr D(A) . 2.2 Minimal Algebras: A DGA, M , i s c a l l e d a minimal algebra i f i t s a t i s f i e s the f o l l o w i n g four p r o p e r t i e s : ( i ) M i s f r e e as a graded al g e b r a , ( i i ) M has a decomposable d i f f e r e n t i a l , ( i i i ) M° = Q , M 1 = 0 , ( i v ) M has cohomology of f i n i t e type; i . e . , f o r each n , H n(M) i s a f i n i t e dimensional v e c t o r space. Note that p r o p e r t i e s ( i i ) - ( i v ) imply - 35 -(v) f o r each n , M n i s a f i n i t e dimensional v e c t o r space. Let M denote the f u l l subcategory of VGA c o n s i s t i n g of a l l minimal algebras and a l l DGA maps between them. By an abuse of n o t a t i o n , we w r i t e M e M f o r "M i s an object of M ." Let M be the f r e e algebra on the (free) generators of n M of degree <_ n ; t h i s i s a subalgebra of M (as a graded a l g e b r a ) , k k 1 and M = M f o r k < n . As M = 0 and the d i f f e r e n t i a l i s n — decomposable, we have d(M ) <=• M , so that M i s a sub-DGA of n n n M , and M e M . Moreover, n d(M k) C Mk+?" f o r k < n . n n-1 — ' I f f : M -»- N i s a map i n M , the f a c t that f i s an algebra map gives f 0^n) *— , so that f = f M : M —> N n 1 n n n i s a l s o a map i n M . So, to each M e M there i s as s o c i a t e d a ca n o n i c a l sequence of sub-DGA's, <D = M 1 < = M 0 < = . . . c : M c= . . . « r II M = M 1 z n „ n n so that M^ e M and has no generators of degree > n . We a l s o have M = M , 0 A ( x n , . . . , x , ) as a graded algebra, and d(x.) e M n +^ . n n-1 n 1 K. ° ° l n-1 2.3 Minimal Models: In the remainder of t h i s chapter, a l l DGA's - 36 -considered are assumed to have cohomology of f i n i t e type. Suppose A i s a simply connected DGA; r e c a l l that t h i s means H°(A) = Q and H 1(A) = 0 . A DGA M = M(A) i s c a l l e d a minimal model f o r A i f ( i ) M e M ( i i ) there i s a DGA map p : M — > A which induces an A A A isomorphism on cohomology; p : H (M) = H (A)' . Note that many such maps p may e x i s t . 2.4 Examples: (a) I f M e M , i t i s i t s own minimal model, as the ' i d e n t i t y map s a t i s f i e s c o n d i t i o n ( i i ) of the d e f i n i t i o n . A (b) Suppose A i s a simply connected DGA and that H (A) i s a f r e e A algebra. Then H (A) w i t h zero d i f f e r e n t i a l i s a minimal model f o r A A . C l e a r l y the zero d i f f e r e n t i a l i s decomposable, so H (A) e M . To v e r i f y c o n d i t i o n ( i i ) , p i c k cocycles z^, z^,... i n A i n such a A way that [z^] , L ^ J ' ••• generate H (A) f r e e l y , and de f i n e A A p : H (A) —> A by P ( [ z ^ ] ) = . Then p i s a DGA map, and p A i s the i d e n t i t y on H (A) . The main r e s u l t of t h i s chapter i s : 2.5 Theorem: Every simply connected DGA (with cohomology of f i n i t e type) has a minimal model which i s unique up to isomorphism. Proof ( E x i s t e n c e ) : Let A be a simply connected DGA. We construct a minimal model M f o r A i n d u c t i v e l y as f o l l o w s : - 37 -Set 'M1 = <Q ( i . e . , M° = Q and = 0 f o r n •> 1) . As Q = H^(A) = Z^(A) c A \ we can define : — > A which maps 0 0 * n n M i d e n t i c a l l y onto Z (A) . Then ' p : H (M^ —> H (A) i s an isomorphism f o r n = 0, 1 and a monomorphism f o r n = 2 . Suppose we have constructed M , e M and p , : M .. — > n-1 n-1 n-1 i n such a way that M ^ has no algebra generators of degree >_ n , * k k " and that p n : H (M 1) — > H (A) i s an isomorphism f o r k < n - 1 n—1 n — i — and a monomorphism f o r k = n . We have a short exact sequence 0 > H n(M ,) > 1 1 - 1 > H n(A) — — » Coker(p* .) — > 0 . n - i n — i n £ Choose z^,...,z^ e Z (A) so that ( e ( [ z ^ ] ) K _ ^ forms a b a s i s f o r * Coker(p n_^) . We a l s o have an exact sequence n IT i * \ „n"*"l/»» \ ^n-1 _ „n+l / A N 0 > Ker(p .,) > H (M -) > H (A) . n—1 n - i Choose w . . . ,w E Zn+"*"(M n) so that {[w.]} 1? n i s a b a s i s f o r i m n — i x x=l * n+1 Ker(p T ) . C l e a r l y w. e M , i s decomposable. As n-1 x n-1 P i ( [ w . ] ) = 0 E "^'"'''(A) , we must have p n(w.) £ Bn+"'"(A) , and we n-1 l n-1 l can choose v.,...,v £ A n so that d(v.) = p , (w.) . 1 m x n-1 x M Define a DGA, M , to be n n = M n - 1 8 A n < x l ' " " V y V " ' y m ) as a graded alg e b r a , and extend the d i f f e r e n t i a l i n M , to M by n-1 n - 38 -s e t t i n g d ( l 8 x^) = 0 and d ( l 8 y_^ ) = w_^  8 1 (and r e q u i r i n g d to be a d e r i v a t i o n ) . The choice of w. shows that M e M . l n Extend P .. to p : M — > A by s e t t i n g n-1 n n J b p (1 0 x.) = z. and p ( 1 8 y.) = v. n x x 'n x x Then dp (1 8 y.) = d(v.) = p n(w.) = P (dy.) , so P i s a map of DGA's. n x x n-1 x n x n By the choice of the z.'s and w.'s , i t i s now c l e a r that x x * k k p^ : H (M ) > H (A) i s an isomorphism f o r k <_ n and a monomorphism f o r k = n + 1 . This completes the i n d u c t i o n step, and s e t t i n g M = l i m M , P = l i m p completes the c o n s t r u c t i o n . Q.E.D. — > N — > N To prove the uniqueness part of Theorem 2.5, we need some other r e s u l t s about the category M . 2.6 Theorem: Suppose f : M N i s a map i n M and f * : H k(M) — > H k i s an isomorphism f o r k <_ m and a monomorphism f o r k = m + 1 . Then f : M — > N i s an isomorphism, m m m r Proof: For m = 1 , the theorem i s c l e a r , as = Q = and f ( l ) = 1 Assume i n d u c t i v e l y that f : M , = N .. , f o r some n-1 n-1 n-1 n <_ m . I t f o l l o w s that f . : Z*(M ) = Z*(N j n-1 n-1 n-1 and f . : B*(M .) = B*(N .) . n-1 n-1 n-1 Moreover, as M e M , we have - 39 -Z n + 1 ( M J = Z n + 1 ( M )<= Z n + 1(M) n-1 n and B n + 1 ( M n) B n + 1 ( M ) = B n + 1(M) , n-1 n and the same i s true f o r N . We f i r s t show that f : B n + 1(M) = B n + 1 ( N ) . The commutative diagram B (M) <= >^ M n-1 v n-1 „n.+l ,„.. „n+l B (N) c= —> N n_ shows that f |Bn+^"(M) i s monic. To see that i t i s a l s o e p i c , choose n~Hl any w e B (N) . As w e B n + 1 ( N ) = B n + 1 ( N )«= Z n + 1 ( N ) = Z n + 1 ( N ,) , n n n-1 and f 1 i s an isomorphism, we can f i n d (a unique) x e Z^^fM ,) n - i a n-1 f o r which f ( x ) = f .(x) = w . Now Z n + 1 ( M J a Z n + 1(M) , so n - l n-1 consider [x] e H n + 1(M) ; we have f * ( [ x ] ) = [ f ( x ) ] = [w] = 0 e H n + 1 ( N ) * n+1 But f i s monic i n dimension h + 1 , so [x] = 0 , and x e B (M) Hence f ( B n + 1 ( M ) ) = B n + 1 ( N ) , and f : B n + 1 ( M ) = B n + 1 ( N ) , as d e s i r e d . We now show that f : Z n(M) = z"(N) . From the commutative square - 40 -B n(M .) B n(M) n-1 n-1 B n(N J B n(N) n-1 we get that f : B n(M) = B n(N) . We have a map of short exact sequences, 0 > B n(M) —> Z n(M) » H n(M) > 0 0 > B (N) -> Z n(N) -» H n(N) "> ,0 , so the 5-lemma gives the r e s u l t f : Z n(M) = Z n ( N ) . We have another map of short exact sequences -> Z n(M) -> Z n(N) -> M f n -> N » B (M) -» B n + 1 ( N ) -> 0 -> 0 , and, again by the 5-lemma, f : M n = N U Define an algebra map g : N —> M as f o l l o w s : s e l e c t a n n set {y^} that generate f r e e l y and set g(y^) = f (y^) • Then c l e a r l y f ° g i s the i d e n t i t y on N , and f i s an epimorphism. J n & n n But M and N are f r e e algebras w i t h the same number of f r e e n n generators of each degree, so t h a t , f o r a l l k >_ 0 , - 41 -k k dim A M = dini N < °° . n d) n Hence f : M = N . O F T ) n n n U..&.D. 2.7 C o r o l l a r y : I f f : M — > N i s a map i n M and f* : H*(M) — > H* i s an isomorphism, then f : M — > N i s an isomorphism. Proof: Theorem 2.6 shows that f : M™ = f o r a l l m , and hence f : M = N . . Q.E.D. To study the problem of induced maps between minimal models, we must f i r s t develop the n o t i o n of homotopy f o r DGA's . 2.8 D e f i n i t i o n s : Define I to be the DGA f r e e l y generated by t i n degree 0 and dt (the d i f f e r e n t i a l of t) i n degree 1 . So 1^ = <Q[t] , a l l r a t i o n a l c o e f f i c i e n t polynomials i n t , 1^ " = Q[t] c o n s i s t s of a l l products p ( t ) d t , where p ( t ) e 1^ , and l n = 0 fo r n >. 2 . For p(t) e 1^ we have d ( p ( t ) ) = p ' ( t ) d t , where p '(t) i s the ordinary d e r i v a t i v e of p ( t ) . Note that I = E*([0,1]) , the de Rham algebra of the u n i t i n t e r v a l . In f a c t , the r o l e played by I i n DGA homotopies i s analogous to that played by [0,1] i n t o p o l o g i c a l homotopies. Note a l s o t h a t , as the i n d e f i n i t e i n t e g r a l of a polynomial * i s again a polynomial, we have H (I) = Q . - 42 -Let A be a DGA, and consider A 9 I to be a DGA as described i n s e c t i o n 0.2. A t y p i c a l element x e (A 0 I ) n = (A n 0 1°) © ( A n _ 1 0 I 1 ) has the form x k £ I a, 0 p (t) + I b. 0 q . ( t ) d t , 1=1 l * x j = l where a. e A n , b. e A n 1 and p . ( t ) , q.(t) £ 1° such an element x we have Q[t] . For k k d(x) = I d(a.) 0 p.(t) + ( - l ) n I a. 0 p'.(t)dt 1=1 x • x 1=1 X X + I d(b.) 0 q . ( t ) d t . j = l 3 3 For each a e Q , define a DGA map e Q : A 8 I — > A by i n s i s t i n g that e a ( a 0 1) = a , f o r a l l a e A , e a ( l 0 t ) = a e Q c A e (1 0 dt) = 0 . a So, f o r t h e - t y p i c a l element x above, we have k Ja ( x ) = I P, ( a) » i = l 1 1 C l e a r l y e^ i s a DGA map. We are e s p e c i a l l y concerned w i t h the cases a = 0 and a = 1 . By the Kunneth formula, we have - 43 -H (A 8 I) = H (A) 8 H (I) = H (A) . * We wish to know that e i s an isomorphism which i s independent of a the choice of ct . 2.9 Lemma: Every element of H n(A 8 I) has a r e p r e s e n t a t i o n as [a 8 1] f o r a unique [a] e H.n(A) . Proof: We f i r s t show that every element x e (A 8 I ) n can be w r i t t e n i n the form k x = I a. 8 p ( t ) + d(y) , i = l where a^ e A n , p^(t) e Q[t] , and y e (A 8 I ) n ^  . For suppose x has a term of the form b 8 q ( t ) d t , where b e A n ^ and q(t) e Q[t] . Choose an i n d e f i n i t e i n t e g r a l r ( t ) e Q[t] f o r the polynomial q(t) ; so r ' ( t ) = q(t) . Then d(b 8 r ( t ) ) = d(b) 8 r ( t ) + ( - l ) n _ 1 b 8 q ( t ) d t , so that b 8 q ( t ) d t = ( - l ) n d(b) 8 r ( t ) + ( - l ) n _ 1 d(b 8 r ( t ) ) , and the r e s u l t f o l l o w s . Now, f o r [x] e H n(A 8 1) , we can w r i t e k x = I a 8 p (t) + d(y) i = l as above, where - 44 -£ p . ( t ) = I c. t m e Q[t] m=0 As d(x) = 0 , we have 0 = d(x) = I d(a.) 8 p.(t) + ( - l ) n I a 8 p'.(t)dt i=l 1 1 i=i As the f i r s t sum i s i n A n + 1 8 I 0 , and the second i n A n sum i s zero, and i n p a r t i c u l a r k k £ x 0 = I a. 8 p ! ( t ) d t = I a. 8 ( I m c. t )dt . - x i . - i , im x=l x=l m=l £ k = I m • ( I c -a ) 8 t m _ 1 d t , . - xm x m=l x=l So, as the powers of t are independent, we must have k j[ c. -a. = 0 f o r 1 < m < £ . . - xm l — — x=l k £ Hence x = / r a . 8 ( I c t m ) + d(y) i = l 1 m=0 i m £ k. = I ( I c . -a.) 8 t m + d(y) „ . , xm x m=0 x=l k = ( I c.n-a.) 8 1 + d(y) i = l u 1 = a 8 1 + d(y) - 45 -k where a = £ °iO* ai ' A l s o , i = l 0 = d(x) = d(a 8 1 ) = d(a) 8 1 , so that a e Z n(A) , and [x] = [a 8 1] e H n(A 8 I) , as d e s i r e d . To show uniqueness, suppose, [a 8 1] = [b 8 1] , so that (a-b) 8 1 e B n(A 0 1 ) . By the f i r s t r e s u l t of t h i s proof, we can f i n d a. e A n ^ and p.(t) e 1^ (l<i<k) such that x x k (a-b) 8 1 = d( I a. 8 p . ( t ) ) i = l 1 1 k k = I d(a.) 0 p.(t) + ( - l ) n 1 I a. 8 p ! ( t ) d t . i = l 1 1 i = l 1 1 As before we have k ( i ) (a-b) 0 1 = I d(a.) 8 p. (t) e A n 8 1° i = l 1 1 k ( i i ) 0 = I a. 8 p ! ( t ) d t e A n 1 0 I X . i = l 1 1 a Again w r i t i n g p . (t) = I c. t f o r c. e Q , equation ( i i ) m=0 becomes - 46 -& k 0 = I a. 0 ( I m-c t 1 1 1" 1) = I m.( I c -a ) 0 t" i - 1 1 m=l i m m=l 1=1 i m 1 k Hence I c. -a. = 0 f o r 1 < m < I , and ( i ) becomes . ., im l ~~ — i = l k I (a-b) 0 1 = 1 d(a.) 0 ( 1 c t m ) i = l 1 m=0 I k I d( I c. -a.) 0 t m " . . ,im x m=0 x=l = d( I c. 0.a.) 0 1 . i = l So a - b e B n(A) , and [a] = [b] e H n(A) . Q.E.D. 2.10 P r o p o s i t i o n : For a l l a, g e <Q , A * * * , e = e D : H (A 0 I) > H (A) * » i s an i s an isomorphism of graded algebras. In p a r t i c u l a r , e^ = e^ isomorphism. Proof: This f o l l o w s e a s i l y from Lemma 2.9, as e*([a 0 1]) = [a] f o r any a e Q . Q.E.D, 2.11 DGA Homotopy: Suppose f, g : A — > B are maps i n VGA . We say that f i s homotopic to g , w r i t t e n f — g , i f there i s a DGA-map F : A — > B 0 I such that e Q ° F = f and e^ ° F = g ; - 47 -l that i s , the f o l l o w i n g diagram commutes: / 7 B F 1 A > B 8 I The question a r i s e s whether DGA-homotopy i s an equivalence r e l a t i o n . I t i s c l e a r l y r e f l e x i v e , as F(a) = f ( a ) 8 1 defines a homotopy from f to i t s e l f . For symmetry, define a DGA map r : B 8 I —> B 8 I by r e q u i r i n g f r ( b 8 1) = b 8 1 f o r a l l b £ B , < r ( l 8 t) = 1 8 (1-t) , r ( l 8 dt) = 1 0.(-dt) . Then e Q 0 r = e^ and e 1 o r = e Q , so i f F : A ^—> B 8 I i s a homotopy from f to g , then r<>F i s a homotopy from g to f . I do not know i f the homotopy r e l a t i o n i s t r a n s i t i v e i n general, but S u l l i v a n [1] has shown that i t i s , when the domain DGA i s i n M . His proof uses our Theorem 2.13, but we omit the d e t a i l s , as we w i l l not use the r e s u l t . The j u s t i f i c a t i o n f o r the name "homotopy" i s t h a t , as i n the case of t o p o l o g i c a l homotopies, we have: - 48 -2.12 P r o p o s i t i o n : I f f — g : A — > B are nomotopic maps i n VGA , then A A A , A f = g : H (A) — > H (B) . Proof: Suppose F : A — > B 8 I i s a homotopy from f to g . Passing to cohomology, P r o p o s i t i o n 2.10 i m p l i e s A A A A A A A A f = (eQ°F) = e Q ° F = e 1 ° F = (e^°F) = g . Q.E.D The reason we intr o d u c e DGA homotopies i s to study induced maps between minimal models. We have the f o l l o w i n g l i f t i n g theorem: 2.13 Theorem: Suppose P : A —;> C and f : M — > C are maps i n A A A VGA, M e M , and p : H (A) —> H (C) i s an isomorphism. Then there i s a DGA map g : M — > A such that f — p•g , so that the diagram M A > C P commutes up to homotopy. Furthermore, i f P i s an epimorphism, we can choose g so that f = P o g . Proof: Assume i n d u c t i v e l y that we have constructed e _ : M — > A °n-l n-1 and a homotopy F 1 : M — > C 8 I from f|M ^ to p ° g •. We n - i n-1 1 n-1 & n - l w i l l extend these maps to one f r e e n-dimensional algebra generator at a time. So assume has only one such generator m e M n , so that M == -M . 8 A (m) as algebras. n n-1 n - 49 -As cl(m) e Mn+?" , we have n—± e.F .(dm) = f(dm) = d(f(m)) e B n + 1 ( C ) . A A A But e^ : H (C 8 I ) = H (C) i s an isomorphism, so F -(dm) e B n + 1 ( C 8 I) . Choose h 1 e (C 8 I ) n f o r which n—1 d(h') = F n (dm) , and set c = e:(h') - f(m) e C n . Then n-1 U d(c) = e.F ,(dm) - f(dm) = 0 as enF . = f|M - , so that 0 n-1 0 n-1 1 n-1 c e Z n(C) . S e t t i n g h = h ' - c 8 1 e ( C 8 I ) n , one v e r i f i e s that d(h) = F (dm) , n - l e Q(h) = f(m) By P r o p o s i t i o n 2.12, f = p ° g n _ 1 ". H (M ±) — > H (C) , so P''([g n_ 1(dm)]) = f*([dm]) = [d(f(m))] = 0 i n H n + 1 ( C ) . As P* i s an isomorphism, g^^dm) E B n + 1 ( A ) , and we can f i n d a e A n such' that d(a) = g .(dm) . Now n-1 d(p(a)) = p(da) = pg n_ 1(dm) = " e l F n - l ( d m ) = e i ( d h > -= d ( e i ( h ) ) , so p(a) - e^(h) e Z n(C) . As P* i s an isomorphism, we can f i n d b £ Z n(A) such that - 50 -P*([b]) = [P(a) - e ^ h ) ] e H n(C) Extend g to g : M — > A by s e t t i n g n-1 n n g (m) = a - b ; n t h i s i s w e l l defined as M i s f r e e . Now we have n d(g (m)) = d(a) = g n(dm) = g (dm) , n n - l n so e i s a map of DGA's. n Observe that Pg n(m) - e^(h) e Z n(C) , as d(pg (m) - e (h)) = Pg (dm) - e. (dh) = n 1 n - l 1 = e F (dm) - e i ( F ,(dm)) = 0 1 n-1 1 n-1 But then we have [pg n(m) - eAh)] = [p(a) - p(b) - e i ( h ) ] = 0 e H n(C) by the choice of b . Hence, we can f i n d x e C n such that d(x) = pg (m) -.e.(h) . n 1 Extend F ^ to F : M — > C 8 I by s e t t i n g n-1 n n ° F (m) = h + d(x 8 t) n . - 51 -Then d(F (m)) = d(h) = F ..(dm) = F (dm) , so F i s a DGA map. n n-1 n n A l s o e F (m) = e n(h) + d(e„(x 8 t ) ) = f(m) , e,F (m) = e.,(h) + d(e_(x 8 t ) ) 1 n 1 1 = eAh) + d(x) = e x(h) + Pg n(m) - e 1(h) = Pg n(m) . So F i s the d e s i r e d homotopy from fIM to P ° g and the n C J 1 n °n i n d u c t i o n continues. Note that the only i n f o r m a t i o n ;about m that we used i s that i t i s a f r e e generator and dm e M , . So i f there n—1 are s e v e r a l n-dimensional generators i n MN , the above c o n s t r u c t i o n can be a p p l i e d one generator at a time to y i e l d g^ and F^ on w i t h the d e s i r e d p r o p e r t i e s . This proves the f i r s t p art of the theorem. Now suppose p i s an epimorphism, and assume i n d u c t i v e l y that we have constructed g .. : M .. —> A so that P ° g , = fIM .. n-1 n-1 n-1 ' n-1 As before, assume that m e i s the only f r e e generator of degree n+1 n+1 n . Now, as above, dm e M ., and g n (dm) e B (A) . Choose n-1 n-1 a e A n such that d(a) = g^_^(dm) . Then d(p(a)) = pg n_ 1(dm) = f(dm) = d(f(m» , so p (a) - f(m) e Z n(C) . * n As p i s an isomorphism, we can f i n d b e Z (A) such that P*([b]) = [P(a) - f(m)] e H n(C) . - 52 -n n—1 Hence p(b) - p(a) + f(m) e B (C) , and we can f i n d x e C such that d(x) = p(b) - p(a) + f(m) . As P i s an epimorphism, we can choose c e A n ^ such that p(c) = x . Now extend g - to g : M — > A by d e f i n i n g °n-l n n g (m) = a - b + d(c) . n Then d(g (m)) = d(a) = g , (dm) = g (dm) , so g i s a DGA map. n n - l n n A l s o , Pg n(m) = p(a) - p(b) + d(p(c)) = p(a) - P(b) + d(x) = f(m) , and p ° g = f|M as d e s i r e d . Q.E.D. °n 1 n We are now i n a p o s i t i o n to prove the uniqueness part of Theorem 2.5. However, we w i l l prove a s l i g h t l y stronger r e s u l t which w i l l be needed i n Chapter 3. 2.14 Theorem: Suppose f : A — > C i s a map of simply connected DGA's such that f : H (A) — > H (C) i s an isomorphism. I f M and N are minimal models f o r A and C , r e s p e c t i v e l y , then M = N . Proof: Suppose p : M — > A and A : N — > C induce isomorphisms on cohomology. Theorem 2.13 gives a map g : M — > N such that the f o l l o w i n g diagram commutes up to homotopy: - 53 -p M - > A I 8| X N > C So f ° P - X ° g , and P r o p o s i t i o n 2.12 i m p l i e s that X X X X - X X f op = X °g : H (M) — > H (C) x x x But f , p and X are isomorphisms, so X X X g : H (M) = H (N) Now C o r o l l a r y 2.7 im p l i e s that g : M = N . Q.E.D. 2.15 Remarks: Throughout t h i s chapter we have considered only DGA's wit h cohomology of f i n i t e type. I f we drop t h i s requirement, and c o n d i t i o n ( i v ) i n the d e f i n i t i o n of a minimal alg e b r a , the r e s u l t s of t h i s chapter remain v a l i d . We inc l u d e d the f i n i t e n e s s c o n d i t i o n to streamline some of the pro o f s , and because one r a r e l y encounters a DGA (or t o p o l o g i c a l space) whose cohomology doesn't have f i n i t e type. For a g e n e r a l i z a t i o n of t h i s theory to the non-simply connected case, see [1]. - 54 -Chapter 3 The Minimal Model i n R a t i o n a l Homotopy Theory In t h i s chapter, we discuss the r e l a t i o n s h i p between the a l g e b r a i c c o n s t r u c t i o n of Chapter 2 and the r a t i o n a l homotopy theory of a t o p o l o g i c a l space. 3.1 D e f i n i t i o n s : By a space we mean a t o p o l o g i c a l space, and maps between spaces are assumed to be continuous. For a space X , we l e t A H (X;G) denote the s i n g u l a r cohomology algebra of X w i t h c o e f f i c i e n t s i n the a b e l i a n group G . I f X i s a s i m p l i c i a l complex, we make fre e use of the n a t u r a l isomorphism between s i n g u l a r and s i m p l i c i a l A cohomology, denoting both by H (X;G) ; i t w i l l be c l e a r from context which theory we are using. Let f : X Y be a map of simply connected spaces. We say that f i s a r a t i o n a l homotopy equivalence i f one (and hence a l l ) of the f o l l o w i n g c o n d i t i o n s h o l d : ( i ) f ^ : Hy,(X;Q) > H J T(Y;Q) i s an isomorphism; A A A ( i i ) f : H ( Y ; Q ) > H (X;Q) i s an isomorphism; ( i i i ) (f^81) : ^ ( X ) 0 Q > TT^(Y) 8 <Q i s an isomorphism. The equivalence of these c o n d i t i o n s f o l l o w s from the mod-C Whitehead theorem [11; Theorem 9.6.22]. Note that i f f i s a r a t i o n a l homotopy equivalence, we are not guaranteed the exi s t e n c e of a map g: Y •+ X A A _ i f o r which g = (f ) Two simply connected spaces, X and Y , are s a i d to have the same r a t i o n a l homotopy type i f there i s a t h i r d simply connected space, Z , and r a t i o n a l homotopy equivalences f : Z — > X and - 55 -g : Z — > Y . R a t i o n a l homotopy type induces an equivalence r e l a t i o n on simply connected spaces; r e f l e x i v i t y and symmetry are c l e a r , and t r a n s i t i v i t y i s a p u l l b a c k argument. Spaces of the same (weak) homotopy type c l e a r l y have the same r a t i o n a l homotopy type. 3.2 Geometric R e a l i z a t i o n : We now o u t l i n e a procedure, due to M i l n o r , which allows us to replace a space w i t h a s i m p l i c i a l complex that contains the same homotopy-theoretic i n f o r m a t i o n . The d e t a i l s can be found i n [ 3 ] , [ 5 ] , [ 6 ] , and [ 7 ] . I f X i s a t o p o l o g i c a l space, l e t S(X) be the graded set of a l l s i n g u l a r s i m p l i c e s i n X ; that i s , ^ n ( ^ ) c o n s i s t s of a l l continuous maps a : A -> X . S(X) becomes a s e m i - s i m p l i c i a l complex n of d e f i n i n g the face and degeneracy operators i n the obvious way (see [ 6 ] ) . From 5(X) we can b u i l d a t o p o l o g i c a l space |S(X)| , c a l l e d the geometric r e a l i z a t i o n of S(X) , as f o l l o w s : to each a e S(X) we as s o c i a t e an n - c e l l \a\ = {a} x A , ' n where {a} i s the set w i t h one element. Then |S(X)| i s obtained from t h e i d i s j o i n t union of a l l such c e l l s by i d e n t i f y i n g {alF.A } x A • ' i n n ' w i t h { a } x F . A , 0 < i < n . We give |S(X)I the i d e n t i f i c a t i o n i n — — topology: a subset W <=• |S(X)| i s open i f f W f\ \o\ i s open i n \a\ f o r every a e S(X) . One v e r i f i e s that |S(X)| i s a CW-complex wi t h one open n - c e l l - 56 -<cr> = {c} x i n t ( A ) n f o r each non-degenerate' simplex a e 5 (X) . A l s o , f o r n > 2 , the n — th n b a r y c e n t r i c s u b d i v i s i o n of |S(X)| , defined i n the obvious way, i s a s i m p l i c i a l complex. ' We now define the n a t u r a l p r o j e c t i o n cox : |S(X) | —> X . I f y e |S(X)| , we have y e <a> = {a} x i n t ( A ^ ) f o r some unique a : A^ -y X , y = (a,t) f o r some t e i n t ( A ^ ) . Define u x ( y ) = o ( t ) . I t i s c l e a r t h a t w i s a continuous s u r j e c t i o n . A l s o , i f A c X i s X a subspace, |S(A) | cz |S(X) | as a subcomplex, and ' co (|S(A)|) = A . X The most important f a c t about co i s that i s an isomorphism f o r a l l n , so that co i s a weak homotopy A. equivalence. Hence, i f X has the homotopy type of a CW-complex the Whitehead theorem i m p l i e s that co i s a homotopy equivalence. The above c o n s t r u c t i o n can be made f u n c t o r i a l . I f f : X — i s a map, we define a map 5 ( f ) | : |S(X)| — > |S(Y) as f o l l o w s : f o r an a r b i t r a r y p o i n t (o,t) e 5(X) , where a : A ->• X n - 57 -and t e i n t ( A ) , we set n |S(f)| (o,t) = (f°a, t) . One v e r i f i e s that |S(-)| i s a functo r from the category of t o p o l o g i c a l spaces to i t s e l f , and that co i s a n a t u r a l transformation from |S(-)| to the i d e n t i t y f u n c t o r . 3.3 The Minimal Model: Let K be a simply connected s i m p l i c i a l * complex w i t h r a t i o n a l cohomology of f i n i t e type. Then E (K) , the de Rham algebra of K , i s a simply connected DGA, and by Theorem 2.5 we can b u i l d i t s minimal model M(K) = M(E*(K)) . M(K) w i l l be c a l l e d the minimal model of K . I f X i s a simply connected space (with r a t i o n a l cohomology of f i n i t e t y p e ) , we may t r i a n g u l a t e the geometric r e a l i z a t i o n |S(X)| of ,S(X) to ob t a i n a s i m p l i c i a l complex |S(X)|' . We define the minimal model, M(X), of X to be the minimal model of |S(X)| : M(X) = M(E*(|S(X)|')) ., Note that i f K i s a s i m p l i c i a l complex, we have two d e f i n i t i o n s of i t s minimal model, namely M(E (K)) and M(E (|S(K)| )) • The f a c t that these DGA's are isomorphic f o l l o w s from the proof of the next theorem. The proof w i l l a l s o show that M(X) doesn't depend on the way we t r i a n g u l a t e |S(X)| . - 58 -3.4 Theorem: Simply connected spaces of the same r a t i o n a l homotopy type have isomorphic minimal models. Proof: Suppose X and Y have the same r a t i o n a l homotopy type. Then there i s a space Z and r a t i o n a l homotopy equivalences f : Z — > X and g : Z — > Y . Triangulate the geometric r e a l i z a t i o n s of these spaces to get a commutative diagram: |S( f ) | . .. |5(g)| |S(X) 5(Z) -> S(Y) X. X Y . As the v e r t i c a l maps are weak homotopy equivalences, we have that |S ( f ) | and |S(g)| are r a t i o n a l homotopy equivalences. By the s i m p l i c i a l approximation theorem (subdividing |S(Z)|' i f necessary) we can f i n d s i m p l i c i a l maps <f> : |S(Z) | — > |S(X) | and .ty : |S(Z)| — > |5(Y)| which are also r a t i o n a l homotopy equivalences. Now applying the de Rham functor E , we obtain DGA-maps <f>* : E*(|S(X)|') — > E*(|S(Z)|') and ty* : E*(|S(Y)|') —> E*(|S(Z)|') which induce isomorphisms on cohomology. Hence, by Theorem 2.14 we have M(X) = M(Z) = M(Y) Q.E.D. - 59 -3.5 Examples: . . . . . . . (a) Let S n denote the n-sphere. I f n i s odd, n >^  3 , the r a t i o n a l cohomology of S n i s f r e e , so by Example 2.4(b) the minimal model i s M(S n) = A (x) , n odd n " ' * 2 w i t h zero d i f f e r e n t i a l . I f n i s even, we must k i l l the cocycle x i n degree 2n , so M(S n) = A (x) 0 A 0 .(y) , n even n zn-1 2 w i t h the d i f f e r e n t i a l given by d(x81) = 0 , d(18y) = x 8 1 . (b) L e t CP n denote complex p r o j e c t i v e n-space. The r a t i o n a l cohomology of CP n i s a truncated polynomial algebra on a s i n g l e generator of degree 2 , truncated a height n + 1 . The minimal model i s M(CP n) = A 2 ( x ) 8 A 2 n + 1 ( y ) , w i t h d i f f e r e n t i a l d(x81) = 0 , d(18y) = x n + 1 8 1 . In both of these examples i t can be shown d i r e c t l y that the given model i s the only minimal algebra w i t h the c o r r e c t cohomology algebra. (c) Let TT be a f i n i t e l y generated a b e l i a n group, and n >_ 2 an i n t e g e r . An Eilenberg-MacLane space of type (Tr,n) i s a space K ( iT,n) f o r which ir k(K(ir,n) ),-=-•{ IT , k = n 0 , k f n - 60 -Any two Eilenberg-MacLane spaces of type ( T , n ) have the same weak homotopy type; furthermore we may choose K ( iT,n) to be a s i m p l i c i a l complex. The r a t i o n a l cohomology of K(ir,n) i s the f r e e algebra on I generators of degree n , where I i s the rank of IT . Hence M(K (TT,n)) = H * (KU,n);Q) = A (x , ,x ) , Jl = rank(ir) , n i a w i t h zero d i f f e r e n t i a l . Notice t h a t , i n a l l these examples, the number of f r e e generators of degree n i n M(X) i s e x a c t l y the rank of TT (X) . The proof of t h i s r e s u l t i n the general case occupies the remainder of the t h e s i s . We f i r s t d iscuss the two main t o o l s to be used: s p e c t r a l sequences and the Postnikov decomposition. 3.6 S p e c t r a l Sequences: In t h i s s e c t i o n we describe the two s p e c t r a l sequences used i n t h i s chapter. The f i r s t i s the Serre s p e c t r a l sequence of a f i b r a t i o n , and the second i s a s p e c i a l case of the s p e c t r a l sequence of a f i l t e r e d DGA. D e t a i l s can be found i n [11; Chapter 9 ] . A l l f i b r a t i o n s considered are o r i e n t a b l e (see [11; S e c t i o n 9.2]). i P Let F > E > B be an o r i e n t a b l e f i b r a t i o n over a connected CW-complex B , A c: B a subcomplex,-and G an a b e l i a n group. The Serre s p e c t r a l sequence of t h i s f i b r a t i o n i s given by E^' q = H P(B,A;H q(F;G)) , - 61 -ft* and the d i f f e r e n t i a l on the E ^ - l e v e l has bidegree ( r , l - r ) . i s the bigraded module a s s o c i a t e d w i t h a decreasing f i l t r a t i o n of * -1 n H (E,E ;G) , where E = p (A) . L e t t i n g B denote the n-skeleton A A —1 n of B and s e t t i n g = p • (A V B ) , the f i l t r a t i o n i s given by: F PH n(E,E A;G) = Ker{H n(E,E A;G) > H n(E ,EA;G) } . We have F n + 1H n(E,E A;G) = 0 , F°H n(E,E A;G) = H n(E,E A;G) and E P ' q = F % P + q ( E , E A ; G ) / F P + V + q ( E , E : G ) . 0 0 A A The map p* : Hn(B,A;G) > H n(E,E A;G) can be fa c t o r e d as H n(B,A;G) = E 2'° » E*'° = F nH n(E,E A;G) ^ — > H n(E,E A;G) . A l s o , i f A = <j) , i " : H n(E;G) > H n(F;G) can be f a c t o r e d as H n(E;G) = F°Hn(E;G) » E ° ' n > > E°' n = H n(F;G) . These f a c t o r i z a t i o n s can be used to derive the Serre exact sequence: i f B i s n-connected and F i s m-connected, there i s an exact sequence ... — > H q(B;G) X H q(E;G) X H q(F;G) - U H q + 1(B;G) —>. . . — > H n + m + 1(F;G) where x i s the t r a n s g r e s s i o n . Now l e t A be a DGA, and set A = A (x.,...,x„) , n > 2 . n n 1 x. — Suppose that the graded algebra C = A 8 ^ i s equipped w i t h a d i f f e r e n t i a l i n such a way that d(a®l) = d(a) ® 1 f o r a E A and - 62 -d(l®xi) e A n + 1 8 1 . We define a decreasing f i l t r a t i o n on C by F P C k = » A q 0 A k" q . n q>p We have 0 = F k + 1 c k - F kC kc= . . . ^ F ° C k = C k A l s o , t h i s f i l t r a t i o n i s preserved by the m u l t i p l i c a t i o n and d i f f e r e n t i a l i n C : F PC ' F qC <= F P + q c , d(F PC) <= F P + 1 C «= F PC Hence there i s a convergent E ^ - s p e c t r a l sequence given by E P , q = H P + q ( F P C / F P + 1 C ) , * p p+1 which converges to some f i l t r a t i o n of H (C) . As d(F C) <=• F r C , we have Ep,q = F P c p + q / F p + 1 c p + q = A P 6 A q . 1 n We wish to compute the E ^ - l e v e l of t h i s s p e c t r a l sequence. In the general case, the elements of E P ' q = H P + q ( F P C / F P + 1 C ) have the form [c + F P + 1 C ] where c e F PC and dc e F P + 1 C . The d i f f e r e n t i a l d1 : E P , q > E P + 1 ' q i s then given by d L ( [ c + F P + 1 C ] ) = [dc + F P + 2 C ] . - 63 -Hence, f o r a 8 b e A P 8 q = E P ' q , d^aQb) i s j u s t the sum of terms of d(a8b) that are i n A P + 1 8 A q ; that i s n d 1(a8b) = d(a) 8 b . Therefore,- E P ' q = H P (A) 8 A^ . 3.7 The Postnikov Decomposition: I f X i s any pointed space, we denote the space of paths to the base,point by PX , and the space of loops at the base p o i n t by ffit . R e c a l l that PX i s c o n t r a c t a b l e , and that there i s the standard path f i b r a t i o n PX'—> X w i t h f i b r e QX . Let X be a simply connected space. The Postnikov decomposition of X i s a tower of spaces and maps: X X w i t h the f o l l o w i n g p r o p e r t i e s : ( i ) p of = f • n n n-1 - 64 -( i i ) \ ( x n ) = 0 f o r k > n > ( i i i ) (f ),, : IT (X) — — > TT (X ) i s an isomorphism f o r k < n ; n ff k k n — ( i v ) p i s a p r i n c i p a l f i b r a t i o n w i t h f i b r e K(TT ( X ) , N ) In f a c t , there i s a p u l l b a c k diagram: K(TT (X),n) n n n-1 fiK(TT (X), n+1) n PK(TT (X),n+1) n P u l l b a c k -> K(TT (X),n+1) n where the r i g h t column i s the path f i b r a t i o n . The above p r o p e r t i e s determine the spaces X^ up to weak homotopy type. As X^ i s a K(^2 (-^ )»2) > we may assume that X^ has the homotopy type of a CW-complex (see [ 8 ] ) . The map k : X , — > K(TT (X) ,n+i) i s c a l l e d n n-1 n th the n k - i n v a r i a n t of X and i s determined up to homotopy. D e t a i l s of the c o n s t r u c t i o n of the Postnikov decomposition can be found i n [9; Chapter 13]. The Postnikov decomposition of X allows us to focus a t t e n t i o n on one homotopy group at a time. We now study the p r i n c i p a l f i b r a t i o n s p^ i n the decomposition and show that the minimal model of the t o t a l space X^ i s j u s t the tensor product of the minimal models of the base X . and the f i b r e K(TT (X) n) w i t h a s u i t a b l e n-1 n d i f f e r e n t i a l . - 65 -3.8 The Main C o n s t r u c t i o n : Let p : E > Y be a f i b r a t i o n w i t h f i b r e K(ir,n) , where Y i s a simply connected s i m p l i c i a l complex, E has the homotopy type of a simply connected CW-complex, TT i s a f i n i t e l y generated a b e l i a n group, and n >^  2 . For each subcomplex B <=• Y , l e t E„ = p _ 1 ( B ) . Then the map p I E _ : E„ > B i s a l s o a C x i an o r i e n t a b l e f i b r a t i o n w i t h f i b r e K( iT,n) . We denote the m-skeleton of Y by Y m , and set E = p \Y™) = E . Note t h a t , f o r each 3 m v ' ym v e r t e x y e Y , E^ i s an Eilenberg-MacLane space of type (ir,n) . As i n 3.2, l e t |S(E)| be the geometric r e a l i z a t i o n of S(E) ; t h i s i s a CW-complex which i s t r i a n g u l a b l e , and the e v a l u a t i o n map LO : |S(E) —>> E i s a homotopy equivalence. For any subcomplex B <=• Y we have E <^  E , and hence 1 5 (E ) I i s • a subcomplex of |S(E)| . Now b a r y c e n t r i c a l l y subdivide |S(E)| twice to get a s i m p l i c i a l complex. Let f : K > Y be a s i m p l i c i a l approximation to the composite i |S(E)| — » E Y , where K i s a f u r t h e r s u b d i v i s i o n of |S(E)| . For a subcomplex B <= Y , l e t denote |S(E^)| w i t h t h i s s u b d i v i s i o n ; c l e a r l y Kg i s a s i m p l i c i a l subcomplex of K , and f|l<- i s a s i m p l i c i a l B approximation to the composite |.S(E B)| — » Eg — > B . A l s o , i f B = Y M , we l e t K^ = K^ . So, f o r each subcomplex B CL Y and ve r t e x y e B ,.we have a diagram - 66 -y B K <— > K y B -> B -> B where the maps i n the bottom row are s i m p l i c i a l , and the v e r t i c a l maps are homotopy equivalences. Choose a minimal model M(Y) f o r Y and a DGA map p : M(Y) > E*(Y> which induces an isomorphism on cohomology. Notice t h a t , as K i s y a K(ir,n) f o r any v e r t e x y e Y , we have H D R ( K y ) S H (iy<Q) - A ^ , . . . ^ ) , the f r e e algebra on generators x ,...,x i n degree n , where SL i s the rank of TT . Hence A (x, ,...,x- ) i s the minimal model of K n 1 SL y We w r i t e A f o r A (x,,...,x.) . Our aim i s to equip E (Y) & A n n 1 SL n w i t h a d i f f e r e n t i a l i n such a way that i t s minimal model i s M(Y) 8 A^ wi t h a s u i t a b l e d i f f e r e n t i a l , and so that there i s a DGA map A : E (Y) 0 A > E' (K) n inducing an isomorphism on cohomology. I t w i l l f o l l o w that M(K) = M(Y) 0 A - 67 -Lemma: For any v e r t e x y e Y , the i n c l u s i o n i : K y > induces an isomorphism i : (K ) = (K ) . DK 1 DK y Proof: Consider the Serre s p e c t r a l sequence w i t h r a t i o n a l c o e f f i c i e n t s i P 1 f o r the f i b r a t i o n E > E, —:——> Y . This i s given by y 1 E P ' q = H P C Y V C E * ) ) y * N P l and converges to H (E^;Q) . As H (Y ) = 0 f o r p > 2 and H*(E ;<D) = A , we have that y n E P ' q = 0 f o r p\> 2 or q f 0 (mod n) Hence a l l the d i f f e r e n t i a l s i n the s p e c t r a l sequence, are zero, and E P ' q = E P ' q . A l s o , on the p + q = n di a g o n a l , the only non-zero O n " n n entry i s E^' . So the edge homomorphism i : H (E^;Q) > H (E^;Q) can be f a c t o r e d as H n( E ; L;Q) = F°H n( E l;Q) = E°Jn - E°2'n - H n(E y;Q) , and hence i i s an isomorphism i n dimension n . The r e s u l t f o l l o w s from the n a t u r a l equivalence of s i m p l i c i a l and de Rham cohomology, and the commutative diagram "> E, K "> K, where co i s a homotopy equivalence. Q.E.D. - 68 I t f o l l o w s from t h i s Lemma that (K ) i s an SL-dimensional DR 1 vector space. For 1 <_ i < £ , choose closed forms OK £ E n(K^) whose cohomology c l a s s e s form a ba s i s f o r HQR(K^ ) . Then, f o r any ver t e x y e Y , the c o l l e c t i o n {[OJ_^ |K ] 1 S a f r e e set of * generators f o r the algebra H „(K ) = A . By P r o p o s i t i o n 1.10, we DR y n can extend co. to a form co. e E n(K) . Then dco. i s a clo s e d form x x x on K which i s zero on K„ . We consider dco. as an element of 1 x -E n + 1 ( K , K Q ) . Lemma: [dcT.] e Im{f* : H ^ C Y . Y 0 ) > HJJ^ CK.KQ)} . Proof: Consider the Serre s p e c t r a l sequence f o r the f i b r a t i o n P 0 E > E > Y r e l a t i v e to Y . I t i s given by y E P ' Q = H P(Y,Y°;H Q(E y;Q)) , and converges to H ( E , E Q , Q ) , . As E P ' Q = 0 f o r p = 0 or q i 0 (mod n) , we see that a l l the d i f f e r e n t i a l s t e r m i n a t i n g at the (n+1,0) • * '• j -i T,n+1,0 ~ „n+l ,0 ,. posxtxon are zero, and hence E^ = E ^ . Now the edge homomorphism p : H n + 1(Y,Y°;Q) > H N + 1 ( E , E 0 , Q ) f a c t o r s as H n + 1(Y,Y;Q) » Ef 1'° - E ; + 1 ' ° = F n + 1 H N + 1 ( E , E o ; Q ) <=^ H N + 1 ( E , E o ; Q ) , and hence Im(p'S) = F n + 1 H N + 1 ( E , E 0 ; Q ) . But on the p + q = n + 1 diag o n a l , the only non-zero e n t r i e s are g n + x » 0 a n ( j E n'^ . Hence oo • 0 0 * n+1 n+1 lm(p ) = F n + i H N + 1 ( E , E 0 ; Q ) = F 2 H N + 1 ( E , E Q ; Q ) = K e r { H N + 1 ( E , E 0 ; Q ) > H ^ V ^ E ^ Q ) } - 69 -Passing to the homotopy equivalent maps K — > K — > Y. we y have Im{f* : H ^ C Y . Y 0 ) - > H ^ V y } = K e r l H ^ 1 (K,K Q) - > ^ ( K ^ ) } . Now the map on the r i g h t i s induced by the r e s t r i c t i o n to of the - . _ x forms on K, and as duu|K^ = 0, we have [duK] e Im(f ) . Q.E.D. Hence we can f i n d cohomology c l a s s e s c. e H?^(Y,Y^) = H^^(Y) i DK UK x - n - j - l such that f (c.) = [dco.] i n H „ (K,K_) . Note that c. i s the 1 1 DR 0 l tr a n s g r e s s i o n of [co_jK y] i n the Serre s p e c t r a l sequence f o r K — > K — > Y, and t h i s i s the case f o r any vert e x y e Y . As p : H (M(Y)) — > H^(Y) i s an isomorphism, we can f i n d m± e Z n + 1 ( M ( Y ) ) such that p*([m ]) = c , 1 < i < t . Set di. = p(m.) e E n +^"(Y), so that [ty.] = c. . Considering ty. as an i i i i ° I n+1 0 * - n+1 element of E (Y,Y ) , we have that [ f (ty±)] = [dco ±] i n H (K,K Q) . Hence there are forms n e En(K,KQ) , 1 £ i _< I, such that * f (.ty*) ~ duK + dn. . A l s o , as i s zero on we have that G± + n.)|K 0 = co.|K 0 . X Now def i n e a d i f f e r e n t i a l i n the algebra E (Y) 8 by s e t t i n g d(? 0 1 ) = d(0 0 1 f o r £ £ E (Y) , • d ( l 0 x i) = ty± ®'l 1 1 1 1 1 » X and extending to a l l of E (Y) 0 A^ by r e q u i r i n g that d be a l i n e a r d e r i v a t i o n . S i m i l a r l y , define a d i f f e r e n t i a l i n M(Y) 0 A n by s e t t i n g - 70 -d(a 0 1 ) = d(a) 0 1 f o r a e M(Y) , { d ( l 0 x ± ) = m 8 1 , 1 < i <_ i . As pCiru) = ty^, the algebra map p 8 1 : M(Y) 0 A —> E (Y) 0 A n n i s a map of DGA's . We w i l l l a t e r show that p 0 1 induces an isomorphism on cohomology. Define a map A : E*(Y) 0 A —> E*(K) by s e t t i n g A(£ 0 1 ) = f*(£) f o r I e E*(Y) A ( l 8 x j = OK + n , 1 < i < i ,. and extending so as to be an algebra map. We have dX(£ 0 1 ) = d f * ( 0 = f*(d?) = A(d£ 8 1) = Ad(£ 0 1) , and d A ( l 0 x ± ) = da5i + d n ± = f * ^ ) = X(ty± 8 1) = Ad( l 8 x ± ) , so A i s a map of DGA's . For each subcomplex B e Y, the map A r e s t r i c t s to a DGA map A B : E*(B) 0 A n —> E*(K B) by s e t t i n g d ( l 0 x ± ) = ( i p j B ) 0 1 i n E^(B) 0 A , and - 71 -X (1 ® x.) = (co. + n.) |K . Then, f o r subcomplexes B <=• C <^ Y, there x x x 13 i s a commutative diagram of DGA maps E (C) 8 A E (B) 0 A C * 2 XT E ( V induced by K -> C The main t e c h n i c a l r e s u l t of the chapter i s : 3.9 Theorem: The induced map X : H (E (Y) 8 A n) —> Hj^OO i s an isomorphism. Proof: We show that X^ i s an isomorphism f o r every f i n i t e dimensional subcomplex B <= Y ; t h i s we do by i n d u c t i o n on the dimension of B . I f y e Y i s a v e r t e x , we have - 72 -E (y) 0 A = Q 8 A = A n n n w i t h zero d i f f e r e n t i a l , and A i s j u s t the map A — > E (K ) that y n y takes x. to (co. + n.) I K = CO.IK . But {[co.lK ] } ^ .. i s a system of 1 i i y i y • ly 1 = i f r e e generators f o r H _, (K ) = A , and hence A i s an isomorphism. DR y n y r I f B i s any o-dimensional subcomplex, then = K , and B „ y yeB 3 K f\ K = 0 f o r d i s t i n c t v e r t i c e s y and z of B . Hence there i s y z . a d i r e c t product decomposition E (K ) = II E (K ) . B yeB y We a l s o have E*(B) 0 A = (JT E*(y)) 0 A = TT (E*(y) 8'A ) , yeB yeB -and Ag i s j u s t the d i r e c t product of the maps A . As cohomology commutes w i t h d i r e c t products and each A y i s an isomorphism, A^ i s an isomorphism. Now assume that A^ i s an isomorphism f o r a l l subcomplexes B <=• Y w i t h dim(B) <_ m - 1, f o r some m >_ 1 . Let B be any m-dimensional subcomplex of Y, and l e t C denote the c o l l e c t i o n of subcomplexes C m-1 * such that B d C B and A^ , i s an isomorphism. By the i n d u c t i o n h y pothesis, B e C, so C i s non-empty. We show that Zorn's Lemma a p p l i e s . Suppose {C : a e J} i s a chain i n C w i t h C cz. Q, 'when a a 3 a <_ 3 i n J . Set C = \J C ; then K = U K • For each a <_ 3, a a a we have a commutative diagram: - 73 -E (C_) 0 A n E (C ) 0 A a' n -> E ( K C ) "> E (Kc ) , a where the v e r t i c a l maps are induced by i n c l u s i o n . So the c o l l e c t i o n {A c : a e J} ±s a map between in v e r s e systems of DGA's . By a . P r o p o s i t i o n 1.12 we have isomorphisms and l i m (E*(C ) 8 A ) = < a n a l i m E (C ) < — a a 0 A E (C) ,0 A n , l i m E*(K ) £ E*(K ) , <— La C a th and the induced map on the l i m i t s corresponds under these isomorphisms * A to A : E (C) 8 A — > E (K ) . Again by 1.12, cohomology commutes wi \> n o . * * i n v e r s e l i m i t s , and as each X i s an isomorphism, X i s a l s o an isomorphism. Hence C e C . We have shown that every chain i n C has an upper bound and hence, by Zorn's Lemma, there i s a maximal element C e C . I f C = B , t h i s completes the i n d u c t i o n step.. So suppose C ^ B . Then there i s an m-simplex a <=• B such that C n a = 9a, . Let D = C u a ; we show that D e C, c o n t r a d i c t i n g . t h e maximality of C . * The d i f f e r e n t i a l i n E (D) 8 A r e s t r i c t s to a d i f f e r e n t i a l n * i n E (D,C) 0 A , and we de f i n e a map of DGA's n - 74 -A' : E (D,C) 0 A n —> E ( K ^ i y by r e q u i r i n g that the f o l l o w i n g diagram be commutative (with exact rows); -> E (D,C) 0 A n -> E (D) 0 A n -» E (C) 8 A n -> 0 (1) -> E ( KD ' V , * -> E «D> -» E (Kc) -> 0 This diagram induces a map of long exact cohomology sequences, and A^ , i s * an isomorphism as C e C . So, by the 5-lemma, to show that A^ i s an isomorphism i t s u f f i c e s to show that (A') i s an isomorphism. S i m i l a r l y f o r the p a i r (a,3a), there i s a DGA-map A" i n the f o l l o w i n g commutative diagram (with exact rows); 0 -> E (a,3a) 0 A -> E (a) 8 A - E ( K a ' K 3 a } -> E (Ka) -» E (3a) 0 A n 3a ~ » E < K 3 o > -> 0 (2) -> 0 A l s o , the i n c l u s i o n s h : (a,3a) — > (D,C) and h. : (K , K. ) — > (K_,K ) J- 2. a do D C induce maps on the r e l a t i v e de Rham algebras, and we have the f o l l o w i n g commutative square: - 75 E (D,C) 0 A n h, 0 1 -> E (o,3a) 8 A A" (3) * A A * C l e a r l y : E (D,C) —> E (o,do\ i s an isomorphism, and hence h^ 0 1 i s a l s o an isomorphism. By [11; Lemma 9.2.2], the i n c l u s i o n (E ,E. ) > (E ,E ) 0 oO D C A A' induces an isomorphism H (E^JE^) s H (E^jEg^) on i n t e g r a l cohomology, A and hence i n diagram (3) induces an isomorphism on de Rham cohomology. A So to prove that (A') i s an isomorphism, i t s u f f i c e s to show that A (A") i s an isomorphism. Diagram (2) induces a map of long exact cohomology sequences and, A' A A again by the 5-lemma, (A") i s an isomorphism i f both A and A are 0 0 0 A isomorphisms. As • 30 i s an (m - 1)-dimensional subcomplex, A i s an 0 0 A isomorphism by t h e , i n d u c t i o n hypothesis. So we need only show that A i s an isomorphism. Let y be a ver t e x of 0 , and denote the i n c l u s i o n s by j n : y — > 0 and i„ : K — > K . Then there i s a commutative J J l J 2 y 0 diagram: E ( 0 ) 8 A n A E (K a) A h 0 1 - ± - -> E (y) 0 A -> E (K ) y (4) As 0 i s c o n t r a c t i b l e , j 2 i s a homotopy equivalence, and j 2 i s an - 76 -A isomorphism on cohomology. Also A i s an isomorphism by the i n d u c t i o n A hypothesis. So to show that A^ i s an isomorphism, i t s u f f i c e s to show A that j 9 1 induces an isomorphism on cohomology. A As i n s e c t i o n 3.6, there are f i l t r a t i o n s on the DGA's E (a) ,6 A . A and E (y) 8 A^ and the a s s o c i a t e d s p e c t r a l sequences are given by i s - " s • A A A A These s p e c t r a l sequences converge to H (E (o) 8 A ) and H (E (y) 8 A^) , A A A r e s p e c t i v e l y . Now j 8 1 : E (a) 8 A^ —> E (y) 8 preserves f i l t r a t i o n s , and hence induces a morphism of s p e c t r a l sequences E —r' E . On the E ^ - l e v e l , t h i s morphism i s j u s t j * 8 1 : H J R ( O ) 8 A q > H P R ( y ) 8 A q , which i s c l e a r l y an isomorphism as a i s c o n t r a c t i b l e . So, by the comparison s p e c t r a l sequence theorem, ( j * 8 1 ) * : H * ( E X(a) 8 A ) — ^ H * (E*(y) 8 A ) i n n i s an isomorphism, as d e s i r e d . A R e t r a c i n g our steps, we have shown that A^ i s an isomorphism and hence D e C, c o n t r a d i c t i n g the maximality of C e C . Hence A C = B, and A i s an isomorphism, as d e s i r e d . This concludes the i n d u c t i o n step, and gives the r e s u l t f o r a l l f i n i t e dimensional subcomplexes of Y . A  , same in v e r s e l i m t argument as i n the m>0 - 77 -. , . , A A A A i n d u c t i o n step shows that X : H (E (Y) 8 A ) — > H (K) i s ap n DR " isomorphism. , Q E D 3.10 P r o p o s i t i o n : The induced map (p 0 1 ) * : H*(M(Y) 0 A ) —>. H * (E*(Y) 0 A ) n n i s an isomorphism. Proof. Again us i n g the r e s u l t s of 3.6, there are f i l t r a t i o n s on the DGA's M(Y) 0 and E (Y) § A , • and the a s s o c i a t e d s p e c t r a l sequences are given by: E P ' q = H P(M(Y)) 0 A q n A A A These s p e c t r a l sequences converge to H (M(Y) 8 A ) and H (E (Y) . 8 A ), r e s p e c t i v e l y . As p 8 1 preserves f i l t r a t i o n s , i t induces a morphism of s p e c t r a l sequences E — > E which, on the E 2 - l e v e l , i s given by p 8 1 : H P(M(Y)) 6 A q —> H P (Y) 8 A q . n DR n A As p i s an isomorphism, so i s p 8 1 . Hence, by the comparison M . A s p e c t r a l sequence theorem, (p 8 1) i s an isomorphism. Q.E.D. S t i l l i n the n o t a t i o n of 3.8, we have: 3.11 C o r o l l a r y : Suppose M(Y) has no algebra generators of degree n + 1 Then M(Y) 8 w i t h the above d i f f e r e n t i a l i s the minimal model of E . - 78 -Proof: By d e f i n i t i o n , M(E) = M(E (K)) . Now the d i f f e r e n t i a l i n M(Y) 8 A i s decomposable because n d ( l 0 x.) = m. 0 1 e M n + 1 ( Y ) 0 A° , i i n and the elements of M n +^(Y) are decomposable by hypothesis. Hence M(Y) 0 i s a minimal algebra. By Theorem 3.9 and P r o p o s i t i o n 3.10, the composite A o (p 0 1) : M(Y) 0 A — > E*(K) n i s a DGA map which induces an isomorphism on cohomology, and the r e s u l t f o l l o w s . Q.E.D. We now come to the main theorem of the t h e s i s . The n o t a t i o n f o r the Postnikov decomposition i s described i n s e c t i o n 3.7 . 3.12 Theorem: Let X be a simply connected space and X n the n ^ space i n i t s Postnikov decomposition. Then M (X) = M(X ) s M U .) 0 A ( x l 5 x ) n n n-1 n 1 Jo w i t h a s u i t a b l y d efined d i f f e r e n t i a l , where H = rank (IT (X)) . Hence the n number of f r e e generators of degree n i n M(X) i s the rank of IT (X) . n Proof: F i r s t , because the map f ^ : X -> X^ induces an isomorphism on homotopy groups through degree n and an epimorphism i n degree n + 1 , the same i s true on homology by the Whitehead Theorem (see [11; Theorem 7.5.9.]) Hence, by the u n i v e r s a l c o e f f i c i e n t theorem, f induces an isomorphism on r a t i o n a l cohomology through degree n and a monomorphism i n degree n + 1 '. - 79 -Now we g e o m e t r i c a l l y r e a l i z e X and X and s i m p l i c i a l l y approximate n f , o b t a i n i n g a homotopy equivalent s i m p l i c i a l map, which we a l s o denote f : X — > X . Let p : M(X) — > E * ( X ) and p' : M(X ) — > E(X ) be n n n n minimal models f o r X and X n . By Theorem 2.13, there i s a DGA map g : M(X ) — > M(X) such that the f o l l o w i n g diagram commutes up to DGA-homotopy: M(X ) n P ' 3 -> E ( X N ) M(X) "> E ( X ) * * * k k As p and (p 1) are isomorphisms, g : H ( M ( X ^ ) ) — > H ( M ( X ) ) i s an isomorphism f o r k <_ n and a monomorphism f o r k = n + 1 . Hence by v Theorem 2.6, M ( X ) = M ( X ) . n n n We now prove the theorem by i n d u c a t i o n on n . For n = 2, X 2 = K ( T T 2 ( X ) , 2 ) , and by Example 3.5(c) we have M ( X 2 ) = A 2 ( X ; L , x 2 ) , £ = rank ( T T ^ X ) ) Hence M 2 ( X ) = M 2(X 2) = M(X 2) as d e s i r e d . Assume i n d u c t i v e l y that M n ( X ) = M(X ,) f o r some n > 3 n-1 n-1 — Let Y denote a t r i a n g u l a t i o n of the geometric r e a l i z a t i o n of X P u l l back the f i b r a t i o n p : X n n -> X n-1 n-1 over the e v a l u a t i o n map co : Y — > : > - x n _ i t o 8 e t a f i b r a t i o n p : E — > Y w i t h f i b r e K(TT ( X ) ,n) As co i s a homotopy equivalence, E has the same homotopy type as X n - 80 -and p i s equivalent to p n i n the homotopy category. We can now apply the c o n s t r u c t i o n i n s e c t i o n 3.8 . As M(Y) s M ( X ) s M (X) , n-1 n-1 there are no generators of degree n + 1 i n M(Y), so C o r o l l a r y 3.11 y i e l d s M(X ) = M(E) n a M(Y) 0 A (x_, x ) n 1 Jo = M(X n ) 0 A (x , x ) , n-x n ± x. where I = rankCiT^CX)) . This a l s o shows that M^X^) has no generators above degree n, so M (X) = M (X ) = M(X ) . n n n n This proves the theorem. Q.E.D. We prove one l a s t r e s u l t about the minimal model. R e c a l l that the Hurewicz homomorphism i s a n a t u r a l transformation from homotopy to homology: h : IT (X) — > H (X) . n n n I f X i s (n - l)- c o n n e c t e d , n'>_ 2, then h^ i s an isomorphism f o r k <_ n and an epimorphism f o r k = n + 1 . As the minimal model contains i n f o r m a t i o n about both homotopy and (co)-homology, i t i s reasonable to expect i n f o r m a t i o n about the r a t i o n a l Hurewicz homomorphism - 81 -h 8 1 : TT (X) 8 Q — > H (X) 8 Q = H (X;Q) . n n ^ n • n Roughly speaking, the rank of h^ 8 1 i s the number of d-closed algebra generators of degree n i n M(X) . As there are many ways to choose a system of generators-, the p r e c i s e statement i s : 3.13 Theorem: Let X be a simply connected space w i t h cohomology of f i n i t e type, and n _^ 2 an i n t e g e r . Then the rank of h 8 1 i s the vect o r space dimension of Z n ( M n ( X ) ) / Z n ( M ^(X)) . Proof: Consider the Postnikov decomposition of X as described i n s e c t i o n 3.7 . Let g : F —> X be the homotopy-theoretic f i b r e of f T : X — > X , . From the a s s o c i a t e d exact sequence i n homotopy we n-1 n-1 have that F i s (n - 1)-connected and g,, : Tr (F) — > IT (X) i s an it n n isomorphism. Therefore the Hurewicz homomorphism h : TT (F) — > H (F) n n i s an isomorphism. By the n a t u r a l i t y of the Hurewicz homomorphism, we have a commutative diagram TT (F) / > TT (X) n = n h n H ( F) > H (X) » H (X ) • > 0 n • g* n ' (f ) n n _ 1 1 1 - 1 * where the bottom row-is a p o r t i o n of the Serre exact sequence f o r the homotopy f i b r a t i o n F —> X — > X - . Hence we have n-1 Im(h n) = Im(g A) = K e r C f ^ - 82 -As t e n s o r i n g w i t h Q i s exact, Im(h 0 1 ) = Im(h ) 0 Q = K e r ( f »). 0 Q n n n-1 * = K e r { ( f ' ) : H (X;Q) — » H (X ;Q)} . n — i K n n n—J . So rank (h 0 1) = dim_H (X;Q) - dim.H (X - ;Q) n Q n Q n n-1 = dim QH n(X;Q) - dim H^X^;*!) , as r a t i o n a l homology and cohomology have the same ( f i n i t e ) dimension. Now H n(X;Q) = H n(M(X)), and by Theorem 3.12, H n(X n_ i ;Q) = H n(M(X n_ 1) = H n(M n_ 1(X)) . I As the d i f f e r e n t i a l i n M(X) i s decomposable, we have B n(M(X)) = B n(M r i .-(X)) , n - l Z n(M(X)) = Z n(M n(X)) . Hence rank(h 0 1 ) = dim Z n(M (X)) - dim Z n(M , (X)) n n n-1 = dim Z n(M (X))/Z n(M ,(X)) . Q.E.D. n n — l 3.14 Remarks: We conclude by n o t i n g some f u r t h e r r e s u l t s concerning the minimal model. By Theorem 3.12, we can i d e n t i f y the free generators of M(X) wi t h the generators of ^ ( X ) 8 Q • The r a t i o n a l Whitehead products are then given by the qu a d r a t i c terms i n the d i f f e r e n t i a l . For example, i f d(x) = a • b f o r f r e e generators x, a, b i n M(X), then x i s the - 83 -Whitehead product of a and b under the above i d e n t i f i c a t i o n , The p r e c i s e statements can be found i n [1]. The proof of t h i s r e l a t i o n s h i p uses the u n i v e r s a l i t y of the Whitehead product on the wedge of two spheres. We have shown how the minimal model can be b u i l t from the Postnikov decomposition. There i s a l s o a c o n s t r u c t i o n a s s o c i a t i n g a r a t i o n a l Postnikov tower to each minimal algebra. Applying t h i s c o n s t r u c t i o n to M(X) y i e l d s the Postnikov decomposition of "tensored w i t h Q ". This can be used to show that isomorphism types of minimal algebras correspond b i j e c t i v e l y w i t h r a t i o n a l homotopy types of spaces. Again, f u r t h e r d e t a i l s may be found i n [ 1 ] . - 84 -BIBLIOGRAPHY [1] P. Deligne, P. G r i f f i t h s , J . Morgan and D. S u l l i v a n , "Real Homotopy Theory of Kahler M a n i f o l d s " , Inventiones Math. 29(1975). [2] E. F r i e d l a n d e r , P. G r i f f i t h s and J . Morgan, "Homotopy Theory and D i f f e r e n t i a l Forms", Lecture Notes, Florence, 1972. [3] S.T. Hu, "Homotopy Theory", Academic Press, 1959. [4] D.W. Kahn, "The Existence and A p p l i c a t i o n s of Anticommutative Cochain Algebras", I l l i n o i s Jour, of Math. 7(1963). [5] A.T. L u n d e l l and S. Weingram, "The Topology of CW-Complexes", Van Nostrand, 1969i [6] J.P. May, " S i m p l i c i a l Objects i n A l g e b r a i c Topology", Van Nostrand, 1967. [7] J . M i l n o r , "The Geometric R e a l i z a t i o n of a S e m i - s i m p l i c i a l Complex", Annals of Math. 65(1957). [8] J . M i l n o r , "On Spaces Having the Homotopy Type of a CW-complex", Trans. Amer. Math. Soc. 90(1959). [9] R.E. Mosher and M.C. Tangora, "Cohomology Operations and A p p l i c a t i o n s i n Homotopy Theory", Harper & Row, 1968. [10] D. Q u i l l e n , " R a t i o n a l Homotopy Theory", Annals of Math 90(1969). [11] E.H. Spanier, " A l g e b r a i c Topology", McGraw-Hill, 1966. [12] M. Spivak, "Calculus on M a n i f o l d s " , Benjamin, 1965. [13] D. S u l l i v a n , " D i f f e r e n t i a l Forms and the Topology of Manifo l d s " , Proceedings of Conference on M a n i f o l d s , Tokyo, 1973. [14] H. Whitney, "On Products i n a Complex", Annals of Math. 39(1938). [15] H. Whitney, "Geometric I n t e g r a t i o n Theory", P r i n c e t o n U n i v e r s i t y P r ess, 1957. 

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