aa r X i v : . [ m a t h . A T ] D ec ON THE BETTI NUMBERS OF A LOOP SPACE
SAMSON SANEBLIDZE
To Tornike Kadeishvili and Mamuka Jibladze
Abstract.
Let A be a special homotopy G-algebra over a commutative unitalring k such that both H ( A ) and Tor Ai ( k , k ) are finitely generated k -modulesfor all i , and let τ i ( A ) be the cardinality of a minimal generating set for the k -module Tor Ai ( k , k ) . Then the set { τ i ( A ) } is unbounded if and only if ˜ H ( A )has two or more algebra generators. When A = C ∗ ( X ; k ) is the simplicialcochain complex of a simply connected finite CW -complex X, there is a similarstatement for the ”Betti numbers” of the loop space Ω X. This unifies existingproofs over a field k of zero or positive characteristic. Introduction
Let Y be a topological space, let k be a commutative ring with identity, andassume that the i th -cohomology group H i ( Y ; k ) of Y is finitely generated as a k -module. We refer to the cardinality of a minimal generating set of H i ( Y ; k ) , denoted by β i ( Y ) , as the generalized i th -Betti number of Y. Theorem 1.
Let X be a simply connected space. If H ∗ ( X ; k ) is finitely generatedas a k -module and H ∗ (Ω X ; k ) has finite type, then the set of generalized i th -Bettinumbers { β i (Ω X ; k ) } is unbounded if and only if ˜ H ∗ ( X ; k ) has at least two algebragenerators. Theorem 1 was proved by Sullivan [11] over fields of characteristic zero andby McCleary [8] over fields of positive characteristic. However, Theorem 1 is aconsequence of the following more general algebraic fact: Let A ′ = { A ′ i } , i ≥ , with A ′ = Z , A ′ = 0 , be a torsion free graded abelian group endowed with ahomotopy G -algebra (hga) structure. Then for A = A ′ ⊗ Z k we have the followingtheorem whose proof appears in Section 4: Theorem 2.
Assume that H ∗ ( A ) is finitely generated as a k -module and that T or A ∗ ( k , k ) has finite type. Let τ i ( A ) denote the cardinality of a minimal generatingset of T or Ai ( k , k ) . Then the set { τ i ( A ) } is unbounded if and only if ˜ H ( A ) has atleast two algebra generators. Let C ∗ ( X ; k ) = C ∗ (Sing X ; k ) /C > (Sing x ; k ) in which Sing X ⊂ Sing X isthe Eilenberg 1-subcomplex generated by the singular simplices that send the 1-skeleton of the standard n -simplex ∆ n to the base point x of X. To deduce Theorem2 from Theorem 1, set A = C ∗ ( X ; k ) , and apply Proposition 2 below together withthe filtered hga model ( RH ( A ) , d h ) → A of A (a special case of the filtered Hirschalgebra [9]). Let BA denote the bar construction of A. When ˜ H ( A ) has at least Mathematics Subject Classification.
Primary 55P35; Secondary 55U20, 55S30.
Key words and phrases.
Betti numbers, loop space, filtered model. two algebra generators, we construct two infinite sequences in the filtered modeland take all possible ⌣ -products of their components to detect a submodule of H ∗ ( BA ) at least as large as the polynomial algebra k [ x, y ] . Each of the sequences mentioned above can be thought of as generalizations ofan infinite sequence ( ∞ - implications of its first component) introduced by Browder[1]. Indeed, this work arose after writing down these special sequences in the hgaresolution of a commutative graded algebra (cga) over the integers via formulas(3.2)–(3.4) below, at which point we realized that their construction mimics thatof Massey symmetric products defined by Kraines [7] (see also [9]). In general, asequence formed from Massey symmetric products is closely related to the one ob-tained from A ∞ -operations in an A ∞ -algebra defined by Stasheff [10] by restrictingto the same variables in question. When a differential graded algebra (dga) A isfree as a k -module, the sequence of A ∞ -operations on the homology H ( A ) wasconstructed by Kadeishvili [5].I am grateful to Jim Stasheff for comments and suggestions and to the refereefor comments that helped to improve the exposition.2. Some preliminaries and conventions
We adopt the notations and terminology of [9]. We fix a ground ring k withidentity, a primary example of which is the integers Z . Let Z k ⊂ Z be the subsetdefined by Z k = { λ ∈ Z | λ k : k → k , κ → λκ, is injective } . Let µ ∈ Z \ Z k denote the smallest integer such that µκ = 0 for all κ ∈ k . Thus if µ = 0, Z k = Z \ k is a field of characteristic zero).A (positively) graded algebra A is 1-reduced if A = k and A = 0 . For a generaldefinition of an homotopy Gerstenhaber algebra (hga) (
A, d, · , { E p,q } ) p ≥ , q =0 , see[3], [4], [6]. The defining identities for an hga are the following: Given k ≥ , (2.1) dE k, ( a , ..., a k ; b ) = P ki =1 ( − ǫ ai − E k, ( a , ..., da i , ..., a k ; b )+ ( − ǫ ak E k, ( a , ..., a k ; db )+ P k − i =1 ( − ǫ ai E k − , ( a , ..., a i a i +1 , ..., a k ; b )+ ( − ǫ ak + | a k || b | E k − , ( a , ..., a k − ; b ) · a k + ( − | a | a · E k − , ( a , ..., a k ; b ) , (2.2) E k, ( a ,..., a k ; b · c )= k X i =0 ( − | b | ( ǫ ai + ǫ ak ) E i, ( a ,..., a i ; b ) · E k − i, ( a i +1 ,..., a k ; c )and(2.3) X k ··· + kp = k ≤ p ≤ k + ℓ ( − ǫ E p, (cid:16) E k ,ℓ ( a , ..., a k ; b ′ ) ,..., E k p ,ℓ p ( a k − kp +1 , ..., a k ; b ′ p ) ; c (cid:17) = E k, ( a , ..., a k ; E ℓ, ( b , ..., b ℓ ; c )) ,b ′ i ∈ { , b , .., b ℓ } , ǫ = p X i =1 ( | b ′ i | + 1)( ε ak i + ε ak ) , b ′ i = 1 ,ε ai = | a | + · · · + | a i | + i. N THE BETTI NUMBERS OF A LOOP SPACE 3 A morphism f : A → A ′ of hga’s is a dga map f commuting with all E k, . Remark 1.
Note that we do not use axiom (2.3) in the sequel.
Below we review the notion of an hga resolution of a cga as a special Hirschalgebra (the existence of such a resolution is proved in [9]). Given a cga H, its hgaresolution is a multiplicative resolution ρ : ( R ∗ H ∗ , d ) → H ∗ , RH = T ( V ) , V = hVi , endowed with an hga structure E k, : RH ⊗ k ⊗ RH → RH, k ≥ , together with a decomposition of V such that V ∗ , ∗ = E ∗ , ∗ ⊕ U ∗ , ∗ , where E ∗ , ∗ = {E < , ∗ p,q } is distinguished by an isomorphism of modules E k, : ⊗ kr =1 R i r H k r O V j,ℓ ≈ −→ E s − k , tk, ⊂ V k − s,t , ( s, t ) = k X r =1 i r + j, k X r =1 k r + ℓ ! . Furthermore, if H is a Z -algebra, its hga resolution ( RH, d ) is automatically en-dowed with two operations ∪ and ⌣ . The first operation ∪ appears because eachcocycle a ⌣ a ∈ E , ∩ R − H j , where a ∈ R H j , is killed by some element in R − H j , denoted by a ∪ a. The second operation arises from the non-commutativityof ⌣ -product in the usual way, and satisfies Steenrod’s formula for the ⌣ -cochainoperation. These two operations are related to each other by the initial relations a ⌣ a = 2 a ∪ a and a ⌣ b = a ∪ b, a = b ∈ U with hUi = U. Note also that a ⌣ a = a ∪ a = 0 for a ∈ U of odd degree. In general, U = T ⊕ N , with anelement of T given by a ∪ · · · ∪ a n , a i ∈ U, n ≥ . The action of the resolutiondifferential d on elements of T such that da i = 0 is(2.4) d ( a ∪ · · · ∪ a n )= X ( i ; j ) ( − | a i | + ··· + | a ik | ( a i ∪ · · · ∪ a i k ) ⌣ ( a j ∪ · · · ∪ a j ℓ ) , where we sum over all unshuffles ( i ; j ) = ( i < · · · < i k ; j < · · · < j ℓ ) of n with( a i , ..., a i k ) = ( a i ′ , ..., a i ′ k ) if and only if i = i ′ and ⌣ denotes E , . In particular,for a = · · · = a n = a = a ∪ and n ≥ da ∪ n = P k + ℓ = n a ∪ k ⌣ a ∪ ℓ , k, ℓ ≥ . And in general d ( a ⌣ b ) = nd ( a ∪ b ) , n ≥ . An hga resolution (
RH, d ) is minimal if d ( U ) ⊂ E + D + κ · V where D ∗ , ∗ ⊂ R ∗ H ∗ denotes the submodule of decomposables RH + · RH + and κ ∈ k is non-invertible; For example, κ ∈ Z \ {− , } when k = Z and κ = 0 when k is a field.Let K = { K j } j ≥ with K j = { a ∈ V − ,j | da = λb, λ = ± , b ∈ V ,j } . Note thata general form of a relation in (minimal) (
RH, d ) starting by variables v i ∈ K ∪ V , ∗ is(2.5) du = X s ≥ λ s P s ( v , ..., v r s ) + λv, λ = ± , λ s = 0 , r s ≥ ,u ∈ [ i ≥ V − i, ∗ , v ∈ [ i ≥ V − i, ∗ \ K, SAMSON SANEBLIDZE where P s ( v , ..., v r s ) is a monomial in D ∗ , ∗ ⊂ R ∗ H ∗ . Let A be an hga and let ρ : ( RH, d ) → H be an hga resolution. A filtered hgamodel of A is an hga quasi-isomorphism f : ( RH, d h ) → ( A, d A )in which d h = d + h, h = h + · · · + h r + · · · , h r : R p H q → R p + r H q − r +1 . The equality d h = 0 implies the sequence of equalities dh + h d = 0 , dh + h d = − h h , dh + h d = − h h − h h , . . . , and h is referred to as a perturbation of d. The map h r | R − r H : R − r H → R H, r ≥ , denoted by h tr , is referred to as the transgressive component of h. The fact thatthe perturbation h acts as a derivation on elements of E implies h tr | E = 0 . For theexistence of the filtered model see [9].In the sequel, A ′ denotes a 1-reduced torsion free hga over Z , while A denotes thetensor product hga A ′ ⊗ Z k . Denote also H = H ∗ ( A ′ ) and H k = H ∗ ( A ) . Assume(
RH, d ) is minimal and let RH k = RH ⊗ Z k ; in particular, RH k = T ( V k ) for V k = V ⊗ Z k . When k is a field of characteristic zero, ρ ⊗ RH k → H ⊗ Z k = H k is an hga resolution of H k , which is not minimal when Tor H = 0. In general, givena filtered model ( RH, d h ) of A ′ , we obtain an hga model f ⊗ RH k , d h ⊗ → ( A, d A ) . for ( A, d A ) . Denote ¯ V k = s − ( V > k ) ⊕ k and define the differential ¯ d h on ¯ V k by therestriction of d + h to V k and obtain the cochain complex ( ¯ V k , ¯ d h ) . Since the map f ⊗ H ∗ ( BA, d BA ) ≈ T or A ( k , k ) and H ∗ ( BC ∗ ( X ; k ) , d BC ) ≈ H ∗ (Ω X ; k ) . Proposition 1.
There are isomorphisms H ∗ ( ¯ V k , ¯ d h ) ≈ H ∗ ( B ( RH k ) , d B ( RH k ) ) ≈ H ∗ ( BA, d BA ) ≈ T or A ( k , k ) . And for A = C ∗ ( X ; k ) we obtain: Proposition 2.
There are isomorphisms H ∗ ( ¯ V k , ¯ d h ) ≈ H ∗ ( BC ∗ ( X ; k ) , d BC ) ≈ H ∗ (Ω X ; k ) . Given (
RH, d ) and x, c ∈ V with dx, dc ∈ D + λV, λ = 1 , let η x,c denote anelement of E > , such that x ⌣ c := η x,c + x ⌣ c satisfies d ( x ⌣ c ) ∈ D + λV. For example, if dx ∈ λV, then η x,c = 0 , and if dx = P i a i b i + λv with da i , db i ∈ λV, then η x,c = P i ( − | a i | E , ( a i , b i ; c ) . Ingeneral, η x,c can be found as follows: Let j : B ( RH ) → RH → ¯ V be the canonicalprojection used by the proof of the first isomorphism in Proposition 1, and choose y ∈ B ( RH ) so that j ( y ) = ¯ x and jµ E ( y ; ¯ c ) = ¯ η x,c + x ⌣ c, where the product µ E : B ( RH ) ⊗ B ( RH ) → B ( RH ) is determined by the hga structure on RH.
The following proposition is simple but useful. Let D k ⊂ RH be a subset definedby D k = D for µ = 0 and D k = { u + λv | u ∈ D , v ∈ V, λ is divisible by µ } for µ ≥ . N THE BETTI NUMBERS OF A LOOP SPACE 5
Definition 1.
An element x ∈ V with d h x ∈ D + λV, λ = 1 , is λ -homologousto zero, denoted by [¯ x ] λ = 0 , if there are u, v ∈ V and z ∈ D such that d h u = x + z + λv ; x is weakly homologous to zero when v = 0 above. Proposition 3.
Let c ∈ V and d h c ∈ D k . If d h c has a summand component ab ∈ D such that a, b ∈ V, d h a, d h b ∈ D k , both a and b are not weakly homologous to zero,then c is also not weakly homologous to zero.Proof. The proof is straightforward using the equality d h = 0 . (cid:3) In particular, for k = Z , under hypotheses of the proposition if [¯ a ] , [¯ b ] = 0 , then[¯ c ] = 0 in H ∗ ( ¯ V , ¯ d h ) . Note that over a field k , Proposition 3 reflects the obvious fact that x ∈ H ∗ (Ω X ; k )is non-zero whenever some x ′ ⊗ x ′′ = 0 in ∆ x = P x ′ ⊗ x ′′ . Formal ∞ -implication sequences Let x be an element of a Hopf algebra over a finite field. In [1], W. Browderintroduced the notion of ∞ -implications (of an infinite sequence) associated with x in the Hopf algebra. The following can be thought of as a generalization of this:Let x ⌣ p denote the (right most) p th -power of x with respect to ⌣ -product withthe convention that x ⌣ = x. Definition 2.
Let x ∈ V k , k ≥ , d h x ∈ D k . A sequence x = { x ( i ) } i ≥ is aformal ∞ -implication sequence (f.i.s.) of x if(i) x (0) = x, x ( i ) ∈ V ( i +1) k − i , and x ( i ) is not µ -homologous to zero for all i ; (ii) Either x ( i ) = x ⌣ ( i +1) or x ( i ) is resolved from the following relation in thefiltered hga model ( RH, d h ) :(3.1) d h b ( i ) = x ⌣ ( i +1) + z ( i ) + µ ′ x ( i ) , b ( i ) ∈ V, z ( i ) ∈ D , µ ′ is divisible by µ. We are interested in the existence of an f.i.s. for an odd dimensional x ∈ V. Proposition 4.
Let x ∈ V be of odd degree with d h x ∈ D k such that x is not µ -homologous to zero. For µ ≥ , assume, in addition, there is no relation d h u = µx mod D , some u ∈ V. Then x has an f.i.s. x = { x ( i ) } i ≥ . Proof.
Suppose we have constructed x ( i ) for 0 ≤ i < n. If x ⌣ ( n +1) is not µ -homologous to zero, set x ( n ) = x ⌣ ( n +1) ; otherwise, there is the relation d h u = x ⌣ ( n +1) + z + µ ′ v for some u, v ∈ V, z ∈ D and µ ′ divisible by µ. Using (2.1)–(2.2)one can easily establish the fact that dx ⌣ ( n +1) contains a summand componentof the form − P k + ℓ = n +1 (cid:0) n +1 k (cid:1) x ⌣ k x ⌣ ℓ , k, ℓ ≥ . We have that v = 0 in theaforementioned relation since Proposition 3 (applied for c = x ⌣ ( n +1) and a · b = − (cid:0) n +1 k (cid:1) x ⌣ k · x ⌣ ℓ , some k ). Clearly, d h v = − µ ′ d h (cid:0) x ⌣ ( n +1) + z (cid:1) ∈ D ; Assuming µ ′ to be maximal v is not λ -homologous to zero. Set x ( n ) = v and b ( n ) = u, z ( n ) = z to obtain (3.1) for i = n. (cid:3) SAMSON SANEBLIDZE
Thus, for µ = 0 (when k is a field of characteristic zero, for example) x = { x ⌣ ( n +1) } n ≥ . Remark 2.
1. The restriction on x in Proposition 4 that no relation d h u = µx mod D exists is essential. A counterexample is provided by the exceptional group F : Let A = C ∗ ( BF ; Z ) be the cochain complex of the classifying space BF . Thenwe have the relation du = 3 x in ( RH, d ) corresponding to the Bockstein cohomologyhomomorphism δx = x on H ∗ ( BF ; Z ) (in the notation of [13] ), but the element x (2) does not exist (see [9] for more details).2. Note that if du = µx in Proposition 4, but [ u ][ x ] = 0 ∈ H k , then one canmodify the proof of the proposition to show that x again has an f.i.s. { x ( i ) } i ≥ . Note that in the above example we just have [ u ][ x ] = 0 ∈ H Z = H ∗ ( BF ; Z ) .
3. The existence of ∞ -implications of x in [1] uses both the ⌣ -product and thePontrjagin product in the loop space (co)homology. In our case each componentof the sequence x is determined by item (ii) of Definition 2 in which the first casecan be thought of as related to the ⌣ -product, and the second with the Pontrjaginproduct. In particular, primitivity of x required in [1] is not issue for the existenceof ∞ -implications of x. In certain cases, a given odd dimensional b ∈ V rises to an infinite sequence b = { b i } i ≥ with b = b in the hga resolution ( RH, d ) . These sequences are builtby explicit formulas and include also the case du = λb, i.e., when the hypothesis ofProposition 4 formally fails (see, for example, Case I of the proof of Proposition 5below). Namely, we have the following cases:(i) For b ∈ V , ∗ and [ b ] = 0 ∈ H (i.e., there exists b ∈ V − , ∗ with db = b ;e.g. b = ab + λ − b ⌣ b for da = λb with λ odd, some a ∈ V − , ∗ ), b = { b i } i ≥ isgiven by(3.2) db n = X i + j = n − b i b j and satisfies the following relation with c i ∈ Vd c n = − ( − n (( n + 1) b n + b ⌣ b n − ) + X i + j = n − ( − i ( c j b i − b i c j ) , n ≥ b ∈ V , ∗ and [ b ] = 0 ∈ H (and b = b ⌣ b ), b = { b i } i ≥ is given by(3.3) db k = X i + j =2 k − b i b j , db k +1 = X i + j = k (2 b i b j + b i − b j +1 ) , and satisfies the following relation with c i ∈ V (below c = 0) d c k = − (2 k + 1) b k − b ⌣ b k − + X i + j = k c j − b i − b i c j − ) − X i + j = k ( c j b i − − b i − c j ) ,d c k +1 = ( k + 1) b k +1 + b ⌣ b k + X i + j =2 k ( − i ( c j b i − b i c j ) , k ≥ N THE BETTI NUMBERS OF A LOOP SPACE 7 (iii) For b ∈ V − , ∗ and db = µc, µ ≥ , c ∈ V , ∗ (below ω := c ), b = { b i } i ≥ isgiven by(3.4) db n = X i + j = n − b i b j + µc n ,c n = − ω ⌣ b n − − X i + j = n − i ≥ j ≥ ( − i ω i ⌣ b j − ( − n ω n , n ≥ c i ∈ Vd c = 2 b + b ⌣ b + µω ∪ b ,d c n = − ( − n (( n + 1) b n + b ⌣ b n − ) + X i + j = n − ( − i ( c j b i − b i c j )+ µ a n , a n = X i + j = n − ( − j (( ω i ∪ b ) ⌣ b j + ω i ⌣ c j +1 )+ ω n − ∪ b ,dω k = X i + j = k − µω i ⌣ ω j , ω k = µ k ω ∪ ( k +1)0 , k ≥ , n ≥ . For example, in view of Proposition 2, the formulas above are enough to calculatethe loop space cohomology algebra with coefficients in k for Moore spaces, i.e., the CW -complexes obtained by attaching an ( n + 1)-cell to the n -sphere S n by a map S n → S n of degree µ. Odd dimensional element l ( a ) . Given m ≥ , let H ( A ) be finitely generatedas a k -module with H i ( A ) = 0 for i > m. Let Z k be the subset of RH defined by Z k = Z ′ k + Z ′′ k + D k , Z ′ k = { v ∈ V | du = λv, u ∈ V, λ ∈ Z k } and Z ′′ k = { v ∈ V | v = λu, u ∈ V, λ ∈ Z \ Z k } . Given x ∈ V with d h x = w ∈ Z k , w = w ′ + w ′′ + z, define˜ x = l.c.m. ( λ ′′ ; µ ) λ ′′ ( λ ′ x − u ) , du = λ ′ w ′ , w ′′ = λ ′′ v ′′ , to obtain d h ˜ x ∈ D k . Regarding (2.5), define also the following subsets K ∗ µ , K ∗ ⊂ V − , ∗ with K ∗ µ ⊂ K ∗ as K µ = { a ∈ K | λ is divisible by µ } ,K = (cid:8) u ∈ V − , ∗ \ E | du ∈ D , ∗ (cid:9) , and assign to a given even dimensional element a ∈ V , ∗ ∪ K µ an odd dimensionalelement l ( a ) ∈ V with dl ( a ) ∈ D k as follows. If a ∈ V , ∗ , let l ( a ) ∈ K be anelement such that dl ( a ) = a k , where k ≥ a ∈ K µ with da = λb consider the relation(3.5) du = − a + λv , dv = 1 λ d ( a ) , u ∈ V − , ∗ , v ∈ V − , ∗ , SAMSON SANEBLIDZE and the perturbation hu = h u + h u . When hu ∈ Z k , set l ( a ) = e u , while when h u / ∈ Z k , consider u = h u | V , ∗ , the component of h u in V , ∗ , and define l ( a )as l ( u ) . When h u / ∈ Z k , and h u ∈ Z k , choose the smallest n > du n = − ah u n − + λv n , dv n = 1 λ d ( ah u n − ) , u n ∈ V − , ∗ , v n ∈ V − , ∗ , with h u n ∈ Z k . (The inequality ( n + 1) | a | > m guarantees the existence of such a relation, since h u i ∈ D + K µ , while K jµ = 0 for j > m in the minimal V ⊂ RH. ) Then set l ( a ) = e u n for h u n ∈ Z k ; otherwise, define l ( a ) as l ( u n ) for u n = h u n | V , ∗ . Proof of Theorem 2
The proof of the theorem relies on the two basic propositions below in which thecondition that ˜ H ( A ) has at least two algebra generators is treated in two specificcases. Proposition 5.
Let H k be a finitely generated k -module with µ ≥ . If ˜ H k hasat least two algebra generators and ˜ H Q is either trivial or has a single algebragenerator, there are two sequences of odd degree elements x k = { x ( i ) } i ≥ and y k = { y ( j ) } j ≥ in V k whose degrees form arithmetic progressions such that all ¯ x ( i ) , ¯ y ( j ) are ¯ d h -cocycles in ¯ V k and the classes (cid:8) [ s − ( x ( i ) ⌣ y ( j ))] (cid:9) i,j ≥ are linearlyindependent in H ( ¯ V k , ¯ d h ) . Proof.
The hypotheses of the proposition imply that K µ defined in subsection 3.1above is non-empty; also by the restriction on ˜ H Q , relation (2.5) reduces to da = λb m , λ = 0 , m ≥ , ( λ, m ) = (1 , , b ∈ V , ∗ for a ∈ V − , ∗ to be of the smallest degree.In the three cases below, we exhibit two odd dimensional elements x, y ∈ V \ E that fail to be µ -homologous to zero.Case I. Let a ∈ K µ be of the smallest degree in K µ ∪ K with da = λb and let | a | be even. Consider the element l ( a ) . If it is not λ -homologous to zero, set x = l ( a );otherwise, we must have relation (2.5) in which v i = a for some i and hu ∈ Z k with | u | < | l ( a ) | , u ∈ S i ≥ V − i, ∗ \ E . By (2.5) choose u to be of the smallest degree with hu ∈ Z k , u = u i , a , where u i is given by (3.5)–(3.6) and da = − ab + λb , db = b . Set x = ˜ u for | u | odd. If | u | is even and u ∈ S i> V − i, ∗ \ E set x = ˜ v ; if u ∈ K and du contains an odd dimensional v i ∈ V , ∗ with [ v i ] = 0 ∈ H Q , set x = v i ; otherwise,for each monomial P s ( v , ..., v r s ) choose a variable v i with a relation du i = µ i v i (forexample, we can choose v i to be odd dimensional for all s ). Let λ be the smallestinteger divisible by all µ i , and replace v i by λµ i u i to detect a new relation in ( RH, d )given again by (2.5): dw = X ≤ s ≤ n λ s λµ i P s ( v , ..., v i − , u i , v i +1 ..., v r s ) + λu, λ s ∈ Z k , w ∈ V − , ∗ . Hence, | w | is odd, and set x = e w for h w ∈ Z k . If h w / ∈ Z k we have the followingtwo subcases:(i1) Assume there exists v ∈ K µ with dv = λh w. If [¯ v ] λ = 0 , set x = v ;otherwise we have a relation d h u ′ = v + z + λ ′ v ′ , some u ′ , v ′ ∈ V, z ∈ D . Clearly,
N THE BETTI NUMBERS OF A LOOP SPACE 9 h tr v ′ = − λλ ′ h w mod D , and set x = λλ ′ w + v ′ . Note that x is not λ -homologous tozero since the component λ λ ′ u in dx. (i2) Assume [ h w ] = 0 ∈ H Q . When r s > v j differentfrom v i in P s ( v , ..., v r s ) to form w ′ entirely analogously to w, and then find x similarly to the above unless [ h w ′ ] = 0 ∈ H Q , in which case set x = αw + βw ′ , some α, β ∈ Z . When k = { s ∈ n | r s = 1 in du } 6 = ∅ , i.e., P s ( v , ..., v r s ) = v m +11 := v m s +1 s , m s ≥ , | v s | is odd for s ∈ k (in particular, µ s is even, since[ v s ] = 0 ∈ H for µ s odd; c.f. (3.2)), then du ′ = P s ∈ k λ s λ ( v s ⌣ v s ) v m s − s + P s ∈ n \ k λ s λµ j P s ( v , ..., v j − , u j , v j +1 ..., v r s ) + λu, k = n, P s ∈ n λ s ( v s ⌣ v s ) v m s − s + 2 u, k = n with u ′ ∈ V − , ∗ , and by considering h u ′ we find x as in item (i1).To find y, consider b and the associated sequence b = { b i } given by (3.2) or(3.3). If hb i ∈ Z k for all i, set y = b and y = { ˜ b i } i ≥ . If h b * Z k , consider thesmallest p > h tr b p / ∈ Z k . Consider t p = h tr b p | V , ∗ , and if h l ( t p ) i λ = 0 , set y = l ( t p ); if h l ( t p ) i λ = 0 and αh u i + βh tr b p = 0 , α, β ∈ Z , for some u i from(3.5)–(3.6), set y = αu i + βb p ; otherwise, we obtain l ( t p ) ∈ K different from l ( a )above; consequently, we must have another relation in ( RH, d ) given by (2.5) inwhich v i = t p for some i and hu ∈ Z k with | u | < | l ( t p ) | , and then y is foundsimilarly to x. Case II. Let a ∈ K µ be of the smallest degree in K µ ∪ K with da = λb and let | a | be odd. Set x = a. Consider l ( b ) ∈ K , and then y is found as in Case I.Case III. Let a ∈ K be of smallest degree in K µ ∪ K with da = λb m , m ≥ , and [ b ] = 0 ∈ H Q . Set x = (cid:26) b, | b | is odd a, | b | is even . To find y consider the following two subcases:(i) Assume λ ∈ Z \ Z k . When both | a | and | b | are odd, set y = a ; otherwise,either | a | or | b | is even, in which case consider l (˜ a ) or l ( b ) respectively, and then y is found as in Case I.(ii) Assume λ ∈ Z k . Since K µ = ∅ , this subcase reduces either to Case I or toCase II.Finally, having found the elements x and y in Cases I-III, consider the f.i.s. x and y in V and the induced sequences x k = { x ( i ) } i ≥ and y k = { y ( j ) } j ≥ in V k . Then the both sequences ¯ x k and ¯ y k consist of ¯ d h -cocycles in ¯ V k whose degreesform an arithmetic progression respectively. Thus, we obtain that [¯ x k ] , [¯ y k ] ⊂ H ( ¯ V k , ¯ d h ) are sequences of non-trivial classes. Moreover, they are linearly indepen-dent and (cid:8) [ s − ( x ( i ) ⌣ y ( j ))] (cid:9) i,j ≥ is the sequence of linearly independent classesin H ( ¯ V k , ¯ d h ) as required. (cid:3) Before proving the second basic proposition we need the following auxiliary state-ment. Given a cochain complex ( C ∗ , d ) over Q , let S C ( T ) = P n ≥ (dim Q C n ) T n and S H ( C ) ( T ) = P n ≥ (dim Q H n ( C )) T n be the Poincar´e series. As usual, we write P n ≥ a n T n ≤ P n ≥ b n T n if and only if a n ≤ b n . The following proposition can bethought of as a modification of Propositions 3 and 4 in [12] for the non-commutativecase.
Proposition 6.
Given an element y ∈ V Q of total degree K µ ≥ such that ¯ d h (¯ y ) =0 , let y ¯ V Q ⊂ ¯ V Q be a subcomplex (additively) generated by the expressions { ¯ y = s − y, s − ( y ⌣ v ) } v ∈ V Q . Then (4.1) S H ( ¯ V Q /y ¯ V Q ) ( T ) ≤ (1 + T k − ) S H ( ¯ V Q ) ( T ) . Proof.
Consider the inclusion of cochain complexes s k ¯ V Q ι → ¯ V Q defined for 1 ∈ Q =( s k ¯ V Q ) k by ι (1) = ¯ y, and for s k (¯ v ) ∈ ( s k ¯ V Q ) >k , v ∈ V > Q , by ι ( s k (¯ v )) = s − ( y ⌣ v ) . Then ι ( s k ¯ V Q ) = y ¯ V Q and there is the short exact sequence of cochain complexes0 → s k ¯ V Q ι → ¯ V Q → ¯ V Q /y ¯ V Q → . Consider the induced long exact sequence · · · → H n − k ( ¯ V Q ) H n ( ι ) −→ H n ( ¯ V Q ) → H n ( ¯ V Q /y ¯ V Q ) → H n − k +1 ( ¯ V Q ) → · · · . Let I = ⊕ I n , where I n = Im( H n ( ι )) , n ≥ , and form the exact sequence0 → I n → H n ( ¯ V Q ) → H n ( ¯ V Q /y ¯ V Q ) → H n − k +1 ( ¯ V Q ) → I n +1 → . Since I = 0 , we have X n ≥ (dim Q I n + dim Q I n +1 ) T n = (1 + T ) S I ( T ) T .
Now apply the Euler-Poincar´e lemma for the above exact sequence to obtain theequality(1 + T ) S I ( T ) T − S H ( ¯ V Q ) ( T ) + S H ( ¯ V Q /y ¯ V Q ) ( T ) − T k − S H ( ¯ V Q ) ( T ) = 0 . Consequently, S H ( ¯ V Q /y ¯ V Q ) ( T ) = (1 + T k − ) S H ( ¯ V Q ) ( T ) − (1 + T ) S I ( T ) T , and since S I ( T ) ≥ , we get (4.1) as required. (cid:3) Proposition 7.
Let H k be a finitely generated k -module. If ˜ H Q has at least twoalgebra generators and A Q = A ′ ⊗ Z Q , the set n τ i ( A Q ) = dim Q T or A Q i ( Q , Q ) o isunbounded.Proof. Consider the first two generators a i ∈ V − , ∗ Q with da i ∈ D , ∗ , i = 1 , . Wehave two cases:(i) Both | a | and | a | are odd. Set x = a and y = a . Then both ¯ x and ¯ y are¯ d h -cocycles and the classes [¯ x ] and [¯ y ] are non-trivial in H ( ¯ V Q , ¯ d h ) . Consequently,the classes(4.2) (cid:8) [ s − (cid:0) x ` i ⌣ y ` j (cid:1) ] (cid:9) i,j ≥ are linearly independent in H ( ¯ V Q , ¯ d h ) . (ii) Either | a | or | a | is even. Denote the (smallest) even dimensional generatorby a and consider da. Then for a, (2.5) reduces to da = uv, u ∈ V , k +1 Q and v ∈ R H ℓ Q , some k, ℓ ≥ . N THE BETTI NUMBERS OF A LOOP SPACE 11
There are the following induced relations in ( RH Q , d ) : db = − u ( a + u ⌣ v ) − au, b ∈ V − , k + ℓ +1) Q and dc = − u ( v ⌣ a + ( u ∪ v ) v + u ( v ∪ v )) − a + bv, c ∈ V − , k + ℓ )+2 Q . Thus we have hc = h c + h c, and in particular, dh c = h b · v. Consider thefollowing two cases:(1) Assume hc ∈ D . Set x = u, y = c, and obtain linearly independent classes in H ( ¯ V Q , ¯ d h ) by formula (4.2).(2) Assume hc / ∈ D . Let ( ¯
W , ¯ d W ) = ( ¯ V Q / ¯ C, ¯ d W ) , where C ⊂ V Q is a subcomplex(additively) generated by the expressions hc and hc ⌣ z for z ∈ V Q . Define x and y as the projections of the elements u and c from V Q under the quotient map V Q → V Q /C, respectively. Then ¯ x and ¯ y are ¯ d W -cocycles in ¯ W .
Once again apply formula(4.2) to obtain linearly independent classes in H ( ¯ W , ¯ d W ) . Finally, Proposition 6implies that S H ( ¯ W ) ( T ) ≤ S H ( V Q ) ( T ) , and an application of Proposition 1 completesthe proof. (cid:3) Proof of Theorem 2.
In view of Proposition 1, the proof reduces to theexamination of the k -module H ( ¯ V k , ¯ d h ) . If ˜ H k has a single algebra generator a, then the set { τ i ( A ) } is bounded since τ i ( A ) = 1 . For example, this can be seenfrom the fact that H ( ¯ V k , ¯ d h ) is generated by a single sequence induced by (3.2) orby (3.3), where x = a or x = l ( a ) for | a | odd or even respectively, and by ⌣ -products of its components. If ˜ H k has at least two algebra generators, then theproof follows from Propositions 5 and 7. References [1] W. Browder, Torsion in H -spaces, Ann. Math., 74 (1961), 24-51.[2] Y. Felix, S. Halperin and J.-C. Thomas, Adams’ cobar equivalence, Trans. AMS, 329 (1992),531-549.[3] M. Gerstenhaber and A.A. Voronov, Higher operations on the Hochschild complex, FunctionalAnalysis and its Applications, 29 (1995), 1–5.[4] E. Getzler and J.D.S. Jones, Operads, homotopy algebra, and iterated integrals for doubleloop spaces, preprint (1995).[5] T. Kadeishvili, On the homology theory of fibre spaces, Russian Math. Survey, 35 (1980),131-138.[6] T. Kadeishvili and S. Saneblidze, A cubical model of a fibration, J. Pure and Applied Algebra,196 (2005), 203-228.[7] D. Kraines, Massey higher products, Trans. AMS, 124 (1966), 431-449.[8] J. McCleary, On the mod p Betti numbers of loop spaces, Invent. Math., 87 (1987), 643-654.[9] S. Saneblidze, Filtered Hirsch algebras, preprint math.AT/0707.2165.[10] J. D. Stasheff, Homotopy Associativity of H -spaces I, II, Trans. AMS, 108 (1963), 275-312.[11] D. Sullivan, Diffrential forms and the topology of manifolds, Manifolds-Tokyo 1973, ed. AkioHattori, Tokyo Univ. Press, 37-51.[12] M. Vigu´e-Poirrier and D. Sullivan, The homology theory of the closed geodesic problem, J.Diff. Geom., 11 (1976), 633-644.[13] H. Toda, Cohomology mod 3 of the classifying space BF of the exceptional group F , J.Math. Kyoto Univ., 13-1 (1973), 97-115.
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