TORSION IN LOOP SPACE HOMOLOGY OF RATIONALLY CONTRACTIBLE SPACES

RAN LEVI

Abstract. Let R be a torsion free principal ideal domain. We study the growth of torsion in loop space homology of simply-connected DGR-coalgebras C, whose homology admits an exponent r in R. Here by loop space homology we mean the homology of the loop algebra construction on C. We compute a bound on the growth of torsion in such objects and show that in general this bound is best possible. Our methods are applied to certain simply-connected spaces associated with classifying spaces of finite groups, where we are able to deduce the existence of global exponents in loop space homology.

1. Introduction and Statement of Results

The growth of torsion in loop space homology is known to be rather hard to control in general. A remarkable example of this was discovered by D. Anick [2], who introduced a finite dimensional CW complex X with the property that the integral loop space homology H_{∗}(ΩX; ℤ) has torsion of all orders. A similar example was also constructed by L. Avramov [3]. On the other extreme, Y. Felix, S. Halperin and J-C. Thomas showed in [6] that if Y is a space of finite Lusternik-Schnirelmann category, with finite type ℤ_{(p)}-homology and such that the dimension of the graded vector space H_{∗}(ΩY; 𝔽_{p}) grows at most polynomially, then the torsion in H_{∗}(ΩY; ℤ_{(p)}) has an exponent, namely there exists an integer r such that p^{r} annihilates all the torsion in the p-local loop space homology of Y.

The spaces constructed by both Anick and Avramov have the additional feature that their integral homology is torsion free. The theorem in [6] on the other hand is concerned with neither spaces of infinite category nor with such, where the mod-p loop space homology growth rate is larger than polynomial.

The motivation for this paper comes from considering spaces of the form BG_{p}^{∧}, where G is a finite p-perfect group and by (−)_{p}^{∧} we denote the 𝔽_{p}-completion functor of Bousfield and Kan [4]. In [10] we show that for such groups G, the integral loop space homology of BG_{p}^{∧} has an exponent, given by the order of the Sylow p-subgroup of G. It is always the case that BG_{p}^{∧} has infinite category when G is finite and in [11, 12] we present examples of finite p-perfect groups G such that the mod-p loop space homology of BG_{p}^{∧} grows exponentially. Thus, in general, spaces of this form do not fit in the context of the theorems mentioned above.

Date: March 31 1995.

1991 Mathematics Subject Classification. Primary 55R35, Secondary 55R40, 55Q52.

The author is supported by a DFG grant.

We specialize to rationally contractible spaces with a reduced integral homology exponent and attempt to understand the growth of torsion in their loop space homology. Clearly one should not expect an exponent result in general, even when restricting to this family of spaces. For example the integral homology of the double loop space on a mod-p^{r} Moore space has p-torsion of arbitrarily high order, even though the homology of the single loop space has an exponent p^{r}. Indeed, our study here may suggest that it is hardly ever the case that an infinite dimensional rationally contractible space X, which admits a homology exponent, still has one in its loop space homology. We refer the reader to Section 9 below for further discussion.

In the preceding we will abbreviate the words “differential graded” by the letters DG. Let R be a commutative ring with a unit. The major tool we use is the loop algebra functor Ω(−) from the category of DGR-coalgebras to DGR-algebras [7] (this functor is more commonly known as the Adams’ cobar construction [1]). We shall generally assume that the ground ring R is a torsion free principal ideal domain (by torsion free we mean that the natural map from the integers to R is injective). For every DGR-module, the underlying graded R-module is assumed to be dimension-wise free and graded by the non-negative integers. A DGR-module (M; d) is said to have a homology exponent if there exists a non-zero element r ∈ R, such that r · H_{i}(M; d) = 0 for all i ≥ 0. All topological spaces considered in this note are assumed to be pointed, connected and to have the homotopy type of a CW complex. For precise definitions of the terminology used below, we refer the reader to section 2.

For a supplemented DGR-coalgebra C, let JC denote the augmentation coideal of C. For a supplemented DGR-algebra A, let IA denote the augmentation ideal of A. A coalgebra C, as above, is said to have a homology exponent r ∈ R if r is an exponent for H_{∗}(JC). Similarly one defines what it means for a supplemented DGR algebra to have a homology exponent. Theorem 1.1. Let C be a c-connected DGR-coalgebra, c ≥ 1. Suppose that C admits a homology exponent r ∈ R. Let d and e be positive integers such that e ≥ \frac{d−c+1}{c}. Then r^{e} annihilates H_{i}(ΩC) for each 1 ≤ i ≤ d.

We construct an example over the integers, which shows that in general the approximation given by Theorem 1.1 is best possible. Indeed if C is precisely (2k − 1)-connected, for some positive integer k and has a homology exponent r ∈ R, then the theorem claims that r^{e−1} annihilates H_{i}(ΩC) for 1 ≤ i ≤ e(2k−1)−1. Thus the least d, for which r^{e−1} might fail to be an exponent for H_{d}(ΩC), is d = e(2k − 1). Theorem 1.2. For each pair of positive integers t and k, there exists a (2k − 1)- connected DGℤ-coalgebra C^{k,t} with a homology exponent t, such that for every positive integer e, H_{e(2k−1)}(ΩC^{k,t}) contains an element of order t^{e}.

An interesting fact about the DGℤ-coalgebra C^{k,t} is that its dual algebra (C^{k,t})^{∗} is commutative and nilpotent of degree 3, in the sense that the third power of its augmentation ideal vanishes. Indeed, it is a triviality that if the dual DGRalgebra C^{∗} of a DGR-coalgebra C is nilpotent of degree 2 (i.e. C has a trivial diagonal) and C has a homology exponent r then so does ΩC. Our example shows that this fails to hold once the nilpotency assumption is relaxed.

It is known and in fact admits a rather easy proof that if X is a rationally contractible finite dimensional CW complex, which admits an integral homology exponent (trivially it does if it has only finitely many cells), then ΩX admits an integral homology exponent. However the usual proof, using the Serre spectral sequence for the path-loop fibration over X, fails to give the best approximation. The methods used in the proof of Theorem 1.1 enables us to improve the existing bound.

Proposition 1.3. Let C be a c-connected DGR-coalgebra of homological dimension d with a homology exponent r ∈ R. Then r^{e} annihilates H_{i}(ΩC) for every e ≥ \frac{d−c−1}{c} and i > 0.

Our bound here should be compared with the corresponding result using the Serre spectral sequence which gives r^{d−c} as a bound, under the same hypotheses. At the end of Section 5 below, we consider finite sub DGR-coalgebra of those constructed in Theorem 1.2, for which the bound given in Proposition 1.3 is best possible.

Next we study an application of the techniques developed here. Recall the construction P_{n}(−) on DGR-coalgebras given in [10]. Given any DGR-coalgebra C, we produce an n-connected quotient DGR-coalgebra P_{n}(C), such that the natural projection from C to P_{n}(C) induces an isomorphism in homology above dimension n. This algebraic construction, being motivated by the Quillen “plus” construction, admits a topological analogue, which in a sense is an easy generalization of Quillen’s idea.

Proposition 1.4. Let X be a connected CW-complex and let n be a positive integer. Then there exists an n-connected CW-complex P_{n}X together with a map ι_{n} : X ⟶ P_{n}X, inducing an isomorphism on integral homology above dimension n. Furthermore, if C = S_{∗}(X), the integral singular chain coalgebra on X, then P_{n}(C) and S_{∗}(P_{n}X) are quasi isomorphic as DGℤ-coalgebras.

With the terminology of Proposition 1.4 we now restrict attention to classifying spaces BG, where G is a finite group.

Theorem 1.5. Let G be a finite group of order N. Then H̃_{∗}(ΩP_{n}BG; ℤ) is annihilated by N^{q}, where q = 3 if n = 1 and q = 2 otherwise.

If G is a finite p-perfect group then (P_{1}BG)_{p}^{∧} is homotopy equivalent to BG_{p}^{∧}. Thus Theorem 1.5 is, in a sense, a generalization of [10, I, thm. 1], although in the last the exponent is given by the order of the group rather than a power of it.

All the results above are motivated by topology. However the loop algebra functor is a purely algebraic construction and so one could consider loop space homology torsion in a context that might have nothing to do with topology. We look at the loop algebra functor Ω(−) together with its left adjoint, the classifying coalgebra functor B(−) [7]. The adjunct morphisms here are equivalences. Thus one could consider a DGR-algebra A, for which the classifying construction B(A) admits a homology exponent and ask whether the same holds for A itself. An example of this is given by the following.

Theorem 1.6. Let A be a nilpotent DGR-algebra of nilpotency rank n. Suppose that B(A) admits a homology exponent r ∈ R. Then r^{n−1} is a homology exponent for A.

In section 8 below we observe that the mod-p homology of a p-local loop space X is a nilpotent algebra if and only if it is finite dimensional, in which case it cannot be rationally contractible. Thus Theorem 1.6 does not correspond to a topological situation.

The paper is organized as follows. Sections 2 and 3 are preliminary in nature and the techniques developed are applied in Section 4 for the proof of Theorem 1.1 and Proposition 1.3. In Section 5 we prove Theorem 1.2. Section 6 is dedicated to the proof of an algebraic analogue of Theorem 1.5. The theorem follows using Proposition 1.4 which we prove in Section 7. In Section 8 we prove Theorem 1.6. We conclude the paper in Section 9 by making some remarks and speculations on the subject.

The author is grateful to the Mathematisches Institut Der Universität Heidelberg for its kind hospitality and support and to Hans-Werner Henn for some useful discussions.

2. Extended Maps On DGR-algebras

Our terminology is mostly borrowed from [7]. We begin this section by recalling the basic definitions we need.

Let R be a commutative ring with a unit. Recall that a DGR-module M is a graded R-module, also denoted by M, together with a differential d_{M} of degree −1. A morphism of DGR-modules is a morphism of graded modules, which commutes with the differentials. Morphisms of graded R-modules (differential or not) will generally be assumed to have degree 0. However we shall frequently use morphisms of non-zero degree, in which case the degree will be explicitly spelled out.

A DGR-module M is said to be positive if M_{n} = 0 for n < 0. The suspension functor Σ(−), given by (ΣM)_{n} = M_{n−1} and (d_{ΣM}) = −(d_{M})_{n−1}, is an automorphism on the category of DGR-modules and preserves the subcategory of positive DGR-modules.

We shall generally assume that R is an integrally torsion free principal ideal domain. Let GRm denote the category of positive graded R-modules M, such that M_{n} is a free R-module for every n. Similarly let DGRm denote the category of DGR-modules such that for every object M the underlying graded R-module is an object of GRm. From this point and on by a GR-module (DGR-module) we shall always mean an object of GRm (DGRm)

A DGR-coalgebra C is a positive DGR-module, also denoted by C, together with a diagonal Δ_{C} : C ⟶ C ⊗ C and a morphism ϵ_{C} : C ⟶ R, such that Δ is associative and ϵ is a counit for Δ. A morphism f : C ⟶ D of DGR-coalgebras is a morphism of DGR-modules, which commutes with the diagonals and such that ϵ_{D}f = ϵ_{C}. Notice that the ground ring R admits a unique structure of a DGR-coalgebra.

A supplemented DGR-coalgebra is a DGR-coalgebra C, together with a map η_{C} : R ⟶ C of DGR-coalgebras called an augmentation. A morphism f : C ⟶ D between two such objects is a map of DGR-coalgebras, which in addition satisfies fη_{C} = η_{D}. For an integer n ≥ 0, a DGR-coalgebra C is said to be n-connected if (ϵ_{C})_{q} is an isomorphism for all q ≤ n. For a supplemented DGRcoalgebra C let the augmentation coideal JC denote the cokernel of η_{C}.

The dual terminology applies to define the concept of a DGR-algebra. Thus the structure maps on a supplemented DGR-algebra A are a multiplication μ_{A}, a unit R ⟶^{η_{A}} A and an augmentation A ⟶^{ϵ_{A}} R. For a supplemented DGR-algebra A, let the augmentation ideal IA denote the kernel of ϵ_{A}. We let DGRc and DGRa denote the categories of supplemented DGR-coalgebras and algebras respectively, such that for each object the underlying graded R-module is an object of GRm. As before, by DGR-coalgebras and algebras we shall always mean, from now on, objects of DGRc and DGRa respectively.

For a GR-module M, let T (M) denote the tensor algebra on M. Let j : M ⟶ T (M) denote the natural inclusion map into elements of tensor filtration 1. For an element x ∈ M let j(x) be denoted by [x]. The module T (M) is naturally a bigraded object. Namely, let bideg([x]) = (−1, deg(x)), and if bideg(y_{i}) = (n_{i}, k_{i}), i = 1, 2, then bideg(y_{1}y_{2}) = (n_{1} + n_{2}, k_{1} + k_{2}).

Definition 2.1. Let M be a GR-module and let F : IT (M) ⟶ IT (M) be a map of GR-modules of some non-negative degree. We say that F is right extended if it is a map of right T (M)-modules. The map F is said to be a right extension of f : M ⟶ IT (M) given by f = F ∘ j.

The following two lemmas are rather elementary. The first appears in [10, I, chp. 4] and the second is an observation made to the author by John C. Moore. Short proofs are included for the convenience of the reader.

Lemma 2.2. For M ∈ GRm, let f : M ⟶ IT (M) be a map of GR-modules of some non-negative degree. Then there exists a unique right extension F : IT (M) ⟶ IT (M) of f.

For x ∈ M, define F [x] = f(x) and inductively

F ([x_{1}| ⋯ |x_{n}]) = F ([x_{1}])[x_{2}| ⋯ |x_{n}].

Using the fact that M_{i} is a free R-module for each i, one observes immediately that F is a well defined right extension of f in the sense given above. Uniqueness is immediate from the definitions.

For any R-module M and a non-zero element r ∈ R, let ϕ_{r} denote the endomorphism of M given by multiplication by r.

Lemma 2.3. Let M be a DGR-module, with a homology exponent r ∈ R. Then ϕ_{r} is null-homotopic on M. Moreover, there is a choice of a null-homotopy s for ϕ_{r}, such that s^{2} = 0.

Since M is a free R-module in each dimension and since R is assumed to be a principal ideal domain, the short exact sequence of DGR-modules

0 ⟶ Z(M) ⟶^{i} M ⟶^{d} ΣB(M) ⟶ 0

is split in GRm. Here Z(M) denotes the submodule of cycles and ΣB(M) means the suspension of the submodule of boundaries, where in both Z(M) and ΣB(M) the differentials are taken to be trivial.

Let σ denote any right inverse for d and let j denote the left inverse for i, given by j(m) = i^{−1}(m − σd(m)). Notice that m − σd(m) is a cycle for every m ∈ M, so i^{−1} is defined on it. Since ij(m) is a cycle for every m ∈ M, one has by hypothesis that r · ij(m) is a boundary. Define

s(m) = σ(r · ij(m)).

The reader can easily verify that s is a null-homotopy for ϕ_{r} and that s^{2} = 0.

Let C be a simply-connected DGR-coalgebra. Recall that the loop algebra functor ΩC is given as a GR-algebra by ΩC = T (Σ^{−1}JC). As a differential graded module, ΩC is bigraded and has two differentials; an internal differential d_{I} of bidegree (0, −1), given by d_{I}[x] = −[dx], where d is the differential of C, and an external differential d_{E} of bidegree (−1, 0), given by d_{E}[x] = ∑_{i}(−1)^{|x′_{i}|}[x′_{i}|x′′_{i}], where Δ̄x = ∑_{i} x′_{i} ⊗ x′′_{i} is the reduced diagonal. Both differentials are required to be derivations of the algebra structure and the total differential d_{T} = d_{I} + d_{E} gives ΩC the structure of a DGR-algebra.

Let C be a simply-connected DGR-coalgebra with a homology exponent r ∈ R. Let s : C ⟶ C be a choice of a null-homotopy for ϕ_{r}. Let s_{1} : IΩC ⟶ IΩC denote the right extension of the composite

Σ^{−1}JC ⟶^{−s} Σ^{−1}JC ⟶^{j} IΩC

in the sense of Lemma 2.2. It is easy to see then that s_{1} is a null-homotopy for ϕ_{r} with respect to the internal differential d_{I} [10, I, chp. 4].

3. The obstruction Map θ and its iterations

In some sense, could the map s_{1}, constructed in the previous section, be a map of DGR-modules with respect to the external differential, it would then be a null-homotopy for ϕ_{r} with respect to the total differential d_{T}. However, this is hardly ever the case, due to the fact that in general the null-homotopy s on C, “inducing” s_{1}, fails to be, in a sense that can be made precise, a map of comodules over C. We now discuss a map θ of DGR-modules on ΩC, which can be considered as the obstruction for s_{1} to be a null-homotopy of ϕ_{r}.

Given the null-homotopy s for ϕ_{r} on C, construct s_{1} on IΩC, as in Section 2, and define θ = θ(s) : IΩC ⟶ IΩC by

θ = d_{E}s_{1} + s_{1}d_{E}.

The following are basic properties of θ. The proofs are straight forward from the definition and are left to the reader.

Proposition 3.1. For any null-homotopy s for ϕ_{r} on C, let θ denote the corresponding map on IΩC. For each k ≥ 1 let θ^{k} denote the k-th iteration of θ. Then

(1) θ^{k} is a map of DGR-modules with respect to both differentials on ΩC.

(2) θ^{k} is homotopic to (−1)^{k}ϕ_{r^{k}} with respect to the total differential on IΩC.

(3) θ^{k} is right extended.

(4) θ^{k} has bidegree (−k, k).

Corollary 3.2. Let k ≥ 1 and suppose that θ^{k} vanishes on elements of tensor filtration 1. Then θ^{k} vanishes on IΩC. Moreover the map ϕ_{r^{k}} is null-homotopic on IΩC if and only if θ^{k} is. Thus if θ^{k} = 0, for some k ≥ 1, then IΩC has a homology exponent r^{k}.

Define s_{k} : IΩC ⟶ IΩC by s_{k} = θ^{k−1} ∘ s_{1}. Define σ_{k} : IΩC ⟶ IΩC inductively by σ_{1} = s_{1} and for every k ≥ 2, σ_{k} = r^{k−1}s_{1}−r^{k−2}s_{2}+⋯+(−1)^{k−1}s_{k}. The following equalities provide some more insight into the structure of θ^{k} and can be proven by an easy induction.

Proposition 3.3. With the notation above, the following relations hold. (1) s_{k}d_{I} + d_{I}s_{k} = rθ^{k−1}.

(2) θ^{k} = ∑_{i=0}^{k}(d_{E}s_{1})^{i}(s_{1}d_{E})^{k−i}.

(3) If s^{2} = 0 then θ^{k} = (d_{E}s_{1})^{k} + (s_{1}d_{E})^{k}.

(4) θ^{k} = d_{E}s_{k} + s_{k}d_{E}.

(5) ϕ_{r^{k}} = dσ_{k} + σ_{k}d + (−1)^{k}θ^{k}.

4. Proof of Theorem 1.1 and Proposition 1.3

Let C be a c-connected DGR-coalgebra, c ≥ 1, such that H_{∗}(JC) has an exponent r ∈ R. Let s be a null-homotopy for ϕ_{r} on C, such that s^{2} = 0. Let T denote the loop algebra construction ΩC and let s_{1} and θ be the maps constructed in the previous sections with respect to s.

Since C is c-connected, T is (c − 1)-connected and so T_{p,q} = 0 for q ≤ −p(c + 1) − 1. In particular T_{−1,q} = 0 if q ≤ c. For each positive integer i let T (i) denote the i-th skeleton of IT.

Lemma 4.1. Suppose e ≥ \frac{d−c+1}{c}. Then θ^{e} vanishes on T (i), for i ≤ d.

By Corollary 3.2 it suffices to check that θ^{e} vanishes on elements of tensor filtration −1, namely on T_{−1,i}, i ≤ d + 1. Since bideg(θ^{e}) = (−e, e), it suffices to verify that T_{−1−e,d+e+1} = 0, for each c ≤ d ≤ c(e + 1) − 1. Indeed, by the connectivity argument above T_{−1−e,q} = 0 if q ≤ ec + e + c, which is satisfied for q = d + e + 1.

Now suppose that C is d-dimensional for some d ≥ c. Then we have T_{−1,q} = 0 if q > d. By the same argument as in Lemma 4.1 it follows that the map θ^{e} vanishes on IT if e ≥ \frac{d−c}{c}. However this can be slightly improved.

Lemma 4.2. Suppose that C is c-connected and d-dimensional for some d > c. Then θ^{e} vanishes on IT for e ≥ \frac{d−c−1}{c}.

It suffices to show that θ^{e} vanishes on T_{−1,d}. Indeed, since E_{−1,d+1}^{0} = 0 and s^{2} = 0, it follows from Lemma 3.3 that θ^{e} = (s_{1}d_{E})^{e}. But (s_{1}d_{E})^{e} = s_{1}((d_{E}s_{1})^{e−1}d_{E}). Hence the restriction of θ^{e} on T_{−1,d} factors through T_{−1−e,d+e−1}, which vanishes since d + e − 1 ≤ ec + e + c. Thus θ^{e} = 0 on T_{−1,∗} and the lemma follows.

We are now ready to prove Theorem 1.1 and Proposition 1.3. Let C be an c-connected DGR-coalgebra with a homology exponent r ∈ R. Let T denote ΩC. By Lemma 4.1, θ^{e} vanishes on T (i) for i ≤ d, if e ≥ \frac{d+c+1}{c}. Hence for any x ∈ T (i), i ≤ d, we have by (5) of Proposition 3.3

ϕ_{r^{e}}(x) = (dσ_{e} + σ_{e}d)(x).

Thus r^{e} · H_{i}(C) = 0 for 0 < i ≤ d. This proves Theorem 1.1.

Next suppose that C_{q} = 0 for q > d. By Lemma 4.2, θ^{e} vanishes on IT, if e ≥ \frac{d−c−1}{c}. Thus ϕ_{r^{e}} is null-homotopic on IT and Proposition 1.3 is proved.

5. Examples

For each pair of positive integers (k, t) define C^{k,t} to be the graded free abelian group, generated by elements {x_{n}}_{n≥2} and {y_{n}}_{n≥3}, where dimensions of the generators are given below.

|x_{2n}| = (2k − 1)(2n − 1) + 1,

|x_{2n+1}| = (2k − 1)(2n − 1) + 2,

|y_{2n+1}| = (2k − 1)2n + 1,

|y_{2n+2}| = (2k − 1)2n + 2.

Define a diagonal map Δ : C^{k,t} ⟶ C^{k,t} ⊗ C^{k,t} by Δ̄x_{2n} = Δ̄y_{2n+1} = 0 Δ̄x_{2n+1} = ∑_{i=1}^{n−1} x_{2i} ⊗ y_{2(n−i)+1} + y_{2(n−i)+1} ⊗ x_{2i} Δ̄y_{2n+2} = ∑_{i=1}^{n} x_{2i} ⊗ x_{2(n−i+1)} where Δ̄ denotes the reduced diagonal and extends to Δ in the obvious way.

Define a differential d on C^{k,t} by

dx_{2n} = dy_{2n+1} = 0, dx_{2n+1} = tx_{2n}, and dy_{2n+2} = ty_{2n+1}

for each i ≥ 1.

Define a map s of degree +1 on C^{k,t} by

sx_{2n+1} = sy_{2n+2} = 0, sx_{2n} = x_{2n+1} and sy_{2n+1} = y_{2n+2}.

Notice that s^{2} = 0.

One verifies by inspection that the triple (C^{k,t}, d, Δ) is a (2k − 1)-connected DGℤ-coalgebra with a homology exponent t and a null-homotopy for ϕ_{t} given by s. Construct s_{1} and θ on IΩC^{k,t} with respect to the null-homotopy s.

Proposition 5.1. For each positive integer n the following equalities hold in ΩC^{k,t}.

(1_{n}) θ^{2n−1}[y_{2n+1}] = (−1)^{n}[x_{2}]^{2n}.

(2_{n}) θ^{2n}[x_{2n+2}] = (−1)^{n}[x_{2}]^{2n+1}.

Since x_{2n} and y_{2n+1} are primitive for all n ≥ 1 it follows from Proposition 3.3 that θ^{k}[x_{2n}] = (d_{E}s_{1})^{k}[x_{2n}] and θ^{k}[y_{2n+1}] = (d_{E}s_{1})^{k}[y_{2n+1}].

One verifies directly that θ[y_{3}] = −[x_{2}]^{2} and that θ^{2}[x_{4}] = −[x_{2}]^{3}. Thus the statement is true for n = 1. Assume that θ^{2n−3}[y_{2n−1}] = (−1)^{n−1}[x_{2}]^{2n−2} and θ^{2n−2}[x_{2n}] = (−1)^{n−1}[x_{2}]^{2n−1}, namely that (1_{n−1}) and (2_{n−1}) hold. We prove (1_{n}) and (2_{n}).

θ^{2n−1}[y_{2n+1}] = −θ^{2n−2}d_{E}[y_{2n+2}] =
−θ^{2n−2} ∑_{i=1}^{n}[x_{2i}|x_{2(n−i+1)}] =
− ∑_{i=1}^{n}(θ^{2n−2}[x_{2i}])[x_{2(n−i+1)}]

By Lemma 4.1 all but the last summand vanish and hence by the inductive hypothesis

θ^{2n−1}[y_{2n+1}] = −(θ^{2n−2}[x_{2n}])[x_{2}] = (−1)^{n}[x_{2}]^{2n}

and (1_{n}) is proved. Next

θ^{2n}[x_{2n+2}] = −θ^{2n−1}d_{E}[x_{2n+3}] =
θ^{2n−1} ∑_{i=1}^{n}[y_{2(n−i+1)+1}|x_{2i}] − [x_{2i}|y_{2(n−i+1)+1}] =
∑_{i=1}^{n}(θ^{2n−1}[y_{2(n−i+1)+1}])[x_{2i}] − (θ^{2n−1}[x_{2i}])[y_{2(n−i+1)+1}].

By Lemma 4.1 all summands vanish except for possibly

(θ^{2n−1}[y_{2n+1}])[x_{2}] and (θ^{2n−1}[x_{2n}])[y_{2(n−i+1)+1}].

But

θ^{2n−1}[x_{2n}] = θθ^{2n−2}[x_{2n}] = (−1)^{n−1}θ[x_{2}]^{2n−1} = 0.

Hence

θ^{2n}[x_{2n+2}] = (θ^{2n−1}[y_{2n+1}])[x_{2}] = (−1)^{n}[x_{2}]^{2n}[x_{2}] = (−1)^{n}[x_{2}]^{2n+1}.

This completes the proof.

Proposition 5.2. For any positive integer m, the element [x_{2}]^{m} represents a non-zero homology class in H_{m(2k−1)}(C^{k,t}).

Theorem 1.2 follows at once from Proposition 5.2. Indeed, set c = 2k − 1. We must show that H_{ec}(ΩC^{k,t}) contains en element of order t^{e}. For e = 2n, consider the class α_{e} ∈ H_{ec}(ΩC^{k,t}), represented by the cycle [y_{e+1}]. Since [y_{e+1}] cannot possibly be a boundary, α_{e} ≠ 0. By propositions 5.1 and 5.2, θ_{∗}^{e−1}α_{e} ≠ 0. Hence by Proposition 3.1, t^{e−1}α_{e} ≠ 0. On the other hand by Theorem 1.1, t^{e}α_{e} = 0. Hence α_{e} is an element of order t^{e}.

For e = 2n + 1, consider the class β_{e} ∈ H_{ec}(ΩC^{k,t}), represented by the cycle [x_{e+1}]. Again β_{e} is evidently non-zero. Propositions 5.1 and 5.2 imply that θ_{∗}^{e−1}β_{e} ≠ 0. Hence t^{e−1}β_{e} ≠ 0. On the other hand, by Theorem 1.1 t^{e}β_{e} = 0. Hence β_{e} is an element of order t^{e}. This completes the proof of Theorem 1.2.

We proceed by proving Proposition 5.2. Let k and t be fixed and let C denote C^{k,t}. Let C^{∗} = Hom(C; ℤ) denote the dual of C. Thus C^{∗} is a DGℤ-algebra with product given by dualizing the diagonal map on C. By abuse of notation, we denote the duals of x_{i}, y_{j} in C^{∗} again by x_{i} and y_{j} respectively. Let C^{∗}(j) denote the sub-algebra of C^{∗} generated by x_{2}, x_{3}, ⋯, x_{j+2}, y_{3}, y_{4}, ⋯, y_{j+1} if j is odd and by x_{2}, ⋯, x_{j+1}, y_{3}, ⋯, y_{j+2} if j is even. Define C(j) to be (C^{∗}(j))^{∗}. Thus C(j) is a quotient DGℤ-coalgebra of C and for every j ≥ 1 there are natural projections

C ⟶^{π_{j+1}} C(j + 1) ⟶^{φ_{j+1}} C(j).

The following lemma is immediate from the definitions.

Lemma 5.3. For each n ≥ 2, the submodule Ker(φ_{n}) of C(n) is n(2k − 1)- connected. Thus so is the submodule Ker(π_{n}) of C.

Since the element [x_{2}] ∈ ΩC is preserved under each of the maps Ωπ_{n}, it suffices to show that [x_{2}]^{n} is not a boundary in ΩC(n) to complete the proof of Proposition 5.2.

Proposition 5.4. The element [x_{2}]^{n} is not a boundary in ΩC(n) but [x_{2}]^{n+1} is. In fact for each m ≥ 1 there exist decomposable elements A_{m} ∈ ΩC(2m) and B_{m} ∈ ΩC(2m − 1) such that

(1_{m}) d(t^{2m−2}[y_{2m+2}] + B_{m}) = [x_{2}]^{2m} in ΩC(2m − 1),

(2_{m}) d(t^{2m−1}[x_{2m+3}] + A_{m}) = [x_{2}]^{2m+1} in ΩC(2m). Furthermore, if z′_{n} ∈ ΩC(n) is such that dz′_{n} = [x_{2}]^{n+1}, then z′_{n} is indecomposable.

We first show that [x_{2}]^{n} is not a boundary in ΩC(n). This certainly holds for n = 1. Thus assume [x_{2}]^{j} is not a boundary in ΩC(j) for j ≤ n − 1 and suppose n = 2m. The kernel of φ_{2m} through dimension (2k−1)2m+2 = |y_{2m+2}| is given by the element y_{2m+1}. Thus the kernel of Ωφ_{2m} through dimension (2k − 1)2m + 1 is generated by the element [y_{2m+1}] if k > 1 and by [y_{2m+1}], [y_{2m+1}|x_{2}] and [x_{2}|y_{2m+1}] if k = 1. Notice that in either case Ker(Ωφ_{2m}) through dimension (2k −1)2m+1 consists of cycles.

Suppose z ∈ ΩC(2m) is such that dz = [x_{2}]^{2m}. Then by naturality dΩφ_{2m}z = [x_{2}]^{2m} in ΩC(2m − 1). But by our induction hypothesis it follows that Ωφ_{2m}z is indecomposable in ΩC(2m − 1). Hence it contains an essential summand of tensor filtration 1. Since Ωφ_{2m} maps the single element of tensor filtration 1 and total dimension (2k − 1)2m + 1, [y_{2m+2}], non-trivially, it follows that z itself is indecomposable. But in ΩC(2m), [y_{2m+2}] is not a cycle with respect to the internal differential. This yields a contradiction. The case n = 2m + 1 follows by analogy.

Next we must show that [x_{2}]^{n+1} is a boundary in ΩC(n). Notice that d[y_{4}] = [x_{2}]^{2} in ΩC(1) and that d(t[x_{5}] − [y_{3}|x_{3}] − [x_{2}|y_{4}]) = [x_{2}]^{3} in ΩC(2). Thus (1_{m}) and (2_{m}) hold for m = 1. Assume that for each j ≤ m−1 there are decomposable elements A′_{j} and B′_{j} in ΩC such that

(1′_{j}) d(t^{2j−2}[y_{2j+2}] + B′_{j}) = [x_{2}]^{2j} − t^{2j−1}[y_{2j+1}],

(2′_{j}) d(t^{2j−1}[x_{2j+3}] + A′_{j}) = [x_{2}]^{2j+1} − t^{2j}[x_{2j+2}].

Notice that (1′_{j}) and (2′_{j}) are satisfied for j = 1 with respect to A′_{1} = −([y_{3}|x_{3}]+ [x_{2}|y_{4}]) and B′_{1} = 0. Proving (1_{m}) and (2_{m}) amounts to showing that there are decomposable elements A′_{m} and B′_{m} in ΩC, such that (1′_{m}) and (2′_{m}) hold. Indeed, reducing (1′_{m}) to ΩC(2m − 1) and (2′_{m}) to ΩC(2m) yields (1_{m}) and (2_{m}). For a graded algebra A and x, y ∈ A, write [x, y] for the difference xy − (−1)^{|x||y|}yx. We write down the formulas and leave it for the reader to verify that the required equations hold.

Define, for m ≥ 2

B′_{m,1} = −([x_{2}]A′_{m−1} + t^{2m−3}([x_{2}|x_{2m+1}] + [x_{2m}|x_{3}])).

B′_{m,2} = ∑_{2≤j< \frac{m+1}{2}} t^{2(m−j)−1}[t^{2j−2}[x_{2j+1}] + tA′_{j−1} + [x_{3}][x_{2}]^{2j−2}, [x_{2(m−j+1)}]].

For m even define B′_{m,3} = 0 and for m odd define

B′_{m,3} = t^{m−2}([x_{3}][x_{2}]^{m−1}[x_{m+1}] + t^{m−1}[x_{m+2}|x_{m+1}] + tA′_{\frac{m−1}{2}} [x_{m+1}]).

Next define

A′_{m,1} = −(t^{2m−2}[x_{2}|y_{2m+2}] + [x_{2}]B′_{m} + t^{2m−2}[y_{2m+1}|x_{3}]).

A′_{m,2} = ∑_{j=2}^{m} t^{2(m−j)}[t^{2j−2}[x_{2j+1}] + tA′_{j−1} + [x_{3}][x_{2}]^{2j−2}, [y_{2(m−j)+3}]].

Let

B′_{m} = B′_{m,1} + B′_{m,2} + B′_{m,3} and A′_{m} = A′_{m,1} + A′_{m,2}.

Finally we must show that if b′_{n} ∈ ΩC(n) is such that db′_{n} = [x_{2}]^{n+1}, then b′_{n} is indecomposable. This is evident for n = 1. Let b_{n} ∈ ΩC(n) denote the element constructed above such that db_{n} = [x_{2}]^{n+1} in ΩC(n). Assume that b′_{n} is a decomposable element with the same property. Then by naturality dΩφ_{n}b′_{n} = [x_{2}]^{n+1}. Hence the element Ωφ_{n}b′_{n} − b_{n−1}[x_{2}] is a cycle in ΩC(n − 1). However |b_{n−1}| = (2k − 1)n + 1 and Ωφ_{n} is an epimorphism in this dimension with kernel given by cycles. Thus b_{n−1} can pulled back to ΩC(n) and the lift is unique up to a cycle. Hence the element b′_{n} − b_{n−1}[x_{2}] is a cycle in ΩC(n). However by (1′_{n}) and (2′_{n}), db_{n−1} = [x_{2}]^{n} −t^{n−1}α in ΩC(n), where α = [y_{n+1}] if n is even and [x_{n+1}] otherwise. Thus d(b′_{n} − b_{n−1}[x_{2}]) = t^{n−1}α[x_{2}] ≠ 0. This yields a contradiction and thus completes the proof.

For each triple of positive integers (k, t, n), we consider two sub-DGℤ-coalgebras A^{k,t,n} and B^{k,t,n} of C^{k,t}, generated as free abelian group by the sets {x_{i}}_{i=2}^{2n+1} ∪ {y_{j}}_{j=3}^{2n} and {x_{i}}_{i=2}^{2n+1} ∪ {y_{j}}_{j=3}^{2n+2} respectively. One immediately observes that A^{k,t,n} ⊂ B^{k,t,n} as a sub-DGℤ-coalgebra, for each n ≥ 1 and that both A^{k,t,n} and B^{k,t,n} are closed under the null-homotopy s for ϕ_{t}. Furthermore, A^{k,t,n} and B^{k,t,n} are ((2n − 1)(2k − 1) + 2) and (2n(2k − 1) + 2)-dimensional respectively. By Proposition 1.3 t^{e} · H_{∗}(A^{k,t,n}) = 0 if e ≥ 2n − 1 and t^{e} · H_{∗}(B^{k,t,n}) = 0 if e ≥ 2n.

On the other hand since both A^{k,t,n} and B^{k,t,n} are sub-DGℤ-coalgebras of C^{k,t} and since [x_{2}] represents an element of infinite height in H_{∗}(ΩC^{k,t}), it follows that the same holds in the loop space homology of A^{k,t,n} and B^{k,t,n}. Thus by the argument above [x_{2n}] represents an element of order t^{2n−1} in H_{(2n−1)(2k−1)}(ΩA^{k,t,n}) and that [y_{2n+1}] represents an element of order t^{2n} in H_{2n(2k−1)}(ΩB^{k,t,n}). Hence the approximation of Proposition 1.3 is best possible in these examples.

6. Transfer type Null-Homotopies

In this section we restrict attention to a particular family of DGR-coalgebras, namely those DGR-coalgebras which appear as a suitable modification of the classifying construction B(A) on a DGR-algebra A, concentrated in degree 0.

We start by recalling the construction P_{n}(−) on DGR-coalgebras as introduced in [10]. Let C be a DGR-coalgebra. For some integer n > 0 consider the differential d_{n} : C_{n+1} ⟶ C_{n}. Since R is assumed a principal ideal domain and C_{n} is free, the image B_{n}(C) of d_{n} is a free submodule of C_{n}. Thus there exists a right inverse ϵ_{n} : B_{n}(C) ⟶ C_{n+1} for d_{n}. Let C′_{n+1} denote the image of ϵ_{n} in C_{n+1}. Let D denote the sub-DGR-module of C given by D_{i} = C_{i} for 0 < i ≤ n, D_{n+1} = C′_{n+1} and D_{i} = 0 otherwise. Define P_{n}C to be the quotient DGR-module C/D. Notice that P_{n}C is a DGR-coalgebra by construction and that the projection ι_{n} : C ⟶ P_{n}C is a map of DGR-coalgebras. The following are elementary properties of P_{n}C and are proven in [10].

Proposition 6.1. Let C be a DGR-coalgebra. Then for every n > 0 (1) the projection ι_{n} is a split epimorphism in DGRm.

(2) ι_{n} induces a homology isomorphism above dimension n.

(3) P_{n}C is n-connected.

Let C be a DGR-coalgebra and let s denote a self map of C of degree +1. Let n be a fixed positive integer. Consider the projection ι = ι_{n} : C ⟶ P_{n}C, where P_{n}C is constructed with respect to some right inverse ϵ for the n-th differential d as above. Denote elements of P_{n}C by ιx, where x ∈ C is some pre-image. A right inverse η of ι in DGRm is given by the identity in dimensions higher than n + 1 and for ιx ∈ (P_{n}C)_{n+1}, define η(ιx) = x − ϵdx. Then s̃ = ιsη can be identified with s in dimensions higher than n + 1 and for ιx ∈ (P_{n}C)_{n+1} one has s̃(ιx) = s(x) − sϵd(x).

The self map of C given by f_{s} = ds + sd is a map of DGR-modules, for which s is a null-homotopy. Suppose that f_{s} = ϕ_{r} for some r ∈ R. Then it is easy to verify that s̃ is a null-homotopy for the self map of P_{n}C, given by f̃_{s} = ιf_{s}η.

We now specialize to the case mentioned above. Let A be a DGR-algebra concentrated in degree 0. Let B(A) denote the classifying construction [7] for A. A self map s of degree +1 on B(A) is said to be a transfer type null-homotopy, if for each n > 1

s([x_{1}|x_{2}| ⋯ |x_{n}]) = ∑_{i} a_{i}[x′_{i}|x′′_{i}|x_{2}| ⋯ |x_{n}],

whenever s([x_{1}]) = ∑_{i} a_{i}[x′_{i}|x′′_{i}]. Notice that transfer type null-homotopies s are a priori unlikely to satisfy s^{2} = 0.

Proposition 6.2. Let A be a DGR-algebra concentrated in degree 0, which is free as an R-module. For some r ∈ R, let s be a transfer type null-homotopy for ϕ_{r} on B(A). For some n ≥ 1 let θ be the corresponding map on ΩP_{n}B(A). Then for every k > n

θ([ι[x_{1}| ⋯ |x_{k}]]) = (−1)^{n}([ιs[x_{1}| ⋯ |x_{n}]|ι[x_{n+1}| ⋯ |x_{k}]]) − [ιsϵd[x_{1}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]]).

Thus θ^{2} = 0 if n > 1 and θ^{3} = 0 if n = 1.

The proof amounts to calculating θ on the algebra generators for ΩP_{n}B(A) given by [ι[x_{1}|x_{2}| ⋯ |x_{k}]], x_{j} ∈ A, observing the fact that θ is right extended. In computing d_{E}s_{1} on a typical generator [x] of homological degree k, we may assume that k ≥ 2n + 1, since otherwise the element s̃(x) is primitive in the coalgebra structure of P_{n}B(A) and thus d_{E} vanishes.

d_{E}s_{1}[ι[x_{1}|x_{2}| ⋯ |x_{k}]] = −d_{E} ∑_{i} a_{i}[ι[x′_{i}|x′′_{i}|x_{2}| ⋯ |x_{k}]] =

∑_{i} a_{i}(−1)^{n}[ι[x′_{i}|x′′_{i}|x_{2}| ⋯ |x_{n}]|ι[x_{n+1}| ⋯ |x_{k}]] +

∑_{i} a_{i} ∑_{j=1}^{k−2n−1}(−1)^{n+j}[ι[x′_{i}|x′′_{i}|x_{2}| ⋯ |x_{n+j}]|ι[x_{n+j+1}| ⋯ |x_{k}]] =

(−1)^{n}[ιs[x_{1}|x_{2}| ⋯ |x_{n}]|ι[x_{n+1}| ⋯ |x_{k}]] +
(−1)^{n+1}[ιs[x_{1}|x_{2}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]] +
∑_{j=2}^{k−2n−1}(−1)^{n+j+1}s_{1}[ι[x_{1}|x_{2}| ⋯ |x_{n+j}]|ι[x_{n+j+1}| ⋯ |x_{k}]].

On the other hand

s_{1}d_{E}[ι[x_{1}|x_{2}| ⋯ |x_{k}]] =
(−1)^{n+1}s_{1}[ι[x_{1}|x_{2}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]] +
∑_{j=2}^{k−2n−1}(−1)^{n+j}s_{1}[ι[x_{1}|x_{2}| ⋯ |x_{n+j}]|ι[x_{n+j+1}| ⋯ |x_{k}]].

Thus adding the two equations we get

θ[ι[x_{1}|x_{2}| ⋯ |x_{k}]] =

(−1)^{n}[ιs[x_{1}|x_{2}| ⋯ |x_{n}]|ι[x_{n+1}| ⋯ |x_{k}]] +
(−1)^{n+1}[ιs[x_{1}|x_{2}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]] +
(−1)^{n+1}s_{1}[ι[x_{1}|x_{2}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]] =
(−1)^{n}[ιs[x_{1}|x_{2}| ⋯ |x_{n}]|ι[x_{n+1}| ⋯ |x_{k}]] +
(−1)^{n+1}[ιs[x_{1}|x_{2}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]] +
(−1)^{n}[ιs[x_{1}|x_{2}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]] +
(−1)^{n+1}[ιsϵd[x_{1}|x_{2}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]] =

(−1)^{n}([ιs[x_{1}| ⋯ |x_{n}]|ι[x_{n+1}| ⋯ |x_{k}]]) − [ιsϵd[x_{1}| ⋯ |x_{n+1}]|ι[x_{n+2}| ⋯ |x_{k}]]),

as claimed.

Notice that θ[ι[x_{1}| ⋯ |x_{k}]] = 0 if k ≤ 2n. Since θ is right extended, it follows that θ^{2} = 0 if n > 1. If n = 1 then θ^{2} might not vanish but then θ^{3} certainly does. This completes the proof.

Corollary 6.3. Let G be a finite group of order p^{r}m with (p, m) = 1. Let B[G] denote the classifying construction for the group ring ℤ_{(p)}[G]. Then p^{2r} annihilates H̃_{∗}(ΩP_{n}B[G]) if n ≥ 2 and p^{3r} annihilates H̃_{∗}(ΩP_{1}B[G]).

For a finite group G of order N there is a canonical null-homotopy s for the map ϕ_{N} on B[G] given by

s([x_{1}|x_{2}| ⋯ |x_{n}]) = ∑_{x∈G}[x|x_{1}|x_{2}| ⋯ |x_{n}].

The map s is easily seen to satisfy the hypotheses of Proposition 6.2. The result follows from Corollary 3.2.

Theorem 1.5 now follows by combining Corollary 6.3 and Proposition 1.4, which we prove in the next section.

7. Geometric Realization of the Construction P_{n}

Throughout this section whenever we say a space, we mean a pointed connected CW complex for which the cellular chain complex admits a natural strictly coassociative diagonal. This can of course always be obtained by geometrically realizing the associated singular simplicial set. This restriction is imposed here for simplicity and has no further significance.

Let X^{(n)} denote the n-skeleton of a given space X. Consider the space P′_{n}X obtained from X by collapsing X^{(n)} to the base point. Obviously the natural map ι′_{n} : X ⟶ P′_{n}X induces an isomorphism on homology groups H_{i}(−; ℤ) for i ≥ n + 2. Furthermore, H_{n}(X^{(n)}; ℤ) is free abelian and thus it follows by inspection of the associated long exact homology sequence that H_{n+1}(P′_{n}X; ℤ) ≅ H_{n+1}(X; ℤ) ⊕ F, where F is a free abelian group and that ι′_{n} induces the obvious split monomorphism in this dimension. Notice that the free abelian group F is no other but the subgroup of boundaries B_{n}(S_{∗}(X; ℤ)), where by S_{∗}(−; ℤ) we denote the cellular chain complex on X.

Next the Hurewicz isomorphism theorem implies that π_{n+1}P′_{n}X ≅ H_{n+1}(X; ℤ)⊕ F. Choose generators {f_{i}}_{i∈I} for F and realize each f_{i} as a map S^{n+1} ⟶^{f_{i}} P′_{n}X. Define P_{n}X to be the cofibre in the sequence

⋁_{i∈I} S_{i}^{n+1} \underrightarrow{⋁ f_{i}} P′_{n}X ⟶ P_{n}X.

It is immediate that the natural map ι_{n} : X ⟶ P_{n}X induces an isomorphism on homology groups in dimensions larger than or equal to n + 1. This completes the proof of the first part of Proposition 1.4.

To prove the second part an intermediate step is needed. For a DGR-coalgebra C, define P̄_{n}S as follows. Let F denote a free R-module on as many generators as the submodule of boundaries B_{n}(C). Without loss of generality we may identify F with B_{n}(C), thus the differential d : C_{n+1} ⟶ F is an epimorphism and one can choose a right inverse ϵ for it. Let P̄_{n}C be the DGR-coalgebra given by dividing C by its n-skeleton with the additional modification that (P̄_{n}C)_{n+2} = C_{n+2} ⊕ F. Define d̄(x, y) = dx + ϵy to be the differential on (P̄_{n}C)_{n+2}. Require that every element in (P̄_{n}C)_{n+2} is primitive. It is thus easy to verify that the obvious map ῑ: C ⟶ P̄_{n}C is a map of DGR-coalgebras inducing an isomorphism on homology in dimensions larger than n. If C = S_{∗}X, then making compatible choices for the attaching maps defining P_{n}X out of P′_{n}X and the right inverse ϵ, we immediately get that there is an isomorphism of DGR-coalgebras P̄_{n}S_{∗}X ≅ S_{∗}P_{n}X.

Next for any DGR-coalgebra C, one can define P_{n}C and P̄_{n}C with respect to the same right inverse ϵ for the differential C_{n+1} ⟶^{d} B_{n}(C). Thus it is immediate that there is a quasi isomorphism of DGR-coalgebras P̄_{n}C ⟶ P_{n}C. Applying this for C = S_{∗}X, completes the proof of Proposition 1.4.

We end this section by pointing out that although P̄_{n}(−) and P_{n}(−) are equivalent constructions on DGR-coalgebras, P_{n}(−) behaves nicer with respect to algebraic maps due to the fact that the projection C ⟶^{ι_{n}} P_{n}C is a split epimorphism in the category of DGR-modules. On the other hand P̄_{n}(−) corresponds more naturally to the geometric analogue.

8. Nilpotent DGR-algebras

In this section we prove Theorem 1.6 and observe that it does not correspond to a geometric situation. We begin with a proof of the theorem.

Let A be a DGR-algebra and assume that the classifying construction B(A) admits a homology exponent r ∈ R. Assume further that A is nilpotent as an algebra, namely that some power of its augmentation ideal IA vanishes.

There is a quasi isomorphism of DGR-algebras [7]

α(A) : ΩB(A) ⟶ A.

Since α(A) is multiplicative, α(A)(IΩB(A))^{n} ⊂ (IA)^{n}, for every n ≥ 1.

Next, by hypothesis B(A) admits a homology exponent r ∈ R. Thus we have a null-homotopy s for ϕ_{r} on B(A). Let θ denote the map constructed on ΩB(A) with respect to s. Then the k-th iteration θ^{k} of θ increases tensor filtration by k and is homotopic to ϕ_{r^{k}} up to a sign. Thus for every k ≥ 1 we get a commutative diagram

[IΩB(A))^{k+1}]IΩB(A)e, tθ^{k}s, r=(IΩB(A))^{k+1}e, tα(A)s, rinc(IA)^{k+1}s, rincIΩB(A)e, tθ^{k}IΩB(A)e, tα(A)IA

By hypothesis (IA)^{n} = 0, which implies Theorem 1.6.

Next we observe that Theorem 1.6 does not refer to a topological situation, but rather to a purely algebraic one. The content of this remark is contained in the following

Proposition 8.1. Let X be a simply-connected, rationally contractible space of finite type. Then H_{∗}(ΩX; 𝔽_{p}) is not a nilpotent algebra.

First observe that a finite loop space cannot possibly have a rationally contractible classifying space. Indeed R. Kane showed [8, pages 256 and 300] that the loop space homology of simply-connected finite H-spaces is torsion free and hence rationally non-trivial. Thus by C-class theory of Serre, a finite H-space cannot have a rationally contractible classifying space. Hence we may assume that ΩX is infinite dimensional.

Consider the Hopf algebra H_{∗}(ΩX; 𝔽_{p}). By assumption it is an infinite dimensional, cocommutative Hopf algebra, which is nilpotent as an algebra. But such Hopf algebras do not exist by Proposition 8.2 below.

Proposition 8.2. Let A be a cocommutative Hopf algebra of finite type and infinite dimension over a field k of characteristic p > 0. Then A is not a nilpotent algebra.

The rest of this section is devoted to the proof of Proposition 8.2. The proposition is rather elementary and should be well known to the expert. We include a proof for the convenience of the reader. Graded vector spaces are always assumed to be of finite type.

Lemma 8.3. Let A be a cocommutative Hopf algebra of finite type over a field k of characteristic p > 0, which is nilpotent as an algebra. Then the underlying algebra of the dual Hopf algebra A^{∗} is locally nilpotent, namely there exists no element x ∈ A of infinite hight.

Let A be a cocommutative Hopf algebra over a field k of characteristic p > 0, which is nilpotent as an algebra. Assume that the dual commutative Hopf algebra A^{∗} is not locally nilpotent as an algebra. Then we claim that A^{∗} contains a primitive element y of infinite hight. Indeed choose an element x of infinite hight and minimal degree. Then, since every element of lower degree is nilpotent, some power y = x^{p^{k}} must be primitive.

Let y ∈ A^{∗} be a primitive element of infinite hight. Then there is a monomorphism of Hopf algebras

P [y] ⟶^{ϕ} A^{∗}.

Dualizing we get an epimorphism of Hopf algebras

A ⟶^{ϕ^{∗}} Γ[y],

where Γ[y] denotes the divided polynomial algebra over k on one generator. In particular Γ[y] contains products of arbitrary length and thus so does A contradicting its nilpotency.

Lemma 8.4. Let A be a commutative Hopf algebra of infinite dimension over a field k of characteristic p > 0, which as an algebra is locally nilpotent. Then the dual Hopf algebra A^{∗} contains non-vanishing products of arbitrary length.

Since A is commutative the Borel structure theorem applies and A is isomorphic as an algebra to a tensor product of monogenic algebras. Assuming that A is locally nilpotent implies that the Borel decomposition for A does not contain polynomial factors. Hence it must be infinitely generated as an algebra. Consequently the module of primitives in the dual Hopf algebra P (A^{∗}) is infinite dimensional. But a product of an arbitrary number of distinct primitives in a Hopf algebra is never zero. The result follows.

Lemmas 8.3 and 8.4 imply that a cocommutative Hopf algebra over k, for which the underlying algebra is nilpotent must be finite dimensional, which completes the proof of Proposition 8.2.

9. speculations, finiteness properties

In the study of loop space homology torsion, it seems reasonable to try and relate certain finiteness conditions on the objects under consideration to the existence of exponents. One may take two different approaches here. Fix a DGRcoalgebra C with a reduced homology exponent r ∈ R.

The first approach is to assume certain finiteness conditions on C and try to gain control on the growth of torsion in loop space homology. In particular one might be interested in finding a loop space homology exponent. Thus one may assume for instance that C is of finite rank over R, in which case we have shown that ΩC admits a homology exponent. If C has a trivial coproduct then it is immediate that C and ΩC have the same homology exponent as is the case when C is the chain coalgebra for a rationally contractible suspension space. One might thus expect that nilpotency of the Hom-dual of a DGR-coalgebra C should imply the existence of a loop space homology exponent. However by Theorem 1.2 this is not nearly the case. Indeed the DGℤ-coalgebras C^{k,t} constructed in the theorem have the property that their duals are nilpotent of degree 3.

A finiteness condition on C, which we did not consider here is the existence of only finitely many primitives in C. Namely one may conjecture that if C contains only finitely many primitives, then ΩC admits a homology exponent. In fact we are not aware of a counter example to this.

The second approach to the problem is taken in Section 8 above and arises by recalling the fact that for every DGR-algebra A, the natural map ΩB(A) ⟶ A is a homotopy equivalence. Thus one may like to consider DGR-coalgebras of the form B(A), which satisfy our basic hypothesis of having a homology exponent and impose additional finiteness conditions on A.

If A is concentrated in degree 0, then the study of its homology doesn’t make too much sense. But in this case one might like to study the associated DGRalgebras ΩP_{n}B(A). If A is of finite rank over R, as is the case if A = R[G] for a finite group G, then it is not hard to construct a condition which imply that B(A) admits a transfer type null-homotopy for ϕ_{r}, where r is the image of the rank of A in R under the canonical map from the integers. However the existence of such a null-homotopy appears to impose severe restrictions on the algebra structure of A. We thus find it reasonable to conjecture that there exist algebras A, concentrated in degree 0, which are finitely generated as algebras or even of finite rank over R, such that B(A) has a homology exponent but ΩP_{n}B(A) does not have one.

References

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[6] Y. Felix, S. Halperin and J-C.Thomas; Torsion in Loop Space Homology; Jour. reine angew. Math. 432 (1992), 77-92.

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[8] R. Kane; The Homology of Hopf Spaces; North Holland (1988).

[9] R. Kane; Implications in Morava K-theory; Memo. A.M.S. 340 (1986).

[10] R. Levi; On Finite Groups and Homotopy Theory; to appear in Memoirs of the A.M.S.

[11] R. Levi; A Counter Example to a Conjecture of Cohen; to appear in the 1994 BCAT proceedings.

[12] R. Levi; On Homological Rate of Growth and the Homotopy Type of ΩBG_{p}^{∧}; preprint.

[13] J. W. Milnor and J. C. Moore; On the Structure of Hopf Algebras; Annals of Math. 81 (1965), 211-264.

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