BETTI NUMBER ESTIMATES FOR NILPOTENT GROUPS

MICHAEL FREEDMAN, RICHARD HAIN, AND PETER TEICHNER

Abstract. We prove an extension of the following result of Lubotzky and Magid on the rational cohomology of a nilpotent group G: if b_{1} < ∞ and G ⊗ ℚ ≠ 0, ℚ, ℚ^{2} then b_{2} > b_{1}^{2}/4. Here the b_{i} are the rational Betti numbers of G and G ⊗ ℚ denotes the Malcev-completion of G. In the extension, the bound is improved when we know that all relations of G all have at least a certain commutator length. As an application of the refined inequality, we show that each closed oriented 3-manifold falls into exactly one of the following classes: it is a rational homology 3-sphere, or it is a rational homology S^{1} × S^{2}, or it has the rational homology of one of the oriented circle bundles over the torus (which are indexed by an Euler number n ∈ ℤ, e.g. n = 0 corresponds to the 3-torus) or it is of general type by which we mean that the rational lower central series of the fundamental group does not stabilize. In particular, any 3-manifold group which allows a maximal torsion-free nilpotent quotient admits a rational homology isomorphism to a torsion-free nilpotent group.

1. The Main Results

We analyze the rational lower central series of 3-manifold groups by an extension of the following theorem of Lubotzky and Magid [15, (3.9)].

Theorem 1. If G is a nilpotent group with b_{1}(G) < ∞ and G ⊗ ℚ ≠ {0}, ℚ, ℚ^{2}, then

b_{2}(G) > \frac{1}{4} b_{1}(G)^{2}

Here b_{i}(G) denotes the ith rational Betti number of G and G ⊗ ℚ the Malcev-completion of G. (We refer the reader to Appendix A, where we have collected the group theoretic definitions.)

This Theorem is an analogue of the Golod-Shafarevich Theorem (see [9] or [14, p. 186]), which states that if G is a finite p-group, then the inequality r > d^{2}/4 holds, where d is the minimal number of generators in a presentation of G, and r is the number of relations in any presentation. It can be used to derive a result for finitely generated nilpotent groups similar to the one above, but with b_{2}(G) replaced by the minimal number of relations in a presentation for G, see [6, p. 121].

The first author was supported in part by an NSF grant and the Guggenheim Foundation.

The second author was supported by the IHES and grants from the NSF.. The third author was supported by the IHES and the Humboldt foundation.

We shall need a refined version of this result in the case where the relations of G are known to have a certain commutator length. In order to make this precise, let

H_{2}(G; ℚ) =: Φ_{2}^{ℚ}(G) ⊇ Φ_{3}^{ℚ}(G) ⊇ Φ_{4}^{ℚ}(G) ⊇ …

be the rational Dwyer filtration of H_{2}(G; ℚ) (defined in Appendix A).

Theorem 2. If G is a nilpotent group with b_{1}(G) < ∞, H_{2}(G; ℚ) = Φ_{r}^{ℚ}(G) and G ⊗ ℚ ≠ {0}, ℚ, ℚ^{2}, then

b_{2}(G) > \frac{(r − 1)^{(r−1)}}{r^{r}} b_{1}(G)^{r}.

The statement that H_{2}(G; ℚ) = Φ_{r}^{ℚ}(G) is equivalent to the statement that in a minimal presentation of the Lie algebra of G ⊗ ℚ, all relations lie in the rth term of the lower central series of the free Lie algebra on the minimal generating set H_{1}(G; ℚ). We know of no analogue of this result for p-groups, although there should be one. The result is a corollary of a more technical and stronger result, Proposition 5. Note that Theorem 1 is the special case of Theorem 2 with r = 2.

The intuitive idea behind these results is that H_{1}(G) corresponds to generators of G and H_{2}(G) to its relations. For example, if G is abelian, then the number of (primitive) relations (≈ b_{2}) grows quadratically in the number of generators (≈ b_{1}) because the commutator of a pair of generators has to be a consequence of the primitive relations. Similarly, if G is nilpotent, then the relations have to imply that for some r, all r-fold commutators in the generators vanish. If H_{2}(G) = Φ_{r}(G), then all relations are in fact r-fold commutators because, by Dwyer’s Theorem (see Appendix A), this condition is equivalent to G/Γ_{r} being isomorphic to F/Γ_{r} where F is the free group on b_{1}(G) generators. Thus b_{2} should grow as the r-th power of b_{1}. But if some of the relations are shorter commutators, then they can imply a vast number of relations among the r-fold commutators. Therefore, one can only expect a lower order estimate in this case.

Example . Let x_{1}, …, x_{4} be generators of H^{1}(ℤ^{4}; ℤ). Then the central extension

1 ⟶ ℤ ⟶ G ⟶ ℤ^{4} ⟶ 1

classified by the class x_{1}x_{2}+x_{3}x_{4} ∈ H^{2}(ℤ^{4}; ℤ) satisfies b_{1}(G) = 4, b_{2}(G) = 5. This shows that the lower bound for b_{2} in Theorem 1 cannot be of the form b_{1}(b_{1} − 1)/2 with equality in the abelian case.

This paper is organized as follows. In Section 2 we give applications of the above results to 3-dimensional manifolds. Section 3 explains how to derive the nilpotent classification of 3-manifolds from Theorem 2. In Section 4 we give further examples and in Section 5 we prove Theorem 2 modulo two key lemmas. These are proven in Sections 6 and 7. Appendices A and B contain background information from group theory respectively rational homotopy theory.

Acknowledgement: All three authors would like to thank the IHES for its hospitality and support as well as for bringing us together.

2. Applications to 3-Manifolds

We will look at a 3-manifold through nilpotent eyes, observing only the tower of nilpotent quotients of the fundamental group, but never the group itself. This point of view has a long history in the study of link complements and it arises naturally if one studies 3- and 4-dimensional manifolds together. For example, Stallings proved that for a link L in S^{3} the nilpotent quotients π_{1}(S^{3} \ L)/Γ_{r} are invariants of the topological concordance class of the link L. These quotients contain the same information as Milnor’s μ-invariants which are generalized linking numbers. For precise references about this area of research and the most recent applications to 4-manifolds see [7] and [8].

Let M be a closed oriented 3-manifold and {Γ_{r}^{ℚ} | r ≥ 1} the rational lower central series (see Appendix A) of π_{1}(M). Similarly to Stallings’ result, the quotients (π_{1}(M)/Γ_{r}^{ℚ})⊗ℚ are invariants of rational homology H-cobordism between such 3-manifolds.

Definition . A 3-manifold M is of general type if

π_{1}(M) = Γ_{1}^{ℚ} ⫌ Γ_{2}^{ℚ} ⫌ Γ_{3}^{ℚ} ⫌ ⋯

and special if, for some r > 0, Γ_{r}^{ℚ} = Γ_{r+1}^{ℚ}. (Following the terminology used in group theory, the fundamental group of a special 3-manifold is called rationally prenilpotent, compare the Appendix A.)

Our nilpotent classification result reads as follows.

Theorem 3. If a closed oriented 3-manifold M is special, then the maximal torsion-free nilpotent quotient of its fundamental group is isomorphic to exactly one of the groups

{1}, ℤ or H_{n}.

In particular, this quotient is a torsion-free nilpotent 3-manifold group of nilpotency class < 3.

Here the groups H_{n}, n ≥ 0, are the central extensions

1 ⟶ ℤ ⟶ H_{n} ⟶ ℤ^{2} ⟶ 1

classified by the Euler class n ∈ H^{2}(ℤ^{2}; ℤ) ≅ ℤ. This explains the last sentence in our theorem because H_{n} occurs as the fundamental group of a circle bundle over the 2-torus with Euler class n.

Since the groups above have nilpotency class < 3, it is very easy to recognize the class to which a given 3-manifold belongs; one simply has to compute π_{1}(M)/Γ_{3}^{ℚ}. Note, in particular, that a 3-manifold M is automatically of general type if its first rational Betti number satisfies b_{1}M > 3.

If b_{1}M = 0, then M is a rational homology sphere, if b_{1}M = 1, then it is a rational homology S^{1} ×S^{2}. In the case b_{1}M = 2, the cup-product between the two 1-dimensional cohomology classes vanishes and one can compute a Massey triple-product to obtain the integer n ∈ ℤ (note that H^{1}(M; ℤ) ≅ ℤ^{2} and thus one can do the computation integrally). It determines whether M is of general type (n = 0) or whether it belongs to one of the groups H_{n}, n > 0. Finally, if b_{1}M = 3, then M is of general type if and only if the triple cup-product between the three 1-dimensional cohomology classes vanishes, otherwise it is equivalent to the 3-torus, i.e., to H_{0}.

One should compare the above result with the list of nilpotent 3-manifold groups from [22]:

finite ℤ/k Q_{2^{n}} × ℤ/(2k + 1) ; infinite ℤ H_{k}

In the case of finite groups, k ≥ 0 is the order of the cyclic group ℤ/k and the quaternion group Q_{k}. In the infinite case, it is the Euler number that determines H_{k}. In Section 3 we will outline why no other nilpotent groups occur.

The reason why we used the rational version of the lower central series is that our Betti number estimates only give rational information. The integral version of Theorem 3 was in the meantime proven by the third author [21] using completely different methods.

The following result is the rational version of Turaev’s Theorem 2 in [23].

Theorem 4. A finitely generated nilpotent group G satisfies

G ⊗ ℚ ≅ (π_{1}(M)/Γ_{r}) ⊗ ℚ

for a closed oriented 3-manifold M if and only if there exists a class m ∈ H_{3}(G; ℚ) such that the composition

H^{1}(G; ℚ) ⟶^{∩ m} H_{2}(G; ℚ) ↠ H_{2}(G; ℚ)/Φ_{r}^{ℚ}(G)

is an epimorphism.

Here Φ_{r}^{ℚ}(G) is the r-th term in the rational Dwyer filtration of H_{2}(G; ℚ) (defined in the Appendix A).

3. Low-dimensional Surgery

In this section we first show how Theorems 4 and 1 imply Theorem 3. Then we recall the proof of Theorem 4 and finally we give a short discussion of nilpotent 3-manifold groups.

First note that by definition Φ_{r}^{ℚ} = 0 if Γ_{r−1}^{ℚ} = {0}. So by Theorem 4, in order for a group G to be the maximal nilpotent quotient of a rationally prenilpotent 3-manifold group, it must possess the following property ∩ ∩ ∩ (here rational coefficients are to be understood):

∩ ∩ ∩ : \begin{cases}There exits m ∈ H_{3}(G) such that \\ ∩ m : H^{1}(G) ⟶ H_{2}(G) is an epimorphism. \end{cases}

Theorem 3 now follows immediately from the following:

Proposition 1. A finitely generated torsion-free nilpotent group satisfying property ∩ ∩ ∩ is isomorphic to exactly one of the groups:

{1}, ℤ or H_{n}.

Observe that H_{0} ⊗ ℚ = ℤ^{3} ⊗ ℚ = ℚ^{3}. When n > 0 each of the groups H_{n} ⊗ ℚ is isomorphic to the ℚ points of the Heisenberg group H_{ℚ}. On the other hand, the groups ℤ^{k} are the only finitely generated torsion-free nilpotent groups whose Malcev completion is ℚ^{k} and, for n > 0, the groups H_{n} are the only finitely generated torsion-free nilpotent groups with Malcev completion H_{ℚ}. Therefore, Proposition 1 follows from the following result and the fact that the Malcev completion of a finitely generated nilpotent group is a uniquely divisible nilpotent group with b_{1} < ∞.

Proposition 2. A uniquely divisible nilpotent group with b_{1} < ∞ satisfying property ∩ ∩ ∩ is isomorphic to exactly one of the groups:

{0}, ℚ, ℚ^{3} or H_{ℚ}.

Property ∩ ∩ ∩ implies b_{1} ≥ b_{2}. This, combined with the inequality b_{2} > b_{1}^{2}/4 from Theorem 1, implies that Proposition 2 follows from the result below. The proof will be given in Section 4.

Proposition 3. Suppose that G is a uniquely divisible nilpotent group with b_{1}(G) < ∞ satisfying property ∩ ∩ ∩. If b_{1} = b_{2} = 2, then G is isomorphic to H_{ℚ}. If b_{1} = b_{2} = 3, then G is isomorphic to ℚ^{3}.

This finishes the outline of the proof of Theorem 3.

Proof of Theorem 4. Given a closed oriented 3-manifold M, we may take a classifying map M → K(π_{1}(M), 1) of the universal covering and compose with the projection π_{1}(M) ↠ π_{1}(M)/Γ_{r} to get a map u : M → K(π_{1}(M)/Γ_{r}, 1) and a commutative diagram

H^{1}(M)  ⟶_{≅}^{∩ [M]}  H_{2}(M)
u^{∗}↑  ^{u_{∗}}↓
H^{1}(π_{1}(M)/Γ_{r}) ⟶^{∩ u_{∗}[M]} H_{2}(π_{1}(M)/Γ_{r})

Clearly u induces an isomorphism on H_{1}, and therefore u^{∗} is an isomorphism. Thus the “only if” part of our theorem follows directly from Dwyer’s theorem (see the Appendix A). Here we could have worked integrally or rationally.

To prove the “if” part, we restrict to rational coefficients. Let u : M → K(G, 1) be a map from a closed oriented 3-manifold with u_{∗}[M] = m, the given class in H_{3}(G). Such a map exists because rationally oriented bordism maps onto homology by the classical result of Thom. Now observe that we are done by Dwyer’s theorem (and the above commutative diagram) if the map u induces an isomorphism on H_{1}. If this is not the case, we will change the map u (and the 3-manifold M) by surgeries until it is an isomorphism on H_{1}: We describe the surgeries as attaching 4-dimensional handles to the upper boundary of M ×I. Then M is the lower boundary of this 4-manifold and the upper boundary is denoted by M′. If the map u extends to the 4-manifold, then the image of the fundamental class of the new 3-manifold (M′, u′) is still the given class m ∈ H_{3}(G).

First add 1-handles D^{1}×D^{3} to M ×I and extend u to map the new circles to the (finitely many) generators of G. This makes u′ an epimorphism (on H_{1}). Then we want to add 2-handles D^{2} × D^{2} to M′ × I to kill the kernel K of

u′_{∗} : H_{1}(M′) ⟶ H_{1}(G).

We can extend the map u′ to the 4-manifold

W := M′ × I ∪ 2-handles

if we attach the handles to curves in the kernel of u : π_{1}(M) → G.

Now observe that the new upper boundary M′′ of W still maps onto H_{1}(G) because one can obtain M′ from M′′ by attaching 2-handles (2 + 2 = 4). But the problem is that Ker{u′′_{∗} : H_{1}(M′′) → H_{1}(G)} may contain new elements which are meridians to the circles c we are trying to kill. If c has a dual, i.e. if there is a surface in M′ intersecting c in a point, then these meridians are null-homologous. But since we only work rationally we may assume all the classes in the kernel to have a dual and we are done.

The more involved integral case is explained in detail in [23] but we do not need it here.

We finish this section by explaining briefly the table of nilpotent 3-manifold groups given in Section 2. The finite groups G in the table have representations into SO(4) such that the corresponding action of G on the 3-sphere is free. Thus the corresponding 3-manifolds are homogenous spaces S^{3}/G.

To explain why only cyclic and quaternion groups can occur, first recall that a finite group is nilpotent if and only if it is the direct product of its p-Sylow subgroups [13]. Now it is well known [2] that the only p-groups with periodic cohomology are the cyclic groups and, for p = 2, the quaternion groups. The only fact about the group we use is that it acts freely on a homotopy 3-sphere, and thus has 4-periodic cohomology.

To understand why only the groups ℤ and H_{n} occur as infinite nilpotent 3-manifold groups, first notice that a nilpotent group is never a nontrivial free product. Except in the case π_{1}(M) = ℤ, the sphere theorem implies that the universal cover of the corresponding 3-manifold must be contractible. In particular, the nilpotent fundamental groups (≠ ℤ) of a 3-manifolds must have homological dimension 3. It is then easy to see that the groups H_{n} are the only such groups. For more details see [22] or [21].

4. Examples

In this section we prove Proposition 3 and give an example concerning the difference between integral and rational prenilpotence of 3-manifold groups.

Proof of Proposition 3. The key to both cases is the following commutative diagram (later applied with r = 2, 3, 4). Here all homology groups, as well as the groups Γ_{r} and Φ_{r}, have rational coefficients, i.e., we supress the letter ℚ from the notation.

Φ_{r+1}(G) ⟶ H_{2}(G) ⟶^{p_{∗}} H_{2}(G/Γ_{r}) ⟶ Γ_{r}(G)/Γ_{r+1}(G)
∩ m↑≅  ^{∩ p_{∗}(m)}↑
H^{1}(G) ⟵_{≅}^{p^{∗}} H^{1}(G/Γ_{r})

The upper line is short exact by the 5-term exact sequence and the definition of Φ_{r+1}. Since we consider cases with b_{1} = b_{2}, the map ∩ m is actually an isomorphism.

Consider first the case b_{1}(G) = b_{2}(G) = 3 and r = 2 in the above diagram. Since H_{3}(G/Γ_{2}) ≅ H_{3}(ℚ^{3}) ≅ ℚ, there are only two cases to consider:

(i) p_{∗}(m) ≠ 0: Then, by Poincaré duality for ℚ^{3}, the map ∩ p_{∗}(m) is an isomorphism and thus p_{∗} : H_{2}(G) → H_{2}(G/Γ_{2}) is also an isomorphism. By Stallings’ Theorem this implies that G ≅ G/Γ_{2} ≅ ℚ^{3}.

(ii) p_{∗}(m) = 0: Then p_{∗} : H_{2}(G) → H_{2}(G/Γ_{2}) is the zero map and thus

Φ_{3}(G) = ℚ^{3}. This contradicts Theorem 2 with r = 3, since 3 ≯ 4. Now consider the case b_{1}(G) = b_{2}(G) = 2. If p_{∗} : H_{2}(G) → H_{2}(G/Γ_{2}) ≅ ℚ is onto, then by Stallings’ Theorem, G would be abelian and thus have b_{2}(G) = 1, a contradiction. Therefore, H_{2}(G) = Φ_{3}(G) and G/Γ_{3} is the rational Heisenberg group H_{ℚ}. This follows from Dwyer’s Theorem by comparing G to the free group F on 2 generators and noting that H_{ℚ} ≅ F/Γ_{3} ⊗ ℚ.

Now consider the above commutative diagram for r = 3. Since H_{3}(H_{ℚ}) ≅ ℚ there are again only two cases to consider:

(i) p_{∗}(m) ≠ 0: Then by Poincaré duality for H_{ℚ}, the map ∩ p_{∗}(m) is an isomorphism and thus p_{∗} : H_{2}(G) → H_{2}(G/Γ_{2}) is also an isomorphism. By Stallings’ Theorem this implies that G ≅ G/Γ_{3} ≅ H_{ℚ}.

(ii) p_{∗}(m) = 0: Then p_{∗} : H_{2}(G) → H_{2}(G/Γ_{2}) is the zero map and thus

H_{2}(G) = Φ_{4}(G). Unfortunately, this does not contradict Theorem 2 with r = 4 (since 2 > 27/16) and we have to go one step further. Again by Dwyer’s Theorem G/Γ_{4} is isomorphic to K := F/Γ_{4} ⊗ ℚ. One easily computes that the cap-product map

∩ : H_{3}(K) ⊗ H^{1}(K) ⟶ H_{2}(K)

is identically zero and thus the above commutative diagram for r = 4 shows that H_{2}(G) = Φ_{5}(G). Now Theorem 2 does lead to the contradiction 2 > 8192/3125.

We believe that the above proof is unnecessarily complicated because Theorem 2 does not give the best possible estimate. In fact, we conjecture that there is no nilpotent group with b_{1} = b_{2} = 2 and Φ_{4} ≅ ℚ^{2}, and that a nilpotent group with b_{1} = b_{2} = 3 is always rationally ℚ^{3} (without assuming property ∩_{ℚ}). However, we found the following

Example . Besides the Heisenberg group H_{ℚ}, there are other nilpotent groups with b_{1} = b_{2} = 2. For example, take a nontrivial central extension

0 ⟶ ℚ ⟶ G ⟶ H_{ℚ} ⟶ 0.

By Proposition 3, G cannot satisfy property ∩_{ℚ}, which can be checked directly.

We finish this section by discussing the 2-torus bundle over the circle with holonomy given by

(z_{1}, z_{2}) ↦ (z̄_{1}, z̄_{2}), z_{i} ∈ S^{1} ⊂ ℂ.

The fundamental group G of this 3-manifold is the semidirect product of ℤ with ℤ^{2}, where a generator of the cyclic group ℤ acts as minus the identity on the normal subgroup ℤ^{2}. One computes that

Γ_{r}(G) = 2^{r−1} · ℤ^{2}

and thus G is not prenilpotent. However, one sees that Γ_{r}^{ℚ}(G) = ℤ^{2} for all r — i.e., G is rationally prenilpotent with maximal torsion-free nilpotent quotient ℤ.

This example illustrates three phenomena. Firstly, the rational lower central series stabilizes more often than the integral one, even for 3-manifold groups. Secondly, going to the maximal torsion-free nilpotent quotient kills many details of the 3-manifold group which can be still seen in the nilpotent quotients. Finally, unlike in the geometric theory, even a 2-fold covering can alter the class (Theorem 3) to which a 3-manifold belongs.

5. The Proof of Theorem 2

The proof of the estimate in Theorem 2 consists of three steps which are parallel to those in Roquette’s proof of the Golod-Shafarevich Theorem, as presented in [14]. We begin by recalling the structure of the proof:

1. The 𝔽_{p}-homology of a p-group G can be calculated using a free 𝔽_{p}[G]- resolution of the trivial module 𝔽_{p} in place of a free ℤ[G]-resolution of ℤ. This puts us in the realm of linear algebra.

2. There is a minimal resolution of 𝔽_{p} over 𝔽_{p}[G]. This is an exact sequence

⋯ ⟶ C_{n} ⟶ C_{n−1} ⟶ ⋯ ⟶ C_{0} ⟶ 𝔽_{p} ⟶ 0

of free 𝔽_{p}[G]-modules C_{i} such that, for all i ≥ 0, one has

rank_{𝔽_{p}[G]} C_{i} = dim_{𝔽_{p}} H_{i}(G; 𝔽_{p}).

3. Let I be the kernel of the augmentation 𝔽_{p}[G] → 𝔽_{p}. Define a generating function d(t) by

d(t) := ∑_{k≥0} dim_{𝔽_{p}}(I^{k}/I^{k+1}) t^{k}

This function is a polynomial which is positive on the unit interval. The proof is completed by filtering the first two pieces of a minimal resolution by powers of I. This leads to an inequality

(a quadratic polynomial)d(t) ≥ 1

for t ∈ [0, 1] from which one deduces the result by looking at the discriminant of the quadratic.

We now consider the analogues of these for a finitely generated torsion free nilpotent group G. We reduce to the case of a uniquely divisible group with b_{1} < ∞ by replacing G by G ⊗ ℚ.

Ad(1) The ℚ-homology of G can be clearly calculated using a free resolution of ℚ over the group ring ℚ[G] rather than over ℤ[G]. More importantly, we can also resolve ℚ as a module over the I-adic completion of ℚ[G], where I is again the augmentation ideal. This step involves an Artin-Rees Lemma proved in this setting in [3]. Let A denote the completion ℚ[G] .

Ad(2) There is a minimal resolution of ℚ over A in the same sense as above. More precisely, there is minimal resolution which takes into account the Dwyer filtration of H_{2}(G). The construction of such a resolution will be given in Section 6. It is crucial that one uses the completed group ring A rather than ℚ[G].

Ad(3) Define a generating function d(t) by

d(t) := ∑_{k≥0} dim_{ℚ}(I^{k}/I^{k+1}) t^{k}

The proof of the following result is at the end of this section.

Proposition 4. With notation as above,

d(t) = ∏_{r≥1} \frac{1}{(1 − t^{r})^{g_{r}}}

where g_{r} is the dimension of Γ_{r}(G)/Γ_{r+1}(G). In particular, d(t) is a rational function.

Define p_{r} to be the dimension of Φ_{r}^{ℚ}(G)/Φ_{r+1}^{ℚ}(G). Recall that, roughly speaking, p_{r} is the number of relations which are r-fold commutators but not (r + 1)-fold commutators. Let b_{i} be the ith rational Betti numbers of G. Set

p(t) := ∑_{r≥2} p_{r} t^{r} − b_{1} t + 1.

The following result will be proven in Section 7 using the resolution constructed in Section 6.

Proposition 5. For all 0 < t < 1 one has the inequality p(t) d(t) ≥ 1.

Now it is easy to prove Theorem 2: First note that when 0 < t < 1, the generating function d(t) is positive, and thus p(t) is also positive in this interval. Now assume that H_{2}(G; ℚ) = Φ_{r}^{ℚ}(G) for some r ≥ 2 and that G ≠ {0}, ℚ. Then b_{1} > 1, b_{2} > 0 and p(t) ≤ q(t) for each t in 0 < t < 1, where

q(t) := b_{2} t^{r} − b_{1} t + 1.

The polynomial q(t) has a minimum at

t_{0} := \sqrt[r−1]{\frac{b_{1}}{r b_{2}}} .

The desired inequality follows from the inequality q(t_{0}) > 0 after a little algebraic manipulation. But in order to do this we need t_{0} < 1, which is equivalent to the condition b_{1} < r b_{2}.

We know that 0 ≤ q(1) = b_{2} − b_{1} + 1, and thus b_{2} ≥ b_{1} − 1. Therefore, the only case where t_{0} < 1 is not satisfied is (b_{1}, b_{2}) = (2, 1) (and r = 2). We claim that (b_{1}, b_{2}) = (2, 1) implies G ≅ ℚ^{2}.

Consider the projection G ↠ G/Γ_{2} ≅ ℚ^{2}. If the induced map on H_{2}( ; ℚ) is onto, then we are done by Stallings’ Theorem. If not then we know that H_{2}(G; ℚ) = Φ_{3}(G), and are in a case where t_{0} < 1 since r = 3. But then q(t_{0}) > 0 leads to the contradiction b_{2} = 1 > 32/27.

Proof of Proposition 4. Set

Gr(G) = ⨁_{r≥1} Γ_{r}(G)/Γ_{r+1}(G).

This is a graded Lie algebra. Observe that Gr(A) ≅ Gr(ℚ[G]). Then, by a theorem of Quillen [18], we have an isomorphism Gr(ℚ[G]) ≅ U(Gr(G)) of graded Hopf algebras. Here U(Gr(G)) denotes the universal enveloping algebra of Gr(G). We shall write G for Gr(G) and G_{r} for Γ_{r}(G)/Γ_{r+1}(G).

By the Poincaré-Birkhoff-Witt Theorem, there is a graded coalgebra isomorphism of UG with the symmetric coalgebra SG on G. Note that we have the isomorphism

SG ≅ ⨂_{r≥1} SG_{r},

of graded vector spaces, where the tensor product on the right is finite as G is nilpotent. Since G_{r} has degree r, the generating function of the symmetric coalgebra SG_{r} is 1/(1 − t^{r})^{g_{r}}. The result follows.

6. The Minimal Resolution

In this section we use techniques from rational homotopy theory to prove the existence of the minimal resolutions needed in the proof of Theorem 2. We obtain the minimal resolution using Chen’s method of formal power series connections, which provides a minimal associative algebra model of the loop space of a space. Chen’s theory is briefly reviewed in Appendix B.

The precise statement of the main result is:

Theorem 5. If G is a nilpotent group with b_{1}(G) < ∞, then there is a free ℚ[G] resolution

⋯ ⟶^{δ} C_{2} ⟶^{δ} C_{1} ⟶^{δ} C_{0} ⟶ ℚ ⟶ 0

of the trivial module ℚ with the properties:

1. C_{k} is isomorphic to H_{k}(G) ⊗ ℚ[G] ;

2. δ(Φ_{m}^{ℚ}) ⊆ H_{1}(G) ⊗ I^{m−1}, where I denotes the augmentation ideal of ℚ[G] .

Denote the nilpotent Lie algebra associated to G by 𝔤. This is a finite dimensional Lie algebra over ℚ as G is nilpotent and the ℚ Betti numbers of G are finite. Denote the standard (i.e., the Chevelley-Eilenberg) complex of cochains of 𝔤 by C^{•}(𝔤). By Sullivan’s theory of minimal models, this is the minimal model of the classifying space BG of G.

Let (ω, ∂) be a formal power series connection associated to C^{•}(𝔤).^{1} Note that there is a natural isomorphism between H̃_{•}(C^{•}(𝔤)) and the reduced homology groups H̃_{•}(G, ℚ). Set

A_{•}(G) = (T (H̃_{•}(G)[1]) , ∂).

Since A_{•}(G) is complete, we may assume, after taking a suitable change of coordinates of the form

Y ↦ Y + higher order terms, Y ∈ H̃_{•}(G)[1],

that

∂Y ∈ I^{k} ⟺ k = max{m ≥ 1 : ∂(Y + higher order terms) ∈ I^{m}}. (1)

Since C^{•}(𝔤) is a minimal algebra in the sense of Sullivan, we can apply the standard fact [1, (2.30)] (see also, [11, 2.6.2]) to deduce that H^{k}(B(C(𝔤))) vanishes when k > 0. It follows from Theorem B.2 that H_{k}(A_{•}(G)) also vanishes when k > 0.

When k = 0, Chen’s Theorem B.1 yields a complete Hopf algebra isomorphism

H_{0}(A_{•}(G)) ≅ U 𝔤 ≅ ℚ[G] .

These two facts are key ingredients in the construction of the resolution. Another important ingredient is the Adams-Hilton construction:

Proposition 6. There is a continuous differential δ of degree −1 on the free A_{•}(G)-module

W := H_{•}(G) ⊗ A_{•}(G)

such that:

1. the restriction of δ to A_{•} = 1 ⊗ A_{•} is ∂;

2. δ is a graded derivation with respect to the right A_{•}(X) action;

3. (W, δ) is acyclic.

1The definition and notation can be found in Appendix B.

Proof. To construct the differential, it suffices to define δ on each Y ⊗ 1, where Y ∈ H_{•}(G). To show that the tensor product is acyclic, we will construct a graded map s : W → W of degree 1, such that s^{2} = 0 and

δs + sδ = id −ϵ.  (2)

Here ϵ is the tensor product of the augmentations of H_{•}(G) and A_{•}(G). Define s(1) := 0 =: δ(1). For Y ∈ H̃_{•}(G), we shall denote the corresponding element of H̃_{•}(G)[1] by \overline{Y}. For Y, Y_{1}, Y_{2}, …, Y_{k} ∈ H̃_{•}(G), define

s(1 ⊗ \overline{Y}_{1}\overline{Y}_{2} … \overline{Y}_{k}) := Y_{1} ⊗ \overline{Y}_{2} … \overline{Y}_{k} and s(Y ⊗ \overline{Y}_{1}\overline{Y}_{2} … \overline{Y}_{k}) := 0.

We shall simply write Y for Y ⊗ 1 and \overline{Y} for 1 ⊗ \overline{Y}. Since δ is a derivation, it suffices to define it on the A_{•}(G)-module generators 1, Y, \overline{Y} of W. We already know how to define δ on the \overline{Y} ’s. In order that (2) holds when applied to Y, we have to define

δY := \overline{Y} − sδ\overline{Y} .

Then (2) also holds when applied to \overline{Y} and this implies δ^{2}Y = 0. One can now verify (2) by induction on degree using the fact that s is right A_{•}(G)-linear.

The way to think of this result is that A_{•}(G) is a homological model of the loop space Ω_{x}BG of the classifying space of G. One can take H_{•}(G) with the trivial differential to be a homological model for BG. The complex (W, δ) is a homological model of the path-loops fibration whose total space is contractible. This picture should help motivate the next step, the construction of the minimal resolution.

Filter the complex (W, δ) by degree in H_{•}(G). This leads to a spectral sequence

E_{p,q}^{1} = H_{p}(G) ⊗ H_{q}(A_{•}(G)) ⟹ H_{p+q}(W, δ).

(This is the algebraic analogue of the Serre spectral sequence for the pathloops fibration.) But since H_{q}(A_{•}(G)) vanishes when q > 0, the spectral sequence collapses at the E^{1}-term to a complex

⋯ → H_{2}(G) ⊗ H_{0}(A_{•}(G)) → H_{1}(G) ⊗ H_{0}(A_{•}(G)) → H_{0}(A_{•}(G)) → ℚ → 0

which is acyclic as (W, δ) is. The existence of the resolution now follows from the fact that

H_{0}(A_{•}(G)) ≅ ℚ[G] .

The ℚ[G] linearity of the differential follows directly from the second assertion of Proposition 6.

It remains to verify the condition satisfied by the differential. Observe that H_{0}(A_{•}(G)) is the quotient of T (H_{1}(G)) by the closed ideal generated by the image of

∂ : H_{2}(G) → T (H_{1}(G)) .

Define a filtration

H_{2}(G; ℚ) = L_{2} ⊇ L_{3} ⊇ ⋯

by L_{k} = ∂^{−1}I^{k}. It is not difficult to verify that in the complex H_{•}(G) ⊗ H_{0}(A_{•}(G)) we have

δ(L_{k} ⊗ ℚ) ⊆ H_{1}(G) ⊗ I^{k−1}.

So, to complete the proof, we have to show that Φ_{k}^{ℚ} = L_{k}.

Consider the spectral sequence dual to the one in the proof of Theorem B.2. It has the property that E_{−1,2}^{r} is a subspace of H_{2}(G; ℚ) and E_{−p,p}^{r} is a quotient of H_{1}(G; ℚ)^{⊗p}. We thus have a filtration

H_{2}(G; ℚ) = E_{−1,2}^{1} ⊇ E_{−1,2}^{2} ⊇ E_{−1,2}^{3} ⊇ ⋯

of H_{2}(G; ℚ). From the standard description of the terms of the spectral sequence associated to a filtered complex, it is clear from condition (1) that when r ≥ 2

L_{r} = E_{−1,2}^{r−1}.

We know from [4, (2.6.1)] that

E_{−p,p}^{p} = E_{−p,p}^{∞} ≅ I^{p}/I^{p+1},

where I denotes the augmentation ideal of ℚ[π_{1}(G)]. Consequently, a group homomorphism G → H induces an isomorphism

ℚ[G]/I^{k} → ℚ[H]/I^{k}

if and only if the induced map H_{2}(G; ℚ)/L_{k} → H_{2}(H; ℚ)/L_{k} is an isomorphism. The proof is completed using the fact (cf. [11, (2.5.3)]) that a group homomorphism G → H induces an isomorphism (G ⊗ ℚ)/Γ_{k} → (H ⊗ ℚ)/Γ_{k} if and only if the induced map ℚ[G]/I^{k} → ℚ[H]/I^{k} is an isomorphism, [17].

7. The Inequality

In this section, we prove Proposition 5. We suppose that G is a nilpotent group with b_{1}(G) < ∞. Denote ℚ[G] by A and its augmentation ideal by I. Recall that

d(t) = ∑_{l≥0} d_{l} t^{l},

where d_{l} = dim I^{l}/I^{l+1}, and that this is a rational function whose poles lie on the unit circle.

Consider the part

H_{2}(G) ⊗ A ⟶^{δ} H_{1}(G) ⊗ A ⟶ A ⟶ ℚ ⟶ 0

of the resolution constructed in Section 6. Define a filtration

E^{0} ⊇ E^{1} ⊇ E^{2} ⊇ ⋯

of H_{2}(G) ⊗ A by

E^{l} := δ^{−1} (H_{1}(G) ⊗ I^{l}) .

Then we have an exact sequence

0 ⟶ E^{l}/E^{l+1} ⟶ H_{1}(G) ⊗ I^{l}/I^{l+1} ⟶ I^{l+1}/I^{l+2} ⟶ 0  (3)

14 MICHAEL FREEDMAN, RICHARD HAIN, AND PETER TEICHNER

for all l ≥ 0. Set e_{l} := dim E^{l}/E^{l+1} and

e(t) := ∑_{l≥0} e_{l} t^{l} ∈ ℝ[[t]].

Proposition 7. The series for e(t) converges to the rational function

\frac{b_{1}(G) td(t) − d(t) + 1}{t}

all of whose poles lie on the unit circle. Proof. Because the sequence (3) is exact, we have e_{l} = b_{1}(G) d_{l} − d_{l+1}. This implies that

te(t) = b_{1}(G) td(t) − d(t) + 1.

from which the result follows.

Our final task is to bound the e_{l}. Our resolution has the property that

δ(Φ_{m} ⊗ I^{l−m+2}) ⊆ H_{1}(G) ⊗ I^{l+1}.

This implies that

∑_{m≥2} Φ_{m} ⊗ I^{l−m+2} ⊆ E^{l+1},

so that the linear map

H_{2}(G) ⊗ A/(∑_{m≥2} Φ_{m} ⊗ I^{l−m+2}) → H_{2}(G) ⊗ A/E^{l+1}

is surjective. This implies that

dim (H_{2}(G) ⊗ A/(∑_{m≥2} Φ_{m} ⊗ I^{l−m+2})) ≥ e_{0} + e_{1} + ⋯ + e_{l}. (4)

To compute the dimension of the left hand side, we apply the following elementary fact from linear algebra. Proposition 8. Suppose that

B = B^{0} ⊇ B^{1} ⊇ B^{2} ⊇ ⋯

and

C = C^{0} ⊇ C^{1} ⊇ C^{2} ⊇ ⋯

are two filtered vector spaces. Define a filtration F of B ⊗ C by

F^{k} := ∑_{i+j=k} B^{i} ⊗ C^{j}.

Then there is a canonical isomorphism

F^{k}/F^{k+1} ≅ ⨁_{i+j=k} A^{i}/A^{i+1} ⊗ B^{j}/B^{j+1}.

Applying this with B = H_{2}(G) with the filtration Φ_{•}, and C = A with the filtration I^{•}, we deduce that

dim (H_{2}(G) ⊗ A/(∑_{m≥2} Φ_{m} ⊗ I^{l−m+2})) = ∑_{m+k≤l+1}(dim Φ_{m}/Φ_{m+1})(dim I^{k}/I^{k+1})

= ∑_{m+k≤l+1} p_{m}d_{k}

where p_{m} = dim Φ_{m}. Combined with (4), this implies that

∑_{m+k≤l+1} p_{m}d_{k} ≥ e_{0} + ⋯ + e_{l}.

Using geometric series, this can be assembled into the following inequality which holds when 0 < t < 1:

\frac{(p_{2} t + p_{3} t^{2} + ⋯ + p_{l+1} t^{l}) d(t)}{1 − t} ≥ \frac{e(t)}{1 − t}.

Plugging in the formula for e(t) given in Proposition 7, we deduce the desired inequality

d(t) (∑_{m≥2} p_{m} t^{m} − b_{1}(G) t + 1) ≥ 1

when 0 < t < 1.

Appendix A.

Here we collect the necessary group theoretic definitions. Let G be a group.

• Γ_{r}(G) denotes the r-th term of the lower central series of G which is the subgroup of G generated by all r-fold commutators. We abbreviate G/Γ_{r}(G) by G/Γ_{r}. We say G is nilpotent if Γ_{r}(G) = {1} for some r and it is then said to have nilpotency class < r. For example, abelian groups have Γ_{2} = {1} and are of nilpotency class 1.

• G is prenilpotent if it’s lower central series stabilizes at some term Γ_{r}, i.e., Γ_{r}(G) = Γ_{r+1}(G). This happens if and only if G has a maximal nilpotent quotient (which is then isomorphic to G/Γ_{r}).

The main homological tool in dealing with nilpotent groups is the following result of Stallings:

Theorem A.1. [20] If f : G → H is a homomorphism of groups inducing an isomorphism on H_{1} and an epimorphism on H_{2}, then the induced maps G/Γ_{r} → H/Γ_{r} are isomorphisms for all r ≥ 1.

• The Dwyer filtration of H_{2}(G; ℤ)

H_{2}(G; ℤ) =: Φ_{2}(G) ⊇ Φ_{3}(G) ⊇ Φ_{4}(G) ⊇ ⋯

is defined by (see [8] for a geometric definition of Φ_{r} using gropes)

Φ_{r}(G) := Ker(H_{2}(G; ℤ) ⟶ H_{2}(G/Γ_{r−1}; ℤ)).

This filtration is used in Dwyer’s extension of Stallings’ Theorem:

Theorem A.2. [5] If f : G ⟶ H induces an isomorphism on H_{1}( ; ℤ), then for r ≥ 2 the following three conditions are equivalent:

1. f induces an epimorphism H_{2}(G; ℤ)/Φ_{r}(G) → H_{2}(H; ℤ)/Φ_{r}(H);

2. f induces an isomorphism G/Γ_{r} → H/Γ_{r};

3. f induces an isomorphism H_{2}(G; ℤ)/Φ_{r}(G) → H_{2}(H; ℤ)/Φ_{r}(H), and an injection H_{2}(G; ℤ)/Φ_{r+1}(G) → H_{2}(H; ℤ)/Φ_{r+1}(H).

There are rational versions of the above definitions.

• Γ_{r}^{ℚ} denotes the r-th term of the rational lower central series of G, which is defined by

Γ_{r}^{ℚ}(G) := Rad(Γ_{r}(G)) := {g ∈ G | g^{n} ∈ Γ_{r} for some n ∈ ℤ}.

It has the (defining) property that G/Γ_{r}^{ℚ} = (G/Γ_{r})/ Torsion. (Note that the torsion elements in a nilpotent group form a subgroup.) G is rationally nilpotent if Γ_{r}^{ℚ}(G) = {1} for some r.

• G is rationally prenilpotent if Γ_{r}^{ℚ}(G) = Γ_{r+1}^{ℚ}(G) for some r. This happens if and only if G has a maximal torsion-free nilpotent quotient (which is then isomorphic to G/Γ_{r}^{ℚ}).

• The rational Dwyer filtration of H_{2}(G; ℚ)

H_{2}(G; ℚ) =: Φ_{2}^{ℚ}(G) ⊇ Φ_{3}^{ℚ}(G) ⊇ Φ_{4}^{ℚ}(G) ⊇ …

is defined by

Φ_{r}^{ℚ}(G) := Φ_{r}(G) ⊗ ℚ = Ker(H_{2}(G; ℚ) ⟶ H_{2}(G/Γ_{r−1}^{ℚ}; ℚ)).

• The Malcev completion [16] G⊗ℚ of a nilpotent group G may be defined inductively through the central extensions determining G as follows: If G is abelian then one takes the usual tensor product of abelian groups to define G ⊗ ℚ. It comes with a homomorphism ϵ : G → G ⊗ ℚ which induces an isomorphism on rational cohomology. Using the fact that the cohomology group H^{2} classifies central extensions, one can then define the Malcev completion for a group which is a central extension of an abelian group. It comes again with a map ϵ as above. The Serre spectral sequence then shows that ϵ induces an isomorphism on rational cohomology. Therefore, one can repeat the last step to define (G ⊗ ℚ, ϵ) for an arbitrary nilpotent group G.

The map ϵ : G → G ⊗ ℚ is universal for maps of G into uniquely divisible nilpotent groups and it is characterized by the following properties:

1. G ⊗ ℚ is a uniquely divisible nilpotent group.

2. The kernel of ϵ is the torsion subgroup of G.

3. For every element x ∈ G ⊗ ℚ there is a number n ∈ ℕ such that

x^{n} is in the image of ϵ.

A version of Stallings’ Theorem holds in the rational setting:

Theorem A.3. [20] If f : G → H is a homomorphism of groups inducing an isomorphism on H_{1}( ; ℚ) and an epimorphism on H_{2}( ; ℚ), then the induced maps (G/Γ_{r}) ⊗ ℚ → (H/Γ_{r}) ⊗ ℚ are isomorphisms for all r ≥ 1.

A good example to keep in mind when trying to understand this theorem is the inclusion of the Heisenberg group H_{1} into the Heisenberg group H_{n}. This induces an isomorphism on rational homology. Both groups are torsionfree nilpotent of class 2.

There is also a rational analogue of Dwyer’s theorem.

Theorem A.4. [5] If f : G ⟶ H induces an isomorphism on H_{1}( ; ℚ), then for r ≥ 2 the following three conditions are equivalent:

1. f induces an epimorphism H_{2}(G; ℚ)/Φ_{r}^{ℚ}(G) → H_{2}(H; ℚ)/Φ_{r}^{ℚ}(H);

2. f induces an isomorphism (G/Γ_{r}) ⊗ ℚ → (H/Γ_{r}) ⊗ ℚ;

3. f induces an isomorphism H_{2}(G; ℚ)/Φ_{r}^{ℚ}(G) → H_{2}(H; ℚ)/Φ_{r}^{ℚ}(H), and an injection H_{2}(G; ℚ)/Φ_{r+1}^{ℚ}(G) → H_{2}(H; ℚ)/Φ_{r+1}^{ℚ}(H).

Appendix B.

In this appendix, we give a brief review of Chen’s method of formal power series connections. Two relevant references are [4] and [10]. There is also an informal discussion of the ideas behind Chen’s work in [12].

Fix a field F of characteristic zero. Suppose that A^{•} is an augmented commutative d.g. algebra over F. Suppose in addition that A^{•} is positively graded (i.e., A^{k} = 0 when k < 0) and that H^{•}(A^{•}) is connected (i.e., H^{0}(A^{•}) = F). For simplicity, we suppose that each H^{k}(A^{•}) is finite dimensional. Typical examples of such A^{•} in this theory are the F (=ℚ, ℝ or ℂ) de Rham complex of a connected space with finite rational betti numbers, or the Sullivan minimal model of such an algebra.

Set

H_{k}(A^{•}) = Hom_{F} (H^{k}(A^{•}), F)

and

H̃_{•}(A^{•}) = ∑_{k>0} H_{k}(A^{•}).

Denote by H̃_{•}(A^{•})[1] the graded vector space obtained by taking H̃_{•}(A^{•}) and lowering the degree of each of its elements by one.^{2}

Denote the free associative algebra generated by the graded vector space V by T (V). Assume that V_{n} is non-zero only when n ≥ 0. Denote the ideal generated by V_{0} by I_{0}. Let T (V) denote the I_{0}-adic completion

lim_{←} T (V)/I_{0}^{m}.

By defining each element of V to be primitive, we give T (V) the structure of a complete graded Hopf algebra. Recall that the set of primitive elements P T (V) of T (V) is the free Lie algebra 𝕃(V), and the set of primitive elements of T (V) is the closure 𝕃(V) of 𝕃(V) in T (V) .

2Here we are using the algebraic geometers’ notation that if V is a graded vectorspace, then V [n] is the graded vector space with V [n]_{m} = V_{m+n}.

Set A_{•} = T (H̃_{•}(A^{•})) . Choose a basis {X_{i}} of H̃_{•}(A^{•}). Then A_{•} is the subalgebra of the non-commutative power series algebra generated by the X_{i} consisting of those power series with the property that each of its terms has a fixed degree. For each multi-index I = (i_{1}, …, i_{s}), we set

X_{I} = X_{i_{1}}X_{i_{2}} … X_{i_{s}}.

If X_{i} ∈ H̃_{p_{i}}(A^{•}), we set |X_{I}| = −s + p_{1} + ⋯ + p_{s}.

A formal power series connection on A^{•} is a pair (ω, ∂). The first part

ω = ∑_{I} w_{I} X_{I}

is an element of the completed tensor product A^{•} ˆ⊗P A_{•}, where w_{I} is an element of A^{•} of degree 1 + |X_{I}|. The second is a graded derivation

∂ : A_{•} → A_{•}

of degree −1 with square 0 and which satisfies the minimality condition ∂(IA_{•}) ⊆ I^{2}A_{•}. Here I^{k}A_{•} denotes the kth power of the augmentation ideal of A_{•}. These are required to satisfy two conditions. The first is the “integrability condition”

∂ω + dω + \frac{1}{2}[Jω, ω] = 0.

Here J : A^{•} → A^{•} is the linear map a ↦ (−1)^{deg a}a. The value of the operators d, ∂ and J on ω are obtained by applying the operators to the appropriate coefficients of ω:

dω = ∑ dw_{I} X_{I}, ∂ω = ∑ w_{I} ∂X_{I}, Jω = ∑ Jw_{I} X_{I}.

The second is that if

∑_{i} w_{i} X_{i}  (5)

is the reduction of ω mod I^{2}A_{•}, then each w_{i} is closed and the [w_{i}] form a basis of H^{>0}(A^{•}) dual to the basis {X_{i}} of H̃_{•}(A_{•}).

Such formal connections always exist in the situation we are describing. To justify the definition, we recall one of Chen’s main theorems. It is the analogue of Sullivan’s main theorem about minimal models.

Theorem B.1. Suppose that X is a smooth manifold with finite betti numbers, x a fixed point of X, and suppose that A^{•} is the F -de Rham complex of X, with the augmentation induced by x. If X is simply connected, the connection gives a natural Lie algebra isomorphism

π_{•}(X, x)[1] ⊗ F ≅ H_{•}(P A_{•}, ∂).

If X is not simply connected, then the connection gives a Lie algebra isomorphism

𝔤(X, x) ≅ H_{0}(P A_{•}, ∂)

and complete Hopf algebra isomorphisms

ℚ[π_{1}(X, x)] ≅ U 𝔤(X, x) ≅ H_{0}(A_{•}, ∂).

Here 𝔤(X, x) denotes the F form of the Malcev Lie algebra associated to π_{1}(X, x) and U 𝔤(X, x) is the completion of its enveloping algebra with respect to the powers of its augmentation ideal.

We can apply the bar construction to the augmented algebra A^{•} to obtain a commutative d.g. Hopf algebra B(A^{•}). (We use the definition in [4].) One can define the formal transport map of such a formal connection. It is defined to be the element

T = 1 + ∑ [w_{I}] X_{I} + ∑ [w_{I}|w_{J}] X_{I}X_{J} + ∑ [w_{I}|w_{J}|w_{K}] X_{I}X_{J}X_{K} + ⋯

of B(A^{•})ˆ⊗A_{•}. It induces a linear map

Θ : Hom_{F}^{cts}(A_{•}, F) → B(A^{•}).

Here Hom_{F}^{cts}(A_{•}, F) denotes the continuous dual

lim_{→} Hom_{F} (A_{•}/I^{k}A_{•}, F)

of A_{•}. It is a commutative Hopf algebra. The map Θ takes the continuous functional ϕ to the result of applying it to the coefficients of T:

Θ(ϕ) =

1+∑ [w_{I}]ϕ(X_{I})+∑ [w_{I}|w_{J}]ϕ(X_{I}X_{J})+∑ [w_{I}|w_{J}|w_{K}]ϕ(X_{I}X_{J}X_{K})+⋯

(Note that this is a finite sum as ϕ is continuous.) The properties of the formal connection imply that Θ is a d.g. Hopf algebra homomorphism (cf. [10, (6.17)]). The basic result we need is:

Theorem B.2. The map Θ induces an isomorphism on homology.

We conclude by giving a brief sketch of the proof. One can filter A_{•} by the powers of its augmentation ideal. This gives a dual filtration of Hom_{F}^{cts}(A_{•}, F). The corresponding spectral sequence has E_{1} term

E_{1}^{−s,t} = [H̃^{•}(A^{•})^{⊗s}]^{t}.

On the other hand, one can filter B(A^{•}) by the “bar filtration” to obtain a spectral sequence, also with this E_{1} term. It is easy to check that Φ preserves the filtrations and therefore induces a map of spectral sequences. The condition on the w_{i} in (5) implies that the map on E_{1} is an isomorphism. The result follows.

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University of California in San Diego, La Jolla, CA, 92093-0112

E-mail address: freedman@euclid.ucsd.edu

Department of Mathematics, Duke University, Durham, NC 27708-0320

E-mail address: hain@duke.math.edu

University of California in San Diego, La Jolla, CA, 92093-0112

E-mail address: teichner@euclid.ucsd.edu