A LOCAL APPROACH TO THE FINITENESS OBSTRUCTION

JOHN EWING, PETER LÖFFLER, AND ERIK KJÆR PEDERSEN

0. Introduction

Given a finitely dominated CW-complex X, Wall [9] defined a finiteness obstruction

σ(X) = ∑_{k}(−1)^{k}[C_{k}(X̃)] ∈ K̃_{0}(Zπ_{1}(X))

and showed that X has the homotopy type of a finite complex if and only if σ(X) = 0. For a fixed group π, it is well-known that any element of K̃_{0}(Zπ) can be realized as the finiteness obstruction for some finitely dominated complex with fundamental group π. If we restrict the class of spaces, however, then in general the set of elements in K̃_{0}(Zπ) that can be realized as finiteness obstructions is also restricted.

One natural class of spaces to consider is the class of complexes on which π = π_{1}(X) acts trivially on rational homology. A special case of this is the class of nilpotent spaces with finite fundamental group – spaces for which π_{1}(X) is nilpotent and acts nilpotently on homology. The question of which elements can be realized by nilpotent spaces has been extensively studied by Mislin [3, 4, 5, 6].

The purpose of this paper is to study the finiteness obstruction in such a way that one can begin to answer some of these questions in a systematic way. The primary tool is a decomposition of the finiteness obstruction into its p-parts for all primes p (Theorem 2.4). To do this, we use the techniques of localization and define p-local Reidemeister torsions whose images under a certain boundary map in K-theory provide the p-parts of the finiteness obstruction. In the case of nilpotent spaces, we are then able to describe (in principle, completely) the set of finiteness obstructions that can be realized.

1. Some algebraic preliminaries

We begin by considering the localization square

\xymatrix{Z \ar[d] \ar[r] & Z_{(p)} \ar[d] \\ Z_{(1/p)} \ar[r] & Q}

1

If we tensor with a ring R, which as an additive group is free abelian, we obtain

\xymatrix{R \ar[d] \ar[r] & R_{(p)} \ar[d] \\ R_{(1/p)} \ar[r] & R_{(0)}}

In the usual way (see, for example Bass [1] or Wojtkowiak [10]) this yields an exact sequence in K-theory:

→ K_{1}(R) → K_{1}(R_{(p)}) ⊕ K_{1}(R_{1/p)}) → K_{1}(R_{(0)}

⟶^{∂^{p}} K_{0}(R_{(p)}) ⊕ K_{0}(R_{1/p)}) → K_{0}(R_{(0)}

Throughout this paper we will be concerned with the case where R = Zπ for a finite group π and hence R_{(0)} = Qπ.

If π is a finite nilpotent group then Qπ splits as follows. For any prime p, let π_{p} denote the p-Sylow subgroup of π. (We allow the case where af, is the trivial group!) Since π is nilpotent, it is isomorphic to the product of its nontrivial p-Sylow subgroups, and hence π_{p} has a complementary subgroup π′_{p}. Consider the central idempotent:

e_{p} = \frac{1}{|π_{p}|} ∑_{g∈π′_{p}} g ∈ Qπ.

Then Qπ splits as:

Qπ ≅ Qπ · e_{p} × Qπ · (1 − e_{p}).

Note that the projection π → π_{p} induces a map Qπ → Qπ_{p}, which in turn produces an isomorphism Qπ ·e_{p} ≅ Qπ_{p}. We shall often simply identify the factor Qπ ·e_{p} with Qπ_{p} using this isomorphism. Of course, in case p does not divide |π| we see that Qπ · e_{p} ≅ Q

Definition 1.1. For a finite nilpotent group and a prime p, let

N_{p}(π) = ∂^{p}(K_{1}(Qπ · e_{p})) ⊆ K_{0}(Zπ).

We shall see in the sequel that the groups N_{p}(π) play a central role in analyzing the finiteness obstructions that can be realized by nilpotent spaces.

2. Reidemeister torsions

In this section we give the definition of p-local Reidemeister torsion and prove the fundamental theorem relating such torsion to the finiteness obstruction. We then show that the p-local Reidemeister torsions of a nilpotent space are restricted.

We begin with some algebra. Let π be a finite group and consider a finitely dominated Zπ-chain complex C_{∗} for which π acts trivially on rational homology. (”Finitely dominated” means that C_{∗} is homotopy equivalent to a finitely generated projective Zπ-chain complex.)

Further assume that we are given a preferred basis \{\underline{e}\} for H_{∗}(C_{∗}; Q), up to simple isomorphism measured in K_{1}(Q) = Q^{∗}.

We define the p-local Reidemeister torsion RT_{p}(C_{∗}\{\underline{e}\}) ∈ K_{1}(Z_{(p)}π) as follows. Consider a finitely generated projective Zπ-chain complex P_{∗} that is homotopy equivalent to C_{∗}. Since the map K̃_{0}(Zπ) → K̃_{0}(Z_{(p)}π) is trivial (see [8]), P_{∗_{(p)}} = P_{∗} ⊗ Z_{(p)} is a finitely generated free Z_{(p)}π- module. Choose a basis for P_{∗_{(p)}} and consider the induced basis for P_{∗_{(p)}} = P_{∗} ⊗ Q. We can use the idempotent

e = \frac{1}{|π|} ∑_{g∈π} g

to split the chain complex into two parts,

P_{∗_{(p)}} = P_{∗_{(0)}} · e × P_{∗_{(0)}} · (1 − e).

Now P_{∗_{(0)}} · (1 − e) is contractible (since π acts trivially on rational homology) and based; hence it has a torsion defined in K_{1}(Qπ · (1 − e)). The complex P_{∗_{(0)}} · e homology which is identified with H_{∗}(C_{∗}; Q) and hence is based by assumption. But a based chain complex with based homology has a well-defined torsion (as in [2]) defined in K_{1}(Qπ · e). We collect the two torsions to define

RT_{p}(C_{∗}, \{\underline{e}\}) ∈ K_{1}(Qπ) ≅ K_{1}(Qπ · e) × K_{1}(Qπ · (1 − e)).

We must now consider the choices involved above in order to compute the indeterminacy. There were two choices: the choice of P_{∗} and the choice of basis in P_{∗_{(0)}}. Different choices, however, vary RT_{p} by an element in the image of K_{1}(Z_{(p)}π) → K_{1}(Qπ). Hence RT_{p}(C_{∗}, \{\underline{e}\}) is well-defined as an element in K_{1}(Qπ)/K_{1}(Z_{(p)}π).

Remarks 2.1. The choice of basis for H_{∗}(C_{∗}; Q) only affects the part of RT_{p}(C_{∗}; \{\underline{e}\}) coming from K_{1}(Qπ · e). We shall see that this means that a different choice of basis changes ∂^{p}(RT_{p}(C_{∗}; \{\underline{e}\})) by an element in the image of the Swan homomorphism. (See Section 5 for the definition.) We will often suppress the specific choice of basis in what follows and simply write RT_{p}(C_{∗}). It is important to note, however, that the choice of basis is necessary in order to define the Reidemeister torsion.

We can now define the p-local Reidemeister torsion of a space.

Definition 2.2. Let X be a finitely dominated space with universal cover X̃. Assume that π_{1}(X) acts trivially on the rational, singular homology of X̃ and fix a basis \{\underline{e}\} for H_{∗}(X̃; Q). Let C_{∗} denote the singular chain complex of X̃. Then

RT_{p}(X) = RT_{p}(C_{∗}, \{\underline{e}\}) ∈ K_{1}(Qπ_{1}(X))/K_{1}(Z_{(p)}π_{1}(X)).

(Note that we have suppressed the dependence on the basis here.)

Lemma 2.3. Given a finitely dominated space X with finite fundamental group π = π_{1}(X) acting trivially on H_{∗}(X̃; Q), and a basis \{\overline{e}\}, for H_{∗}(X̃; Q), then RT_{p}(X) = o for all but a finite number of primes.

Proof. Suppose p||π| Then the idempotent e = (1/|π|)∑_{g∈π} g is defined over Z_{(p)}π and there

is a splitting

Z_{(p)}π = Z_{(p)}π · e × Z_{(p)}π · (1 − e).

Now let P_{∗} be a finitely generated projective chain complex homotopy equivalent to the singular chains of X̃. Then the above splitting induces a splitting of P_{∗_{(p)}}. If H_{∗}(X̃; Z) has no p-torsion, then P_{∗(p)} · (1 − e) is contractible since P_{∗(0)} · (1 − e) is contractible.

The part of the Reidemeister torsion lying in K_{1}(Qπ·(1−e)) thus lifts to K_{1}(Z_{(p)}π·(1−e)) and is consequently zero in the quotient. The remaining part of the Reidemeister torsion lies in K_{1}(Qπ · e) ≅ K_{1}(Q) and is defined at once for all primes. Excluding the primes in this rational number, we see that RT_{p}(X) = 0 for all but a finite number of primes.

We can now state the fundamental theorem of this paper, which determines the finiteness obstruction in terms of the p-local Reidemeister torsions.

Theorem 2.4. Let X be a finitely dominated space with finite fundamen- tal group π = π_{1}(X) and universal cover X̃. Assume that π acts trivially on H_{∗}(X; Q) and choose a basis for H_{∗}(X̃; Q) . Then the finiteness obstruction σ(X) is given by

σ(X) = ∑_{p prime} ∂^{p}(RT_{p}(X)).

(Note that by Lemma 2.3 this is a finite sum).

Notation. : Given a space X as in Theorem 2.4, we write σ_{p}(X) = ∂^{p}(RT_{p}(X)) and call σ_{p}(X) the p-part of the finiteness obstruction. (Of course, σ_{p}(X) is only defined modulo a choice of basis for H_{∗}(X̃; Q) We can then write

σ(X) = ∑_{p prime} σ_{p}(X)

Before proving Theorem 2.4 we require an algebraic lemma.

Lemma 2.5. Let π be a finite group and let S = {p_{1}, p_{2}, …, p_{k}} be a finite set of primes. Consider the Zπ-module A which is the pullback of the following diagram:

[figure: (Z_{(p_{1})}π)^{n}; (Z_{(p_{2})}π)^{n}; ···; (Z_{(p_{3})}π)^{n}; (Z_{(p_{4})}π)^{n}; (Q_{π})^{n}; α_{1}; ≅; (Q_{π})^{n}; ···; α_{2}; ≅; (Q_{π})^{n}; α_{k}; ≅; (Q_{π})^{n}; α_{k+1}; ≅; (Q_{π})^{n}; =; (Q_{π})^{n}; = ··· =; (Q_{π})^{n}; =; (Q_{π})^{n}]

Then A is a projective Zπ-module, and as an element in K_{0}(Zπ),

A = ∑_{i=1}^{k} ∂^{p_{i}}[α_{i}],

where [α_{i}] is the class of the isomorphism α_{i} in K_{1}(Qπ).

Proof. Denote the pullback of the diagram

[figure: (Z_{(p_{1})}π)^{n}; (Z_{(p_{2})}π)^{n}; ···; (Z_{(p_{3})}π)^{n}; (Z_{(1/S)}π)^{n}; (Q_{π})^{n}; α_{1}; ≅; (Q_{π})^{n}; ···; α_{2}; ≅; (Q_{π})^{n}; α_{k}; ≅; (Q_{π})^{n}; α_{k+1}; ≅; (Q_{π})^{n}; =; (Q_{π})^{n}; = ··· =; (Q_{π})^{n}; =; (Q_{π})^{n}]

by A(α_{1}, α_{2}, …, α_{k}, α_{k+1}). We want to compute A(α_{1}, α_{2}, …, α_{k}, 1).

It is clear that A(1, 1, …, 1, α_{i}, 1, …, 1) is projective and equal to ∂^{p_{i}}[α_{i}] ∈ K_{0}(Zπ), by definition of the boundary map ∂^{p_{i}}. It is also immediate from the definition of ∂^{p_{i}} that ∂^{p_{i}}([α_{i}] + [α_{i}^{−1}]) = 0. Thus, adding

A(α_{1}^{−1}, 1, 1, …, 1)⊕A(1, α_{2}^{−1}, 1, 1, …, 1) ⊕ ⋯ ⊕
A(1, 1, …, α_{k}^{−1}, 1) ⊕ A(α_{1}, α_{2}, …, α_{k}, 1),

we are reduced to the case where ∂^{p_{i}}[α_{i}] = 0; we need to show in this case that A(α_{1}, α_{2}, …, α_{k}, 1) is stably free.

Given an isomorphism β : (Qπ)^{n} → (Qπ)^{n}, one sees that

A(βα_{1}, …, βα_{k}, α_{k+1}) ≅ A(α_{1}, …, α_{k}, β^{−1}α_{k+1})

Also if γ : Z_{(p_{i})}π → Z_{(p_{i})}π is an isomorphism inducing an isomorphism of (Qπ)^{n} then

A(α_{1}, …, α_{i}γ, …, α_{k+1}) ≅ A(α_{1}, …, α_{i}, …, α_{k+1})

Now consider one of the isomorphisms, say α_{1}. The fact that ∂^{p_{1}}[α_{1}] = 0 means that after possible stabilization (which may be performed correspondingly on the pullback diagram), α_{1} can be written as β_{1} · γ_{1} where γ_{1} comes from an isomorphism of ((Z_{(p_{1})}π)^{n} and β_{1} is an isomorphism from (Z_{(1/p_{1})}π)^{n}. We thus have

A(α_{1}, α_{2}, …, α_{k}, 1) ≅ A(β_{1} · γ_{1}, α_{2}, …, α_{k}, 1)
≅ A(β_{1}, α_{2}, …, α_{k}, 1)

by the above remark. But also

A(β_{1}, α_{2}, …, α_{k}, 1) ≅ A(1, α_{2} · β_{1}^{−1}, …, α_{k} · β_{1}^{−1}, β_{1}^{−1})
≅ A(1, α_{2}, …, α_{k}, 1)

since β_{1}^{−1} lifts to an isomorphism of (Z_{(p)}π)^{n} for p ≠ p_{1}. Repeating this argument for each p_{i}, we find that A(α_{1}, …, α_{k}, 1) is stably isomorphic to A(1, 1, …, 1), which is clearly free.

Proof of Theorem 2.4. Let P_{∗} be a finitely generated, projective chain complex homotopy equivalent to the singular chains of X̃. Consider the pullback diagram

[figure: P_{∗}; P_{∗_{(p_{1})}}; P_{∗_{(p_{2})}}; ···; P_{∗_{(p_{k})}}; P_{∗_{(p_{(1/S)})}}; P_{∗_{(0)}}]

where S = {p_{1}, …, p_{k}} is the set of primes p for which RT_{p}(X) ≠ 0. Note that all chain complexes P_{∗_{(p_{i})}} are free because K̃_{0}(Zπ) → K̃_{0}(Z_{(p)}π) is the zero map. The complex P_{∗_{(1/S)}} is free because the Euler characteristic of X is zero (since we may assume π is nontrivial).

The fact that RT_{p}(X) = 0 for p ∉ S means that we can choose a basis for P_{∗_{1/S)}} so that for the induced basis of P_{∗_{(0)}} the torsion with respect to the given basis for rational homology is zero.

By choosing a basis for each chain complex P_{∗_{(p_{i})}}, and considering the induced basis for P_{∗_{(0)}}, we may reinterpret the pullback diagram above follows:

[figure: P_{∗}; P_{∗_{(p_{1})}}; P_{∗_{(p_{2})}}; ···; P_{∗_{(p_{k})}}; P_{∗_{(1/S)}}; P_{∗_{(0)}}; α_{1}; P_{∗_{(0)}}; α_{2}; ···; P_{∗_{(0)}}; α_{k}; P_{∗_{(0)}}; P_{∗_{(0)}}; =; P_{∗_{(0)}}; = ··· =; P_{∗_{(0)}}; =; P_{∗_{(0)}}]

Clearly the torsion of α_{i} is RT_{p_{i}}(X) by definition. The theorem now follows by applying Lemma 2.5.

We are now in a position to show that for a finitely dominated, nilpotent space, the p-local Reidemeister torsions (and hence the p-parts of the finiteness obstruction) are restricted.

Let X be a finitely dominated nilpotent complex with finite fundamental group π = π_{1}(X). Since π is nilpotent, it splits into the product of its p-Sylow subgroups π_{p}. As in Section 1, for a fixed prime p let π′_{p} denote the complementary group to π_{p} and let

e_{p} = \frac{1}{|π′_{p}|} ∑_{g∈π′_{p}} g

The idempotent e_{p} induces a splitting

Qπ = Qπ · e_{p} × Qπ · (1 − e_{p}).

Proposition 2.6. Let p be a prime. For a finitely dominated nilpotent space X with finite fundamental group π, the image of RT_{p}(X) ∈ K_{1}(Qπ)/K_{1}(Z_{(p)}π) is zero in K_{1}(Qπ · (1 − e_{p}))/K_{1}(Z_{(p)}π · (1 − e_{p})). Hence σ(X) ∈ N_{p}(π). (See Definition 1.1.)

Proof. Let P_{∗} be a finitely generated, projective chain complex homotopy equivalent to the singular chains of the universal cover X̃. The idempotent e_{p} induces a splitting

P_{∗} = P_{∗} · e_{p} × P_{∗} · (1 − e_{p}).

Since X is nilpotent, any element of π′_{p} must act trivially on the homology of P_{∗_{(p)}}. (Such an element must act trivially on p-torsion and on rational homology.) It follows that an element of π′_{p}, acts trivially on the homology of P_{∗_{(p)}} · (1 − e_{p}) and hence P_{∗_{(p)}} · (1 − e_{p}) is a contractible chain complex.

Now e_{p} splits the chain complex P_{∗_{(0)}} similarly, and the argument above shows that the torsion of P_{∗_{(0)}} · (1 − e_{p}) comes from the torsion of P_{∗_{(p)}} · (1 − e_{p}). Hence the projection of RT_{p}(X) into K_{1}(Qπ · (1 − e_{p}))/K_{1}(Z_{(p)}π · (1 − e_{p}) is zero.

We can summarize our results as follows. For any finitely dominated complex X with finite fundamental group π, for which π acts trivially on rational homology, the finiteness obstruction can be written as

σ(X) = ∑_{p prime} σ_{p}(X)

where σ_{p}(X) = ∂^{p}(RT_{p}(X)). If X is nilpotent, then σ(X) ∈ N_{p}(π) = ∂^{p}(K_{1}(Qπ · e_{p})).

3. Realizing Reidemeister torsions

In the previous section we showed that the finiteness obstruction for a finitely dominated complex can be computed as the sum of its p-parts. For a nilpotent complex the p-part of the finiteness is restricted by Proposition 2.6 and must lie in N_{p}(π).

Following Mislin [5], we define N(π) to be the set of elements in K_{0}(Zπ) that can be realized as finiteness obstructions of finitely dominated, nilpotent complexes with fundamental group π. From the preceding remarks, we conclude that

N(π) ⊆ ∑_{p prime} N_{p}(π).

We now show that they are in fact equal.

Theorem 3.1. For a finite nilpotent group π

N(π) = ∑_{p prime} N_{p}(π).

Proof. We need only show that every element of the right hand sum can be realized. We give a direct geometric proof of this, although an algebraic proof can be given following Mislin’s work in [5].

First, one needs to show that 0 ∈ N(π). This is equivalent to the fact that π occurs as the fundamental group of a finite complex X. To see this, choose a faithful unitary representation of π. This induces a free action of π on SU(n) for some n (by left multiplication), and we can take X = SU(n)/π. The space X is clearly a nilpotent complex.

Given the finite complex X with π_{1}(X) = π, we now need to show that we can modify X so that the p-local Reidemeister torsion is an arbitrary element in K_{1}(Qπ · e_{p}). (See Definition 1.1.)

The complex X is the homotopy pullback to the localization square:

\xymatrix{X \ar[d] \ar[r] & X_{(1/p)} \ar[d] \\ X_{(p)} \ar[r] & X_{(0)}}

Let C_{∗} denote the cellular chain complex of the universal cover X̃. Applying a lemma of Wojtkoviac [10], we see that C_{∗_{(p)}} is homotopy equivalent to the singular chain complex of X_{(p)} × Bπ′_{p}, the fiberwise p-localization of X → Bπ. (As before, π′_{p} is the complement of the p-Sylow subgroup π_{p} in π.) However, we have seen that C_{∗}_{(p)} splits into a Z_{(p)}(π_{p})-chain complex and a contractible Z_{(p)}(π′_{p})-chain complex via the idempotent e_{p}. The complex C_{∗_{(p)}} is stably free (since K̃_{0}(Zπ) → K̃_{0}(Z_{(p)}π is the zero map) and so C_{∗_{(p)}} is homotopy equivalent to a free Z_{(p)}(π_{p})-chain complex, which as a Z_{(p)}(π)-chain complex is projective. This is, however, exactly the chain complex of X_{(p)} since the universal cover of Bπ′_{p} is contractible.

We now modify X_{(p)} so as to vary the p-local Reidemeister torsion. Suppose we are given an element [α] ∈ K_{1}(Qπ · e_{p}) ≅ K_{1}(Qπ_{p}). It is easy to see that α may be written as α = β · γ^{−1}, where β and γ lift to maps (but not necessarily isomorphisms!) of (Z_{(p)}π_{p})^{n} to itself. Hence, we may assume that α itself lifts to a self-map of (Z_{(p)}π_{p})^{n}.

Replacing X_{(p)} by

Z_{(p)} = X_{(p)} ∨ ∑_{i=1}^{n} S_{(p)}^{2k} ∪_{α} e^{2k+1},

adds

0 → (Z_{(p)}π_{p})^{n} ⟶^{α} (Z_{(p)}π_{p})^{n}

to the chain complex of X_{(p)}. The space Z_{(p)} is again p-local, and since α is a rational homology isomorphism, X_{(p)} and Z_{(p)} are rationally equivalent. It is now clear that we have added [α] to the p-local Reidemeister torsion.

The pullback

\xymatrix{Z \ar[d] \ar[r] & X_{(1/p)} \ar[d] \\ Z_{(p)} \ar[r] & X_{0}}

will be nilpotent, since nilpotent spaces are characterized by the fact that the fundamental group is nilpotent and its p-Sylow subgroups can act non-trivially only on p-torsion. Away from the prime p we have not changed anything there; at the prime p we have only introduced p-torsion.

Finally, Z is finitely dominated since it is nilpotent and H_{∗}(Z; Z) is a finitely generated abelian group. (See [4])

4. Pullbacks of pullbacks

In this section we present some preliminary algebra that will be used to analyze the groups N(π) and N_{p}(π) in the next section.

Consider a pullback diagram of pullback diagrams of rings:

[figure: A; A_{2}; B; B_{2}; A_{1}; A_{0}; B_{1}; B_{0}; C; C_{2}; D; D_{2}; C_{1}; C_{0}; D_{1}; D_{0}]

Such a pullback can also be written as the pullback:

[figure: A; B; A_{2}; B_{2}; C; D; C_{2}; D_{2}; A_{1}; B_{1}; A_{0}; B_{0}; C_{1}; D_{1}; C_{0}; D_{0}]

We will assume throughout this section that the following two conditions hold:

(i) The two pullback diagrams

[figure: A; A_{2}; A; B; and; A_{1}; A_{0}; C; D]

have Milnor exact sequences in K-theory.

(ii) D_{0} = O.

Remarks 4.1. It is standard that condition (i) holds if one of the maps to the lower corner is onto. The condition holds, however, in some cases where neither map is onto; for example, in localization squares. The second condition implies that C_{o} × B_{0} = A_{0} and D_{1} × D_{2} = D.

Theorem 4.2. Given a pullback of pullbacks as above, satisfying conditions (i) and (ii) above, the following diagram commutes:

[figure: [K_{1}(C_{1}) × K_{1}(C_{2})] × [K_{1}(B_{1}) × K_{1}(B_{2})]; μ; j_{C}×j_{B}; [K_{1}(C_{1}) × K_{1}(B_{1})] × [K_{1}(C_{1}) × K_{1}(B_{2})]; j_{1}×j_{2}; K_{1}(C_{0}) × K_{1}(B_{0}); K_{1}(D_{1}) × K_{1}(D_{2}); K_{1}(A_{0}); ∂_{N}; K_{0}(A); ∂_{L}; K_{1}(D)]

where μ = (1, [ \begin{matrix}0 & −1 \\ −1 & 0 \end{matrix} ] , 1) and where j_{C}, j_{B}, j_{1}, j_{2},∂_{N},∂_{L} all are the appropriate maps from the Milnor exact sequence. (The letters N and L here stand for “number” and “letter”.)

Remarks 4.3. Such pullbacks occur more often than one might think at first. In fact, most Milnor squares

\xymatrix{A \ar[d] \ar[r] & A_{2} \ar[d] \\ A_{1} \ar[r] & A_{0}}

can be thought of as the result of applying a functor to the identity map A ⟶^{1} A in the category of “rings under A”; that is, in the category of rings R together with a homomorphism A → R. In such a situation, it is enough to specify the two pullback squares:

[figure: A; A_{2}; A; B; and; A_{1}; A_{0}; C; D]

The rest of the diagram can be obtained from these by applying the functors. (See the examples in Section 5.)

Proof of Theorem 4.2. By symmetry it is sufficient to consider an element [α] ∈ K_{1}(C_{1}) where α is an isomorphism of C_{1}^{n}. Following the diagram along the left side, let [β] ∈ K_{1}(A_{0}) denote the image of [α]; the representative β is an isomorphism of A_{0}^{n} which under the identification A_{0} = B_{0} × C_{0} is the identity on B_{0}^{n}.

Now ∂_{N}[β] is represented by a projective module P given by the pullback:

\xymatrix{P \ar[dd] \ar[r] & A_{2}^{n} \ar[d] \\  & A_{0}^{n} \ar[d] \\ A_{1}^{n} \ar[r] & A_{0}^{n}}

We wish to show that P can be described as well by following the diagram around the right side.

To do this, we first note that P ⊗_{A} A = P, and so P can also be described as the pullback:

\xymatrix{P \ar[d] \ar[r] & P ⊗_{B} B \ar[d] \\ P ⊗_{C} C \ar[r] & P ⊗_{D} D}  (∗)

(To see that this is a pullback, note that it is when P is free. For an arbitrary projective P, add a projective to obtain a free module. Now use the fact that summands of pullback diagrams are pullback diagrams themselves.)

By assumption B_{0}^{n} = A_{0}^{n} ⊗_{A_{o}} B_{0} = B_{0}^{n} is the identity, thus producing an isomorphism

P ⊗_{A} B ≃ B^{n}. The isomorphism C_{0}^{n} = A_{0}^{n} ⊗_{A_{0}} C_{0} ⟶^{β⊗1} A_{0}^{n} ⊗_{A_{0}} C_{0} = C_{0}^{n} lifts to α : C_{1}^{n} → C_{1}^{n},

producing an isomorphism P ⊗_{A} C ≃ C^{n} The pullback diagram (∗) can therefore be written as:

\xymatrix{P \ar[dd] \ar[r] & C^{n} \ar[d] \\  & D^{n} \ar[d] \\ B^{n} \ar[r] & D^{n}}

It is now easy to verify that the compatibility isomorphism

D^{n} = C^{n} ⊗_{C} D ≃ P ⊗_{A} C ⊗_{C} D = P ⊗_{A} D
≃ P ⊗_{A} B ⊗_{B} D ≃ B^{n} ⊗_{B} D = D^{n}

is induced from the given isomorphism α of C_{1}^{n}. Hence P is the image of [α] around the right side of the diagram.

Starting with an element [α] ∈ K_{1}(C_{2}) or K_{1}(B_{1}), the same argument applies, but the compatibility isomorphisms have opposite directions, thus introducing a minus sign.

5. More about N(π) and N_{p}(π)

In this section we complete the general discussion about the groups N(π) and N_{p}(π). We show that N(π) = D(π) for a p-group, and recover as a corollary the result of Mislin and Varadarajan [6] that N(π) ⊆ D(π) for any finite nilpotent group. (Recall that D(π) is the kernel of the map K_{0}(Zπ) → K_{0}(M) where M is a maximal order in Qπ. We also fit N_{p}(π) into an exact sequence.

We begin by relating the ∂^{p} homomorphisms of Section 1 to the Swan homomorphism SW, which can be described as the boundary homomorphism in the Milnor sequence associated to the pullback:

\xymatrix{Zπ \ar[r] \ar[d] & Z \ar[d] \\ Zπ/Σ \ar[r] & Z/|π|}

where Σ = ∑_{g∈π} g. To do this, we need to compare the boundary homomorphisms in two

Milnor sequences.

To be precise, let π be a finite group and consider the idempotent.

e = (1/|π|) ∑_{g∈π} g.

As usual Qπ splits as Qπ · e × Qπ · (1 − e). We can define an embedding ϕ : Q^{∗} → Qπ^{∗} by ϕ(x) = x · e + (1 − e).

We wish to compute ∂^{p} on the image of ϕ. Since ϕ is multiplicative, it clearly suffices to compute ∂^{p} on the image of each prime.

Proposition 5.1. Let π be a finite group of order p^{n}·q^{m}·r, where p and q are distinct primes not dividing r. (We allow n = 0 or m = 0 here!) Let ϕ : Q^{∗} → (Qπ)^{∗} be the embedding defined above. Then

∂^{p}(ϕ(p)) = −Sw(x), where x = \{ \begin{matrix}1 & mod & p^{n} \\ p & mod & q^{m}r \end{matrix}

and

∂^{p}(ϕ(p)) = −Sw(x), where x = \{ \begin{matrix}1 & mod & q^{m} \\ q & mod & p^{n}r \end{matrix}

Remarks 5.2. Since we allow n = 0 or m = 0, this provides a complete determination of ∂^{p} on the image of ϕ in terms of the Swan homomorphism.

Proof of Proposition 1.5. According to Remark 4.3, the two pullback diagrams

[figure: Zπ; Z_{(p)}; Zπ; Z; and; Z_{(1−p)}π; A_{0}; Zπ/Σ; D]

determine a pullback of pullback diagrams. Applying Theorem 4.2 to this situation, one obtains a commutative diagram:

[figure: K_{1}(Z_{(1/p)}) × K_{1}(Z_{(p)}); i×i; j_{α}−j; K_{1}(Q); ϕ; K_{1}((Z/|π|)_{(1/p)}) × K_{1}((Z/|π|)_{(p)}); K_{1}(Qπ); ∂^{p}; K_{1}(Z/|π|); Sw; K_{0}(Zπ)]

where i and j are the natural inclusions of units. One now immediately reads off the answer from this diagram, starting with the element in K_{1}(Q) and tracing both ways around.

Theorem 5.3. Let π be a finite p-group. Then N(π) = D(π), where as usual D(π) = ker{K_{0}(Zπ) → K_{0}(M)} for a maximal order M in Qπ.

Proof. For q ≠ p, N_{p}(π) is contained in the image of the Swan homomorphism by Proposition 5.1. However, the same proposition now shows that the image of the Swan homomorphism is contained in N_{p}(π). Hence N(π) = N_{p}(π).

Now the image of ∂^{p} contains D(π). To see this, let M be a maximal order in Qπ and note that M_{(1/p)} = Z_{(1/p)}π since π is a p-group. The sequence

K_{1}(Qπ) → K_{0}(Zπ) ⟶^{i} K_{0}(Z_{(1/p)}π)

is exact, and the map factors as:

[figure: K_{0}(Zπ); i_{∗}; K_{0}(Z_{(1/p)}π); K_{0}(M); K_{0}(M_{(1/p)})]

Hence (N(π) ⊆ D(π).

To obtain the other inclusion, note that by [7, Chap. 22] the map K_{0}(M) → K_{0}(M_{(1/p)}) is monic. Hence any element in D(π) is contained in the kernel of i_{∗}, showing that D(π) ⊆ N(π).

As a corollary of the preceding result, we recover the following result of Mislin and Varadarajan [6].

Corollary 5.4. For any finite nilpotent group π, N(π) ⊆ D(π).

Proof. It is enough to show that N_{p}(π) ⊆ D(π) for any prime p. Consider the diagram:

\xymatrix{M \ar[rrr] \ar[ddd] &  &  & M_{(1/p)} \ar[ddd] \\  & Zπ \ar[ul] \ar[r] \ar[d] & Z_{(1/p)}π \ar[d] \ar@{-->}[ur] \\  & Z_{(p)}π \ar@{-->}[dl] \ar[r] & Qπ \ar[dr]^{≃} \\ M_{(p)} \ar[rrr] &  &  & M_{(0)}}

where M is a maximal order in Qπ.

Now N_{p}(π) is the image of ∂^{p} restricted to K_{1}(Qπ ·e_{p}), where e_{p} is the idempotent defined in Section 1. We need to show that this image maps to zero in K_{0}(M). That is equivalent to showing that the boundary map in the Milnor sequence of the outer diagram maps the Qπ · e_{p} component of M_{(0)} = Qπ to zero.

The outer diagram splits, however, and the component we must consider is precisely the component corresponding to the maximal order of π_{p}, the p-Sylow subgroup of π. This case was considered in Theorem 5.3.

For a finite nilpotent group π of composite order, the group N(π) is in general not equal to all of D(π). To obtain further information about N(π) it is necessary to study the groups N_{p}(π) individually.

Consider the nilpotent group π and a prime p dividing the order of π. As usual we write π = π_{p} ×π′_{p} where π_{p} is the p-Sylow subgroup and π′_{p} is the complementary subgroup. There is then a Milnor square:

\xymatrix{Zπ \ar[d] \ar[r] & Zπ_{p} \ar[d] \\ Zπ/Σ \ar[r] & Zπ_{p}/|π′_{p}|}

where Σ′ = ∑_{g∈π′_{p}} g. We denote the boundary map in the Milnor sequence for this square by

∂^{f_{p}}. (The “f”” stands for ”finite” here.)

Theorem 5.5. For any finite nilpotent group π there is an exact sequence:

K_{1}(Z_{1/p)}π_{p}) ⟶^{ψ_{p}} N_{p}(π) ⟶^{j_{p}} D(π_{p}) → 0

where j_{p} is induced from the projection π → π_{p}, and ψ_{p} is the composition

K_{1}(Z_{(1/p)}π_{p}) → K_{1}(Z_{(1/p)}π_{p}/|π′_{p}|) = K_{1}(Zπ_{p}/|π′_{p}|) ⟶^{∂^{fp}} K_{0}(Zπ)

Proof. According to Remark 4.3, the two pullback squares

[figure: Zπ; Z_{(1/p)}; Zπ; Zπ_{p}; and; Z_{(p)}π; Qπ; Zπ/Σ; Zπ_{p}/|π′_{p}|]

determine a pullback of pullback diagrams. Applying Theorem 4.2 to this situation, we obtain a commutative diagram:

[figure: K_{1}(Z_{(1/p)}π_{p}); (i,0); K_{1}(Qπ · e_{p}) × K_{1}(Qπ · (1 − e_{p})); K_{1}(Zπ_{p}/|π′_{p}|; =; K_{1}(Z_{(1/p)}π_{p}/|π′_{p}|); K_{1}(Qπ); ∂; K_{0}(Zπ); K_{1}(Qπ_{p}); ∂^{p}; K_{0}(Zπ_{p})]

The lower square shows that projection induces an epimorphism N_{p}(π) → N_{p}(π_{p}), and by Theorem 5.3 we know that N_{p}(π_{p}) = D(π_{p}).

Commutativity of the diagram shows that the image of ψ_{p} is contained in N_{p}(π). Since

K_{1}(Z_{(p)}π_{p}) ⊕ K_{1}(Z_{(1/p)}π_{p}) → K_{1}(Qπ_{p})

is part of a Milnor exact sequence, we also conclude that the image of ψ_{p} is contained in the kernel of j_{p}.

We need only show, therefore, that if x ∈ N_{p}(π) and j_{p}(x) = 0 then x is in the image of ψ_{p}. Since x ∈ N_{p}(π) we know that x = ∂^{p}y for some y ∈ K_{1}(Qπ · e_{p}) = K_{1}(Qπ_{p}). But in the exact sequence for π_{p}, the boundary of y is zero in K_{0}(Zπ_{p}). Hence there are elements z_{1} ∈ K_{1}(Z_{(1/p)}π_{p}) and z_{2} ∈ K_{1}(Z_{(p)}π_{p}) such that (z_{1}, z_{2}) maps to y.

Now the idempotent e_{p} is also an idempotent in Z_{(p)}π, since the denominators involve only primes different from p. Hence Z_{(p)}π splits as

Z_{(p)}π = Z_{(p)}π · e_{p} × Z_{(p)}π · (1 − e_{p})

and the element (z_{2}, 1) defines an element in K_{1}(Z_{(p)}π) that maps to z_{2} under projection. Let y_{2} denote the image of z_{2} in K_{1}(Qπ). Then y − y_{2} lies in K_{1}(Qπ · e_{p}) and maps to x under the boundary map, and is also the image of z_{1} ∈ K_{1}(Z_{(1/p)}π_{p}). By commutativity of the diagram we conclude that x is in the image of ψ_{p}.

6. A computation

Let C_{n} denote the cyclic group of order n. In [5] Mislin has shown that N(C_{21}) is either 0 or Z/2. To illustrate Proposition 5.4 we will show that N(C_{21}) is in fact Z/2 by showing that N_{3}(C_{21}) ≠ 0. Since D(C_{3}) = 0, we must show that the map ψ_{3} has nontrivial image. This involves a small computation in number theory.

The Milnor square defining ∂^{f_{3}} is now

\xymatrix{Z[C_{21}] \ar[d] \ar[r] & Z[ζ][C_{3}] \ar[d] \\ Z[C_{3}] \ar[r] & F_{7}[C_{3}]}

where ζ = e^{2πi/7} and F_{7} is the field of 7 elements (a less cumbersome notation that Z/7.) The portion of the K-theory exact sequence we are interested in is

Z[C_{3}]^{∗} × Z[ζ][C_{3}]^{∗} ⟶^{j} F_{7}[C_{3}]^{∗} ⟶^{∂^{f_{3}}} K_{0}(Z[C_{21}]).

Since we have a splitting F_{7}[C_{3}]^{∗} = F_{7}^{∗} × F_{7}^{∗}[ω]^{∗}, where ω = e^{2πi/3}, we will write our units accordingly.

Now we claim that the unit (1, ω − 1) ∈ F_{7}[C_{3}]^{∗} is not in the image of j and hence ∂^{f_{3}}(1, ω − 1) ≠ 0. Since (x − 1) + \frac{1}{3}(1 + x + x^{2}) ∈ Z_{(1/3)}[C_{3}]^{∗} maps to (1, ω − 1), we can conclude that N_{3}(C_{21}) ≠ 0. To prove the claim we need to do a little work.

First, note that since Z[C_{3}]^{∗} ⊆ Z[ζ][C_{3}]^{∗}, it is enough to show that (1, ω − 1) is not in the image of Z[ζ][C_{3}]^{∗}. To analyze this group we use the standard pullback

\xymatrix{Z[ζ][C_{3}] \ar[d] \ar[r] & Z[ζ] \ar[d] \\ Z[ζω] \ar[r] & F_{3}[ζ]}

yielding the Milnor sequence:

1 → Z[ζ][C_{3}]^{∗} → Z[ζω]^{∗} × Z[ζ]^{∗} → F_{3}[ζ]^{∗}

We now require two lemmas.

Lemma 6.1. The image of Z[ζ]^{∗} → F_{3}[ζ]^{∗} is generated by (ζ − 1)^{2}.

Proof. Fortunately, one knows that the units of Z[ζ]^{∗} are generated by the cyclotomic units −ζ^{−1},(ζ^{2} − 1)/ζ − 1), and (ζ^{3} − 1)/(ζ − 1).

Repeated cubing shows that

(ζ − 1)^{3} = ζ^{3} − 1 mod 3
(ζ − 1)^{3} = ζ^{9} − 1 mod 3

and

(ζ)^{27} = ζ^{6} − 1 mod 3

Hence,

(ζ^{3} − 1)/(ζ − 1) = (ζ − 1)^{2} mod 3
(ζ^{2} − 1)/(ζ − 1) = (ζ − 1)^{8} mod 3

and

(ζ^{6} − 1)/(ζ − 1) = −ζ^{−1} = (ζ − 1)^{26} mod 3

Lemma 6.2. Given a unit u ∈ Z[ζω]^{∗} such that u ≡ 1 mod (ζ − 1), the projection of u in F_{3}[ζ] is an even power of ζ − 1).

Proof. We consider the commutative square

\xymatrix{Z[ζω]^{∗} \ar[d] \ar[r]^{ϕ_{1}} & F_{3}[ζ]^{∗} \ar[d] \\ Z[ζ]^{∗} \ar[r]^{ϕ_{2}} & F_{3}[ζ]^{∗}}

where ϕ_{1} is reduction mod (ω − 1), ϕ_{2} is reduction mod 3, N is the norm map, and Sq is the squaring map.

Since u ≡ 1 mod (ζ − 1), we see that N(u) ≡ 1 mod (ζ − 1) as well. But as mentioned above, we know the units of Z[ζ]^{∗} and hence

N(u) = (−ζ^{−1})^{a} (\frac{ζ^{2} − 1}{ζ − 1})^{b} (\frac{ζ^{3}}{ζ − 1})^{c}

for some integers, a, b, and c. Reducing mod (ζ − 1) we have

(−1)^{a}(2)^{b}(3)^{c} ≡ 1 mod 7.

Now write 2 ≡ 3^{2} mod 7 and −1 ≡ 3^{3} mod 7. We then obtain

3^{3a} · 3^{2b} · 3^{c} ≡ 1 mod 7

showing that 3a + 2b + c ≡ 0 mod 6.

From the proof of Lemma 6.1 we also know that

ϕ_{2}N(u) = (ζ − 1)^{26a}(ζ − 1)^{8b}(ζ − 1)^{2c} = (ζ − 1)^{26a+8b+2c}.

Since c = −3a − 2b + 6q for some integer q, we can substitute to see that

ϕ_{2}N(u) = (ζ − 1)^{20a+4b+12q} = Sqϕ_{1}(u).

We can conclude that ϕ_{1}(u) is (ζ − 1) to an even power.

To complete the proof of the claim, suppose (1, ω − 1) is the image of u ∈ Z[ζ][C_{3}]^{∗}. Let (u_{1}, u_{2}) ∈ Z[ζ]^{∗} × Z[ζω]^{∗} be the projection of u; we know, of course, that u_{1} = u_{2} in F_{3}[ζ]. Now u_{2}(ζω − 1)^{−1} ≡ 1 mod (ζ − 1). Projecting into F_{3}[ζ] and applying Lemma 6.1, we see that u_{2} = (ζ − 1)^{2k+1} in F_{3}[ζ] for some k. Hence u_{1} = (ζ − 1)^{2k+1} in F_{3}[ζ] contradicting Lemma 6.2.

With a small additional effort it is possible to show that N_{3}(C_{21}) is precisely Z/2 (although this follows from Mislin’s result). The type of argument employed here can be used more generally for cyclic groups. The number theoretic problems, however, rapidly become more difficult.

References

[1] H. Bass, Algebraic K-theory, Benjamin, 1968.

[2] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. (N.S.) 72 (1966), 358–426.

[3] G. Mislin, Wall’s obstruction for nilpotent spaces, Topology 14 (1975), no. 4, 311–317.

[4] , Finitely dominated nilpotent spaces, Ann. of Math. (2) 103 (1976), 547–556.

[5] , The geometric realization of Wall obstructions by nilpotent and simple spaces, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 2, 199–206.

[6] G. Mislin and K. Varadarajan, The Finiteness obstruction for nilpotent spaces lie in D(Z[π]), Invent. Math. 53 (1979), 185–191.

[7] I. Reiner, Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press Oxford University Press, Oxford, 2003, Corrected reprint of the 1975 original, With a foreword by M. J. Taylor.

[8] R. Swan, Induced representations and projective modules, Ann. of Math. (2) 71 (1960), 267–291.

[9] C. T. C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. (2) 81 (1965), 56–69.

[10] Z. Wojtkowiak, Localizations, finiteness obstructions and Reidemeister torsions, Quart. J. Math. Oxford Ser. (2) 35 (1984), no. 138, 223–241.

Department of Mathematics, Indiana University, Bloomington Indiana 47405, U. S . A.

Mathematisches Institut, Göttingen D-3400, FRG

Department of Mathematics, Odense Universitet, DK-5230 Odense M, Denmark