CONTROLLED ALGEBRAIC K-THEORY, A SURVEY

ERIK KJÆR PEDERSEN

0. Introduction

The purpose of this paper is to give the reader an impression of various techniques used in controlled algebra. Controlled algebra was introduced by Connell-Hollingsworth and developed by Quinn [3, 9] in connection with foundational studies in topological manifold theory. Suppose given a ring R and a space X, possibly with some extra structure such as a metric. The objects studied in controlled algebra are free based R-modules together with a map π from the basis to the metric space X. One usually requires that the image of π is nowhere dense in X, and that π^{−1}(x) is finite for all x in X. For every point x in X we denote the free R-module generated by π^{−1}(x) by A_{x}. A morphism φ between objects A and B can now be given a matrix decomposition as φ = {φ_{x}^{y}} where φ_{x}^{y} is the composition A_{x} → A ⟶^{φ} B → B_{y}. Control is then a requirement that φ_{x}^{y} = 0 if x is “too far away” from y. In this paper we shall consider X exhibited with a metric and require the existence of a k such that φ_{x}^{y} = 0 if the distance from x to y is bigger than k. These are the bounded categories introduced in [6]. Notice that even in the case when X is compact, this category is a little more that just the category of R-modules, since the hom-sets come exhibited with a filtration.

Controlled algebra is used to guide geometric constructions. Suppose given a manifold or CW-complex with a reference map to a metric space. Typically the cellular chains of the CWcomplex or the cellular chains of a handle-decomposition of the manifold, can be interpreted as being the chain complex in a controlled category, by subdividing the cells sufficiently to get the control condition on the boundary homomorphisms satisfied. Of course the cells also have to be so small that associating the cell with a generator in the reference space produces a canonical object up to isomorphism in the controlled category.

Geometrically there are two basic moves. The first one is handle addition, consisting of sliding one cell (or handle in the manifold case) across another cell in a cell-decomposition. Algebraically this corresponds to changing the boundary map in the cellular chain complex by an elementary matrix. Sliding a cell across another cell will typically increase the diameter of the cell. The purpose of controlled algebra is to keep track of how much this diameter increases. The second basic move geometrically is used among other things to avoid increasing these diameters too much, namely by introducing a pair of canceling cells at an appropriate spot, and then using these extra cells in the cell-slidings. Algebraically, introducing a pair of extra canceling cells or canceling handles corresponds to stabilization – introducing an extra generator in adjacent dimensions letting the boundary map send generator to generator. In the cell-complex situation this is obtained simply by wedging on a disc at an appropriate point.

The author was partially supported by NSF DMS 9104026.

We can perform many handle additions simultaneously as long as we do it locally finitely. We also have to ensure that we do not reuse handles that have already been increased in size in each step. This means that we need to divide the basis in two groups, and only slide handles corresponding to elements in the first group across handles in the second group. These moves are then expressed algebraically by a 2 × 2 elementary block matrix e = ( \begin{matrix}1 & A \\ 0 & 1 \end{matrix} ). We define the size of e as the size of A, the size of A being an expression of how far away a generator is sent. This size is clearly a measure of the increase in the size of the cells. Repeating this procedure a finite number of times will preserve control, so in general we define a deformation E to be a sequence of such elementary deformations e and we define the size of E to be the sum of the size of e_{i} when E is the product of the e_{i}’s. We say that α can be δ-deformed to β if α · E = β and the size of E is less than δ. A very useful algebraic result due to Quinn is that once an automorphism is sufficiently small, then we may stabilize and deform in a small way to an arbitrarily small automorphism i. e. there exists an ε_{X} only depending on the metric space such that for every ε < ε_{X} there is a δ so that if α is δ-controlled, then after stabilization, α can be ε-deformed to an automorphism of arbitrarily small control. The crucial point is that δ only depends on ε and X, not on how small we want the automorphism to be. We present a new proof of this result here.

In a final section, we give a new proof of topological invariance of Whitehead torsion. Here is how controlled algebra comes into the picture: If h : K → L is a homeomorphism between finite polyhedra, we can use simplicial approximation to approximate h by a simplicial homotopy equivalence with simplicial homotopy inverse g. If f and g are chosen close enough to h and h^{−1}, the tracks of the homotopies f ∘ g ≃ id and g ∘ f ≃ id will be small. Following the usual Whitehead prescription represents τ(h) by a controlled automorphism. The point here is that since polyhedra are locally simply connected, it is unnecessary to specify paths between the barycenters of simplices, provided that these simplices are close together. Thus, the usual passage to ℤπ_{1}L-modules is replaced by a passage to geometric algebra. We now use a standard algebraic trick to pass from a K_{1}-problem in ε-δ controlled algebra to an equivalent K_{2} problem in Pedersen-Weibel’s bounded algebra. It turns out we can show the Whitehead torsion is in the image from a certain K_{2}-group. We compute this K_{2}-group to be ±π which is precisely what is divided out in K_{1}(ℤπ) in the definition of the Whitehead group. Basically we prove that the Whitehead torsion of a homeomorphism is in the image of the K-theory assembly map, and hence trivial.

Other algebraic approaches to the topological invariance of Whitehead torsion are given by Quinn in [8], and Ranicki and Yamasaki in [10]. They use an ε controlled K_{1}-group, K_{1}^{ε}, where we use an ordinary K_{2} of an additive category in the sense of Bass [1].

1. A Category of metric spaces and eventual Lipschitz maps

We shall work in the category of metric spaces and eventual Lipschitz maps. There are several similar, slightly conflicting concepts in the literature, such as the uniformly bornologous maps in [11] and eventual Lipschitz maps in [7] and in [12]. The following definitions seem appropriate for the purposes of this paper:

Definition 1.1. The category M of metric spaces and eventual Lipschitz maps has as objects arbitrary metric spaces. An eventual Lipschitz map f : M → N is a not necessarily continuous map satisfying:

(i) The inverse image of every bounded subset of N is bounded in M. (ii) There exist r and k, depending only on f, so that for all x, y ∈ M d(f(x), f(y)) <

k · d(x, y) + r.

Subspaces are given the induced metric and product spaces are given the sup metric. A homotopy of morphisms f_{0}, f_{1} : M → N is a morphism F : M × I → N which restricts to f_{0} and f_{1} on the ends. It follows immediately that maps f_{0} and f_{1} are homotopic if and only if d(f_{0}(x), f_{1}(x)) is uniformly bounded.

To relate this to similar notions, we recall Gromov’s notion of quasi-isometry [4]:

Definition 1.2. Let X and Y be metric spaces and let F : X → Y be a function. F is a (K, c)-quasi-isometry (c ≥ 0 and K ≥ 1) provided that for all x, y ∈ X

(1/K)d(x, y) − c ≤ d(F (x), F (y)) ≤ Kd(x, y) + c

Metric spaces X and Y are quasi-isometric if there are quasi-isometries in both directions. This is clearly equivalent to saying that there is a quasi-isometry F from X to Y and a constant C such that every point of Y is within C of the image of X.

A homotopy equivalence in the category M is therefore a quasi-isometry in the sense of Gromov and homotopy equivalent spaces are quasi-isometric.

Remark 1.3. All notions of this section make equally good sense if we replace the metric spaces by pseudo-metric spaces, since the condition d(x, y) = 0 implies x = y is not really needed. This is useful, for instance, in case we have a map p : M → X and we want to use the pseudo-metric ρ(m_{1}, m_{2}) = d(p(m_{1}), p(m_{2})) to measure distances in M.

2. Bounded algebraic categories

Given a ring R and a pseudo-metric space M, we define a category C_{M}(R). The model case of our definition is the case in which M is the infinite open cone O(K) = {t · x ∈ ℝ^{n+1}|t ∈ [0, ∞), x ∈ K} on a complex K ⊂ S^{n} ⊂ ℝ^{n+1} and R = ℤπ, with π a finitely presented group.

Definition 2.1. An object A of C_{M}(R) is a collection of finitely generated free based right R-modules A_{x}, one for each x ∈ M, such that for each ball C ⊂ M of finite radius, only finitely many A_{x}, x ∈ C, are nonzero. A morphism φ : A → B is a collection of morphisms φ_{y}^{x} : A_{x} → B_{y} such that there exists k = k(φ) such that φ_{y}^{x} = 0 for d(x, y) > k.

The composition of φ : A → B and ψ : B → C is given by (ψ ∘ φ)_{y}^{x} = ∑_{z∈M} ψ_{y}^{z}φ_{z}^{x}. The composition (ψ ∘φ) satisfies the local finiteness and boundedness conditions whenever ψ and φ do.

This makes C_{M}(R) into a filtered additive category, where F_{d} hom(A_{1}, A_{2}) consists of morphisms having bound ≤ d.

Definition 2.2. The idempotent completion C_{M̂}(R) of C_{M}(R) is the category whose objects are pairs (A, p) where A is an object of C_{M}(R) and p : A → A is idempotent. A morphism φ : (A_{1}, p_{1}) → (A_{2}, p_{2}) is a morphism φ : A_{1} → A_{2} in C_{M}(R) such that φ = p_{2}φp_{1}. The filtration degree of φ is the smallest d such that φ = p_{2}fp_{1} for some f ∈ F_{d} hom(A_{1}, A_{2}) with fp_{1} = p_{2}f.

Theorem 2.3. (Pedersen-Weibel [7]) If P is a finite polyhedron and ∗ ≥ 0, then

H_{∗−1}(P; K^{−∞}(R)) ≅ K_{∗}(C_{Ô(P)}(R)).

The term on the left is the homology of P with coefficients in the algebraic K-theory spectrum of the ring R, the spectrum including the negative K-groups, while the term on the right is the algebraic K-theory of the category C_{Ô(P)}(R)).

It is easy to see that an additive category is cofinal in its idempotent completion. It thus follows that the idempotent completion only affects the K-groups in degree 0, so we have

H_{∗−1}(P; K^{−∞}(R)) ≅ K_{∗}(C_{O(P)}(R))

for ∗ ≥ 1. In particular, we have

K_{1}(C_{O(P)}(R)) ≅ H_{0}(P; K^{−∞}(R)).

The group K_{1}(C_{M}(R)) has a “classical” description: it consists of equivalence classes of pairs (A, α), where A is an object of C_{M}(R) and α : A → A is an automorphism. If B is another object in C_{M}(R), then α is equivalent to α ⊕ id : A ⊕ B → A ⊕ B. In addition, id is declared to be equivalent to α if α is an elementary automorphism as in Definition 3.1 below. We shall also be discussing K_{2} classically as the limit of the endomorphism ring of an object under inclusion of direct summands.

One purpose of this paper is to relate the bounded K-theory of O(P) to the controlled or “ε-δ” K-theory on P in the sense of Chapman and Quinn. We will begin by showing that for each finite polyhedron P there is a critical size ε_{P} so that automorphisms in C_{P} (R) which have bounds less than ε_{P} can be deformed by small moves to automorphisms with arbitrarily small bounds.

3. Squeezing K_{1} and K_{2}

Consider a finite simplicial complex K. As an additive category, C_{K}(R) is equivalent to the category of finitely generated free R-modules, since K is eventual Lipschitz equivalent to a point. However, as a filtered category the situation is different. We consider the filtration of the hom-sets given by the bounds on the morphisms. It is the aim of this section to prove that given K there is an ε so that if an automorphism and its inverse are both bounded by ε, then the automorphism may be given a small deformation to an automorphism with arbitrarily small bounds. We introduce some definitions:

Definition 3.1. Let X be a metric space.

(i) If A is an object in C_{X}(R), a morphism η : A → A is strictly triangular with respect to an internal direct sum decomposition A = A_{1} ⊕ A_{2} if η factors as A → A_{1} ⟶^{η′} A_{2} → A. We will denote the elementary automorphism id +η : A → A by e_{η} when η is strictly triangular.

(ii) If η : B → C is a morphism in C_{O(K)}(R), η_{≤k} will denote the morphism such that:

(η_{≤k})_{y}^{x} = \begin{cases}η_{y}^{x} & if ||x||, ||y|| ≤ k \\ 0 & if ||x|| > k or ||y|| > k. \end{cases}

(iii) If e_{η} : A → A is an elementary automorphism in C_{O(K)}(R), then e_{η}^{≤k} will denote the

elementary automorphism id +η_{≤k}. (iv) We will say that an object A in C_{X}(R) has support in Y ⊂ X if A_{x} = 0 for all x ∉ Y.

We will say that a morphism α : A → B in C_{X}(R) has support in Y if α_{y}^{x} = 0 whenever x or y ∉ Y. We will say an elementary automorphism e_{η} has support in Y if η has support in Y.

Remark 3.2. In the definition above, since the direct sum decomposition is required to be internal, and A_{i} are objects in the category we have an internal direct sum decomposition A_{x} = (A_{1})_{x} ⊕ (A_{2})_{x} for each x ∈ M. Notice that there are two different kinds of “cutting off” in the definition above, one by morphisms and one by objects.

Remark 3.3. If {e_{η_{i}}} is a collection of elementary automorphisms on C_{O(K)}(R) and ∏_{i=1}^{n} e_{η_{i}} = 1, then ∏_{i=1}^{n} e_{η_{i}}^{≤k}, k large, will be equal to 1 for ||x|| sufficiently large or sufficiently small, but may be nontrivial in the band k − ∑ bound(e_{i}) ≤ ||x|| ≤ k + ∑ bound(e_{i}). Compressing this band onto a single copy of K, say ||x|| = k, we obtain an object in C_{K}(R) and an automorphism of this object. By choosing k to be large, we can force this automorphism to have bound as small as we like, in the original metric on K, since it is bounded in the metric which has been multiplied by k. This construction will be used in the proof of Theorem 3.7, which is our main squeezing theorem.

Lemma 3.4. If α : A → B and β : B → C are morphisms in C_{O(K)}(R),

(i) bound(β ∘ α) ≤ bound(β) + bound(α).

(ii) bound(α_{≤k}) ≤ bound(α).

(iii) If e = (\begin{matrix}1 & η \\ 0 & 1 \end{matrix}), then bound(e) = bound(η).

(iv) If α, ᾱ : A → B and β, β̄ : B → C are morphisms with α_{y}^{x} = ᾱ_{y}^{x} and β_{y}^{x} = β̄_{y}^{x} for x, y ∈ S ⊂ O(K), then α ∘ β = ᾱ ∘ β̄ away from a (bound(α) + bound(β))-neighborhood of boundaryS.

For the rest of this section, we will assume that ℝ^{n} has the max metric so that the unit ball is ∏_{i=1}^{n}[−1, 1]. By a cubical subcomplex of the boundary of the unit ball in ℝ^{n}, we will mean a complex consisting entirely of faces of ∏_{i=1}^{n}[−1, 1]. To see that every finite polyhedron is PL homeomorphic to such a cubical complex, take the standard n-simplex Δ^{n} to be the convex hull of unit vectors e_{1}, …, e_{n+1} in ℝ^{n+1}. A PL homeomorphism from Δ^{n} to a cubical subcomplex of the cube ∏_{i=1}^{n+1}[0, 1] in ℝ^{n+1} is given by sending each barycenter { \atop ⟨e_{i_{0}}, …, e_{i_{j}}⟩} to e_{i_{0}} + ⋯ + e_{i_{j}} and extending linearly. This PL homeomorphism takes each simplex of Δ^{n} onto a cubical subcomplex of the cube. Since every finite simplicial complex K is isomorphic to a subcomplex of some Δ^{n}, composing with the homeomorphism above produces a cubical subcomplex of the cube which is PL homeomorphic to K. Closer inspection shows that the cubes are obtained by amalgamating simplexes of the first barycentric subdivision of K.

Recall the following basic matrix identity

Lemma 3.5. (6 term identity)

(\begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix}) = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ −1 & 1 \end{matrix}) (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ α & 1 \end{matrix}) (\begin{matrix}1 & −α^{−1} \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ α & 1 \end{matrix})

Proof. This is of course well known. A graphical presentation is given as follows where all the horizontal arrows are identities and matrix multiplication is given by adding all ways through the diagram:

\xymatrix{• \ar[r] \ar[dr]^{α} & • \ar[r] & • \ar[r] \ar[dr]^{α} & • \ar[r] & • \ar[r] \ar[dr]^{−1} & • \ar[r] & • \\ • \ar[r] & • \ar[r] \ar[ur]_{−α^{−1}} & • \ar[r] & • \ar[r] \ar[ur]_{1} & • \ar[r] & • \ar[r] \ar[ur]_{1} & •}

Here is the main technical result leading to the K_{1}-squeezing theorem.

Theorem 3.6. (K_{1}-vanishing) Let K be a cubical subcomplex of the boundary of the unit ball in ℝ^{n}. If A is supported outside of the unit ball in O(K) and α : A → A is an automorphism in C_{O(K)}(R) such that α and α^{−1} are bounded by δ < \frac{2}{3} · 6^{−(dim K)} in the max metric, then there exist an object B in C_{O(K)}(R), and a deformation of size δ · 6^{dim K+1}, ∏ e_{η_{i}} on A ⊕ B in the category C_{O(K)}(R) such that :

(i) B is supported outside of the unit ball in O(K).

(ii) (α ⊕ id_{B}) · ∏ e_{η_{i}} = id_{A⊕B}.

Before proving Theorem 3.6, we will show how it leads to the squeezing theorem.

Theorem 3.7. (K_{1}-Squeezing Theorem) Let K be a cubical subcomplex of the boundary of the unit ball in ℝ^{n}. If ε < 6^{− dim K} and α : A → A is an automorphism in C_{K}(R) such that α and α^{−1} are bounded by ε in the max metric, then for each μ > 0 there is an object C in C_{K}(R) and an automorphism β : A ⊕ C → A ⊕ C such that

(i) β and β^{−1} are bounded by μ. (ii) There is a deformation of size 6^{dim K+1} · ε of α ⊕ id to β

Proof. We have K a subset of the boundary of the unit cube, hence a subset of O(K). Let α : A → A be an automorphism as in the statement of the theorem. If ε is small, Theorem 3.6 guarantees that we can find B and a deformation α ⊕ id_{B} · ∏ e_{η_{i}} = id_{A⊕B}.

Let L > 0 be large and consider (α ⊕ id)^{≤L} ∏ e_{η_{i}}^{≤L}. For L sufficiently large, (α ⊕ id)^{≤L} is simply α ⊕ id_{C} where C denotes the modules of B sitting inside a region a little larger than L. Since each e_{η_{i}}^{≤L} is an elementary matrix, we will write e_{η_{i}}^{≤L} = eη̄_{i}. Obviously the product (α ⊕ id_{C}) · ∏ eη̄_{i} is no longer the identity. In a region around radius L it will be an automorphism, and we do of course have to be careful to make C so big that it supports this automorphism. Now restrict the attention to a band from radius 1 out to a radius which is L plus the bound of the deformation. This region is being preserved, and since the automorphism is bounded, independently of L, and only different from the identity in a band around radius L. This automorphism, when measured only in K, forgetting the radial direction by using the radial projection x → \frac{x}{|x|} can now be made as small as wanted by choosing L large. The domain of this radial projection is the region between a sphere of radius 1 and a sphere of radius a little larger than L intersected with O(K) mapped to K = O(K)∩S^{n}. (Remember we are using the max metric, so the unit sphere is the boundary of the unit cube). This radial projection is a map of compact spaces so we do get an induced map at the category level. Under this map L · K is sent to K by a map decreasing distances by the factor L, so we get the desired result.

Proof of Theorem 3.6. We begin by considering the case K = ∗ even though this case does not exhibit the typical behavior. In this case, O(K) is a ray and we have the picture below:

A  0  0  0  0

A is supported outside the unit ball. In the picture we illustrate the case where A is supported on an embedded copy of K, but it is obvious that the arguments to follow only need A supported outside the unit ball. We now take B to be an infinite direct sum of copies of A, one at each “integral” point n ∈ O(K) for n ≥ 2. We represent α ⊕ id schematically in the picture below:

[figure: A; α; A; id; A; id; A; id; A; id]

The 6 term identity allows us to multiply by a product ∏ e_{η_{i}} of elementary matrices to obtain the picture:

[figure: A; α; A; α^{−1}; A; α; A; α^{−1}; A; α]

Shifting and using the 6 term identity again, we see that multiplication by a suitable ∏ e′_{η_{i}} gives us:

[figure: A; id; A; id; A; id; A; id; A; id]

We have therefore obtained that (α ⊕ id) · (∏ e_{η_{i}})^{−1} · ∏ e′_{η_{i}})^{−1} = id, as desired. In this too simple case α is bounded by 0, but evidently we have used 12 elementary operations, six of which have ±α and ±α^{−1} and six of which have ±1 off the diagonal. The latter are bounded by 1 because we placed the modules at the integral lattice points, but by stabilizing more, i. e. by placing the modules closer together these may be as controlled as one may want. This way we only lose control by a factor of 6, which in this 0-dimensional case is precisely 6^{dim(K)+1} even though in this too simple situation the control of α is 0, so nothing much happens in this case.

We now move to the case K = S^{1}, which is more representative of the general case. We would like to perform the same sort of Eilenberg swindle, but now pushing the module A to infinity in all directions at once increases the bound unacceptably.

The solution to this difficulty is to apply a variation of the swindle rigidly to the topdimensional faces and induct on dimension. Quinn points out that it is the interplay between Cartesian and polar coordinates that is being exploited. Let {F_{i}} be the collection of topdimensional faces of K. We stabilize using infinitely many copies of A|F_{i} with the identity automorphism. Doing this to all of the top-dimensional faces at once, we have the picture below

S⋮ S _{···}  _{···} S S⋮

Let α|(A|F_{i}) be the composition (A|F_{i}) → A ⟶^{α} A → (A|F_{i}) and define α^{−1}|(A|F_{i}) similarly. Note that bound(α|F_{i}) ≤ bound(α). Since α|(A|F_{i}) need not be invertible, we cannot use the earlier identities directly. We can, however, consider a similar product of elementary matrices:

∏ e_{η_{i}} = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ −1 & 1 \end{matrix}) (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) ((\begin{matrix}1 & 0 \\ α|F_{i} & 1 \end{matrix}) (\begin{matrix}1 & −(α^{−1}|F_{i}) \\ 0 & 1 \end{matrix}) (\begin{matrix}1 & 0 \\ α|F_{i} & 1 \end{matrix})

We can choose the copies of A|F_{i} close together so that ∑ bound(e_{η_{i}}) < 3δ and so that ∏ e_{η_{i}}

is equal to (\begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix}) outside of a 3δ-neighborhood of ∪_{i}∂F_{i}. Applying the swindle as before,

we have ∏ e_{η_{i}} · ∏ e′_{η_{i}} · (α ⊕ id) = 1 outside of a 6δ-neighborhood of ∪_{i}∂F_{i}, and therefore in a neighborhood of the rays denoted S, provided that δ < \frac{1}{6}. The \frac{2}{3}-factor ensures that we actually have the identity in the middle third of the band around the ray S. The operations on the separate faces are disjoint, so corresponding operations can be combined into single elementary matrices and the bounds need not be summed over the faces. Destabilizing, we excise the area where ∏ e_{η_{i}} ·∏ e′_{η_{i}} ·(α⊕id) = 1, obtaining modules in four separate quadrants and an automorphism that is forced to preserves each quadrant by the control condition, at least after a further application of an elementary operation that does not increase control.

To be precise, let B be the union of the two axis in ℝ^{2}. we write ℝ^{2} = B ∪ ⋃_{i=1}^{4} Q_{i} where Q_{i} are the four quadrants. On A|B the automorphism is now the identity, actually it is the identity in a little neighborhood, and the control conditions ensure that the automorphism preserves A|Q_{i}. We can eliminate A|Q_{i} by performing the infinite repetition trick in the four directions of 45 degree in each quadrant. That is, we stabilize by making infinitely many

identical copies of the nonzero modules and then we apply the ((\begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix}) identity twice to

write α ⊕ id (on the nose, this time) as a product of elementary matrices. This completes the case K = S^{1}. Since the last set of elementary matrices is constructed using modified α’s, which lead to a loss of control by factor of 6, they are bounded by 6^{2} · δ. This is because in each dimension we lost a factor of 6.

Suppose, now, that K is a cubical subcomplex of S^{2}. We begin by performing the infinite repetition trick along a ray perpendicular to each top-dimensional face. Destabilizing, this eliminates all of the modules inside a distance of \frac{1}{3} of the rays perpendicular to the topdimensional faces.

Note that we have eliminated all modules A_{x} such that x has coordinates (x_{1}, x_{2}, x_{3}) with |x_{i}| < 1 − 6δ for two values of i. Next, we perform the infinite repetition trick along the 45 degree angled arrows to eliminate all modules A_{x} such that x has coordinates with |x_{i}| < 1−36δ for some value of i. After this operation, the nonzero modules are concentrated into separate octants provided, of course, that δ < \frac{2}{3} · \frac{1}{36} and can be eliminated by doing swindles in directions (δ_{1}, δ_{2}, δ_{3}), where |δ_{i}| = 1.

The general induction proceeds in an entirely similar fashion. We begin by eliminating all modules A_{x} where x has n − 1 coordinates with absolute value < 1 − 6δ. This is done by swindling in directions with one coordinate equal to ±1 and the rest equal to 0. Next, we eliminate all A_{x} so that x has n − 2 coordinates with absolute value < 1 − 36δ. This is done by pushing in directions with two coordinates equal to ±1 and with the rest equal to 0. Continuing, we eventually have no remaining nonzero modules A_{x} with any coordinates of x having absolute values < 1 − 6^{(n−1)}δ. At this point, the remaining nonzero modules are isolated in the 2^{n} “octants” and can be eliminated by swindling in directions with all coordinates equal to ±1. Notice we may do this process to all of S^{n}, not just the cubical subcomplex of S^{n}. In the end the modules will have support in O(K) since we are stabilizing with the zero modules on the faces of S^{n} that are not in K. A good example to think of is the 1-skeleton of S^{2} where the first step in the argument is an empty step.

Addendum 3.8. As mentioned before, the argument we have given above actually works for any automorphism α : A → A in C_{O(K)}(R) such that α and α^{−1} are bounded by δ and such that A_{x} = 0 for x inside of the unit ball. The point is that the induction given above naturally starts with any α such that A_{x} = 0 for all x with all coordinates < 1.

We want to prove that the squeezing is unique. For this, we will need a K_{2} analog of Theorem 3.6. We begin by recalling Bass’ definition of K_{2} of an additive category. [1] Definition 3.9. Let A be an additive category with a set of isomorphism classes of objects. Assume A = A_{1} ⊕ A_{2}, then we get a ring homomorphism End(A_{1}) → End(A) sending η to η ⊕ 0. Bass defines

K_{2}(A) = lim_{A} K_{2}(End(A))

where the limit is taken over objects in A and inclusions as described above.

Conjugating by an automorphism of A induces the identity on K_{2}(End(A)) so there is no trouble in defining the above limit. C. Weibel proved in [13] that Bass’ definition agrees with Quillen’s definition of K_{2} of the symmetric monoidal category obtained by restricting morphisms to isomorphisms. Since endomorphisms in a sum of r copies of A can be written as an r × r matrix with entries in End(A),

End(A^{⊕r}) = M(r, End(A)),

we can think of an element of K_{2}(A) as a product of Steinberg symbols in End(A), for some object A in A which when evaluated as a product of elementary matrices gives the identity. More commonly we will be given a product of automorphisms of A = A_{1} ⊕ A_{2} ⊕ … ⊕ A_{r} which are the identity except for a component sending A_{i} to A_{j}, i ≠ j i. e. elementary automorphisms with product equal to the identity, and we want to think of this as an element in K_{2}(A), but this may be done by stabilizing each A_{i} by ⊕_{j≠i}A_{j} to get an element in K_{2}(End(A)). Whenever we have a product of elementary matrices equal to the identity, we thus get an element of K_{2} of the category by replacing the elementary automorphisms by Steinberg symbols in the obvious way.

Theorem 3.10. (K_{2}-vanishing) Let K be a cubical subcomplex of the boundary of the unit ball in ℝ^{n}. If {e_{η_{i}}} are elementary automorphisms of a module A in C_{O(K)}(R) with A^{<1} = 0,

and such that ∏_{i=1}^{k} e_{η_{i}} = 1 and ∑_{i=1}^{k} bound(e_{η_{i}}) < δ < 6^{−(dim K−1)} in the max metric, then the

corresponding element in K_{2}(C_{O(K)}(R) is trivial.

Proof. The proof is very similar to the proof of the squeezing theorem. We shall illustrate the proof by the case K = S^{1} = ∂([−1, 1] × [−1, 1]).

Consider the four regions B_{1} = [1, ∞) × [−1, 1], B_{2} = [−1, 1] × [1, ∞), B_{3} = (−∞, −1] × [−1, 1] and B_{4} = [−1, 1] × (−∞, −1]. The strategy of the proof will be to write the element in K_{2} as an element with support in B_{i} multiplied by an element with support in the complement and such that the control conditions ensure that the elements in the four quadrants C_{i} whose union is the complement are independent. This will then ensure the element is trivial since

K_{∗}(C_{B_{i}}(R)) = K_{∗}(C_{C_{i}}(R)) = 0

by Eilenberg swindles.

Consider a product of elementary matrices e_{η_{1}} · e_{η_{2}} · . . . · e_{η_{k}} = 1 and let x_{η_{1}} · x_{η_{2}} · . . . · x_{η_{k}} be the corresponding element in the Steinberg group giving a K_{2} element in C_{O(K)}(R). Write O(K) = B_{1} ∪ D_{1} as a disjoint union, and write the modules on which the e′_{η_{i}}s are realized as

A = (A|B_{1})⊕(A|D_{1}). In this direct sum decomposition η_{k} may be written as η_{k} = \{ \begin{matrix}η_{11}^{k} & η_{12}^{k} \\ η_{21}^{k} & η_{22}^{k} \end{matrix} \}, where η_{11}^{k} preserves (A|B_{1}) and η_{22}^{k} preserves (A|D_{1}). Clearly

η_{k} = (\begin{matrix}η_{11}^{k} & 0 \\ 0 & 0 \end{matrix}) + (\begin{matrix}0 & η_{12}^{k} \\ η_{21}^{k} & η_{22}^{k} \end{matrix})

so a Steinberg relation allows us to replace x_{η_{k}} by

x_{\{ \begin{matrix}η_{11}^{k} & 0 \\ 0 & 0 \end{matrix} \}} · x\{ \begin{matrix}0 & η_{12}^{k} \\ η_{21}^{k} & η_{22}^{k} \end{matrix} \}

The morphism e\{ \begin{matrix}0 & η_{12}^{k} \\ η_{21}^{k} & η_{22}^{k} \end{matrix} \} will only reach a tiny amount into the B_{1}-region, and we choose

B′_{1} to be [1, ∞) × [−1 + μ, 1 − μ] where μ is chosen as small as possible so that e\{ \begin{matrix}0 & η_{12}^{k} \\ η_{21}^{k} & η_{22}^{k} \end{matrix} \}

does not reach into the region B′_{1}. We now decompose using a Steinberg relation

x_{η_{k−1}} = x_{\{ \begin{matrix}0 & η_{12}^{k−1} \\ η_{21}^{k−1} & η_{22}^{k−1} \end{matrix} \}} · x_{\{ \begin{matrix}η_{11}^{k−1} & 0 \\ 0 & 0 \end{matrix} \}}

based on B′_{1} and its complement, but since η_{11}^{k−1} preserves B′_{1} and \{ \begin{matrix}0 & η_{12}^{k} \\ η_{21}^{k} & η_{22}^{k} \end{matrix} \} does not reach into B′_{1} we may commute x_{\{ \begin{matrix}η_{11}^{k−1} & 0 \\ 0 & 0 \end{matrix} \}} and x\{ \begin{matrix}0 & η_{11}^{k} \\ η_{21}^{k} & η_{22}^{k} \end{matrix} \} using a Steinberg relation. Continuing

this process we get

x_{η_{1}} · x_{η_{2}} ⋯ x_{η_{k}} =

x\{ \begin{matrix}0 & η_{12}^{1} \\ η_{21}^{1} & η_{22}^{1} \end{matrix} \} ⋯ x\{ \begin{matrix}0 & η_{12}^{k} \\ η_{21}^{k} & η_{22}^{k} \end{matrix} \} · x_{\{ \begin{matrix}η_{11}^{1} & 0 \\ 0 & 0 \end{matrix} \}} ⋯ x_{\{ \begin{matrix}η_{11}^{k} & 0 \\ 0 & 0 \end{matrix} \}}

in the Steinberg group. The product of the corresponding elementary matrices is of course still the identity, but

e_{\{ \begin{matrix}η_{11}^{1} & 0 \\ 0 & 0 \end{matrix} \}} ⋯ e_{\{ \begin{matrix}η_{11}^{k} & 0 \\ 0 & 0 \end{matrix} \}}

is an automorphism, say α satisfying the same control as the original deformation, with support in a small band around [1, ∞)×{−1, 1}. Such an automorphism may be deformed to the identity using the usual α, α^{−1} trick along the ray [1, ∞), and putting these deformations in the middle leads to a description of the given K_{2}-element as a product of 2 elements, one with support in B_{1} and one with support in a small neighborhood of the complement. We loose a factor of 6 in the control as usual when we deform α to the identity. The above process may be done simultaneously on B_{1}, … B_{4} in the end writing the given K_{2}-element as a product of 8 elements each with support in B_{i} or C_{i} and hence all 0.

The general case proceeds as in the squeezing theorem generalizing this idea, starting with the top-dimensional simplices, noticing that on the faces not belonging to K nothing is done precisely as in the squeezing theorem.

4. Bounded algebra and geometric groups

Definition 4.1. ([3] and [8]) Let K be a finite polyhedron. A geometric ℤπ -module on K is an object A in C_{K}(ℤπ). A deformation is a composable string e_{1}, …, e_{n} of elementary isomorphisms. The bound of a deformation is the sum ∑ bound(e_{η_{i}}). A δ-isomorphism is an isomorphism α : A → B such that bound(α) and bound(α^{−1}) are both less than δ. A δ-isomorphism α : A → B is geometric if α is induced by a bijection of basis sets. We will identify α with α ⊕ id : A ⊕ C → B ⊕ C.

Here is Quinn’s Stability Theorem. For clarity, we state the theorem for finite polyhedra. The generalization to locally compact ANR’s is not difficult.

Theorem 4.2. ([9, p. 381]) Suppose that K is a finite polyhedron represented as a cubical subcomplex of the boundary of the unit cube in ℝ^{n}. Then there is an ε_{K} > 0 such that for every δ < ε_{K}, and automorphism α in the category C_{K}(R), bounded by δ, there is a 6^{dim K+1} · δ-deformation to the identity if and only if an invariant σ(α) ∈ H^{1}(K; K^{−∞}(R)) vanishes. Any element in H^{1}(K; K^{−∞}(R)) may be realized by such an α with arbitrarily small prescribed control. We may take ε_{K} = \frac{2}{3} · 6^{− dim K}.

Quinn’s proof is a torus argument, and extracting explicit bounds would be painful. Working directly with bounded topology avoids the torus and makes the argument remarkably concrete.

Proof. Let K^{+} be the disjoint union of K and a basepoint. Given α as above, we will now describe an element σ(α) ∈ K_{2}(C_{O(K+)}(R)) for δ sufficiently small. Recall that Theorem 2.3 tells us that K_{2}(C_{O(K+)}(R) = H_{1}(K; K^{−∞}(R)). By Theorem 3.6, α ⊕ id = ∏ e_{η_{i}}, where the e_{η_{i}}’s are elementary automorphisms in C_{O(K)}(R) which are supported outside of B_{1} ∩ O(K), B_{1} being the unit ball in ℝ^{n}. On the other hand, the usual swindle through the origin and “out the tail” O(+), writes α⊕id = ∏ e_{ξ_{i}}, where the e_{ξ_{i}} are supported on O(+)∪(O(K)∩B_{1}). Thus, (∏ e_{ξ_{i}})^{−1} · ∏ e_{η_{i}} = id. We define

σ(α) = (∏ x_{ξ_{i}})^{−1}(∏_{i=1}^{12} x_{η_{i}}) ∈ K_{2}(C_{O(K^{+})}(R)).

We need to see this is well defined, so suppose given two such deformations. The difference will then be supported on O(K) outside K, and the difference is 0 by Theorem 3.10. To see that the vanishing of σ(α) ensures a small deformation to the identity, suppose that σ(α) represents 0 ∈ K_{2}(C_{O(K_{+})}(R)). This means that using Steinberg relations, σ(α) can be written as a product of Steinberg relations. We do not a priori know how many of these relations are needed. But starting with the original representative σ(α), we may cut down to a band from 1 to L, L large, to obtain a small deformation from α to some much smaller automorphism α′. At this point we have not used any Steinberg relations, so we have precise control of the size of the deformation. Operating on σ(α) by Steinberg relations and cutting at L gives deformations of α′, but we know how many Steinberg relations are needed to trivialize σ(α), so we may choose L so large that the deformation from α′ to a trivial automorphism can be made as small as we wish. This two-stage process is necessary to keep control since there is no known upper bound on the number of Steinberg relations needed to trivialize a given element in K_{2}, and we only know the elementary automorphisms corresponding to the Steinberg relations are bounded not a priory what the bound is. The key point here is that cutting a Steinberg relation produces an elementary automorphism, and we only have to choose L after we know the reason for the K_{2}-element being 0.

We thus obtain the deformation to the identity by first using the deformation of α to α′, where α′ is obtained by cutting at a very large L, and then use the fact that Steinberg relations produce elementary automorphisms to show that α′ can be written as a product of elementary automorphisms of controlled size. To see that all obstructions are realized, assume an element β ∈ K_{2}(C_{O(K_{+})}(R)) is given. Cutting at a large L produces an automorphism α as small as we wish hence we get an element of K_{1}^{δ}(C_{K}(R)) by choosing L sufficiently large. We also get a product of elementary matrices with support outside L whose product is the identity except in a region around L where we get α. We want to show that σ(α) = β. The difference however can be written as a product of an element with support inside L and a sufficiently bounded element with support outside L and these are both 0, so σ([α]) = β. (see the final section for more details of this type of argument).

5. An algebraic proof of topological invariance of Whitehead torsion

In this section we give a quick proof that a homeomorphism of finite PL-complexes has trivial Whitehead torsion, using the techniques developed in this paper. Let K and L be finite complexes π = π_{1}(K), and let h : L → K be a homeomorphism. The Whitehead torsion is given as follows: approximate h by a PL map f and consider the induced map f_{#} : C_{#}(L̃) → C_{#}(K̃), a chain homotopy equivalence of based ℤπ-chain complexes. Let C_{#} denote the mapping cylinder of f_{#}, then C_{#} is a contractible chain complex of based ℤπmodules so α = s + ∂ : ⊕C_{even} → ⊕C_{odd} is an isomorphism of based modules. The modules must have the same rank, so we can identify the modules by identifying basis. This way we obtain an automorphism, whose torsion in K_{1}(ℤπ) represents the Whitehead torsion of h. Since there are choices involved this is only well defined in Wh(π) = K_{1}(ℤπ)/±π, where the ± should be interpreted as K_{1}(ℤ). (We obviously have to divide out by all permutations of the generators, but these permutations generate K_{1}(ℤ)).

We want to make a couple of inessential reinterpretations of this. The category C_{K}(ℤπ) is equivalent to the category of finitely generated free ℤπ-modules, and associating the barycenter of each simplex to the generator the simplex represents, the chain complexes C_{#}(K) and C_{#}(L) can be thought of as chain complexes in C_{K}(ℤπ) and we can compute the torsion in K_{1} of this category. Refining a bit further consider the universal cover of K, K̃. Consider the category Cπ ; K̃(ℤ) the subcategory of π-invariant ℤ-modules parameterized by K̃. This is actually the most obvious way to think of the chain complexes C_{#}(K) and C_{#}(L). We are using the chain complexes of the universal cover anyway, and we remember the point in K̃ where the barycenter of a simplex is (or is mapped to in the case of C_{#}(L)).

The method we intend to use is to refine the torsion of a homeomorphism to the point where it lives in a group which algebraically maps to zero in the Whitehead group. First we want to refine the element to a K_{2}-element in an associated category. Consider C_{+}Cπ ; K̃(ℤ)^{∞}. Objects are π-invariant ℤ-modules parameterized equivariantly by K̃ × ℤ_{+}, and morphisms are germs at infinity of bounded morphisms invariant under the π-action. This means we identify morphisms if they agree on objects with ℤ_{+}-coordinate sufficiently large. Given an automorphism in Cπ ; K̃(ℤ), (A, α), consider the object (A, A, A, …) with the i’th copy of A placed at the i’th coordinate in ℤ_{+}. The automorphism

(α, 1, 1, …)

represents the identity in C_{+}Cπ ; K̃(ℤ)^{∞} because the germ at infinity is the identity. There is an obvious deformation of (α, 1, 1, 1, …) (which represents the identity in the category) to the identity through (α, α^{−1}, α, α^{−1}, …) and further to (1, 1, 1 …), using the six term identity, setting the parenthesis two different ways. This defines an element in K_{2}(C_{+}Cπ ; K̃(ℤ)^{∞}) which we shall denote by [α].

Lemma 5.1. There is an epimorphism K_{2}(C_{+}Cπ ; K̃(ℤ)^{∞}) → K_{1}(Cπ ; K̃(ℤ)) sending [α] to [(A, α)].

Proof. The map we define is actually an isomorphism, but we have no need for that in the present argument. Given an element in K_{2} it is represented by a product of elementary automorphisms in the germ category with product the identity. Choosing representatives we get a product of elementary automorphisms with product the identity except on modules with ℤ_{+}- coordinate smaller than some number l. Cutting the product of elementary automorphisms at some k larger than l plus the bound r of the deformation produces an automorphism which is the identity on modules with ℤ_{+} coordinate less than k − r and bigger than k + r, so summing the modules in the region k − r to k + r produces an object in Cπ ; K̃(ℤ) together with an automorphism. It is easy to see that a Steinberg relation gives rise to an elementary automorphism, thus the 0-element in K_{1}, and that products are sent to products, direct sums to direct sums, so we have a well defined map from K_{2}(C_{+}Cπ ; K̃(ℤ)^{∞}) → K_{1}(Cπ ; K̃(ℤ)).

Inspection shows that [α] is sent to [(A, α)]. More precisely consider the diagram

\xymatrix{A \ar[d]_{1} \ar[dr]^{α} & A \ar[d]^{1} & A \ar[d]_{1} \ar[dr]^{α} & A \ar[d]^{1} & A \ar[d]_{1} \ar[dr]^{α} & A \ar[d]^{1} \\ A \ar[d]_{1} & A \ar[dl]_{−α^{−1}} \ar[d]^{1} & A \ar[d]_{1} & A \ar[dl]_{−α^{−1}} \ar[d]^{1} & A \ar[d]_{1} & A \ar[dl]_{−α^{−1}} \ar[d]^{1} \\ A \ar[d]_{1} \ar[dr]^{α} & A \ar[d]^{1} & A \ar[d]_{1} \ar[dr]^{α} & A \ar[d]^{1} & A \ar[d]_{1} \ar[dr]^{α} & A \ar[d]^{1} \\ A \ar[d]_{1} & A \ar[dl]_{1} \ar[d]^{1} & A \ar[d]_{1} & A \ar[dl]_{1} \ar[d]^{1} & A \ar[d]_{1} & A \ar[dl]_{1} \ar[d]^{1} \\ A \ar[d]_{1} \ar[dr]^{−1} & A \ar[d]^{1} & A \ar[d]_{1} \ar[dr]^{−1} & A \ar[d]^{1} & A \ar[d]_{1} \ar[dr]^{−1} & A \ar[d]^{1} \\ A \ar[d]_{1} & A \ar[dl]_{1} \ar[d]^{1} & A \ar[d]_{1} & A \ar[dl]_{1} \ar[d]^{1} & A \ar[d]_{1} & A \ar[dl]_{1} \ar[d]^{1} \\ A & A & A & A & A & A}

This is the deformation of the identity to (α, α^{−1}, …) (the matrix multiplication is computed by adding all the ways one can get from one point in the diagram to another). Cutting at a certain level consists of replacing all diagonal arrows by 0 to the left of a certain point. Consider the identity as the composite

\xymatrix{A \ar[d]_{α} & A \ar[d]_{α^{−1}} & A \ar[d]_{α} & A \ar[d]_{α^{−1}} & A \ar[d]_{α} & A \ar[d]_{α^{−1}} \\ A \ar[d]_{α^{−1}} & A \ar[d]_{α} & A \ar[d]_{α^{−1}} & A \ar[d]_{α} & A \ar[d]_{α^{−1}} & A \ar[d]_{α} \\ A & A & A & A & A & A}

and replace the upper arrows by the diagram above, and the lower arrows by the diagram above shifted. We see that cutting in the middle we get

\xymatrix{A \ar[d]_{1} & A \ar[d]_{1} & A \ar[d]_{1} & A \ar[d]_{1} & A \ar[d]_{α} & A \ar[d]_{α^{−1}} \\ A \ar[d]_{1} & A \ar[d]_{1} & A \ar[d]_{1} & A \ar[d]_{α} & A \ar[d]_{α^{−1}} & A \ar[d]_{α} \\ A & A & A & A & A & A}

which is the identity except on the fourth module where it is α as claimed.

Since we started with a homeomorphism h and we can triangulate as finely as we need, and PL-approximate as closely as needed, we have

Lemma 5.2. We can find representatives of the Whitehead torsion of h, (A_{n}, α_{n}) satisfying the following conditions:

(i) α_{n} and α_{n}^{−1} are bounded by \frac{1}{n}. (ii) There is a deformation of (A_{n} ⊕ A_{n+1}, α_{n} ⊕ α_{n+1}^{−1}), and hence also of (A_{n} ⊕ A_{n+1}, α_{n}^{−1} ⊕

α_{n+1}) bounded by \frac{1}{2n}.

Proof. By triangulating and PL-approximating finely we may obtain an isomorphism β_{n} : A_{n} → B_{n} smaller than any given r, so choose r < \frac{1}{n}. Here A_{n} is the sum of the odd dimensional cellular chains, and B_{n} the sum of the even-dimensional cellular chains in the mapping cylinder of the chain complex of the cellular chains K and L. Let δ = \frac{1}{n} −r. We may construct an isomorphism γ_{n} from B_{n} to A_{n} with bound smaller than δ, sending generators to generators at least after stabilizing, but stabilizing is no problem, we just replace A_{n} and B_{n} by the stabilized modules. To see this consider a generator g_{b} in B_{n} and a generator g_{a} in A_{n}. Assume g_{a} is at the point x ∈ K̃ and g_{b} at the point y ∈ K̃. We may now choose a finite sequence of points in K̃, x = x_{0}, x_{1}, …, x_{m} = y such that the distance from x_{n−1} to x_{n} is less than δ. Stabilize A_{n} and B_{n} by generators g_{i} at x_{i} , i = 1, …, m − 1 and the isomorphism β_{n} by the identity, and define γ_{n}(g_{i}) = g_{i−1}. At this point we only need to define g_{n} on a free module with one less generator than before, and the proof is completed by induction. We may now define α_{n} = γ_{n}β_{n}, an automorphism that clearly represent the torsion of h. To see the second part, notice that if we have two fine PL-approximations they are homotopic by a small homotopy. We may change the homotopy to be PL relative to endpoints. Denote the PL homotopy by H. We may now use C_{#}(H) to compute the torsion, but the torsion of C_{#}(H) is connected to α_{n} and α_{n+1} by a sequence of collapses. It is however easy to see that the inverse of a collapse algebraically consists of stabilizing and multiplying by an elementary matrix. Since the collapses can be made as small as we wish we are done.

This means we can find another K_{2} element in C_{+}Cπ ; K̃(ℤ)^{∞} representing the torsion of h namely given by the deformation first from (α_{1}, 1, 1, …) to (α_{1}, α_{2}^{−1}, α_{3}, …), and then to (1, 1, 1, …). Once again we see that if we cut this deformation at some large k we get (A_{n}, α_{n}) hence a representative for α. The advantage of this representation is that it remains a bounded deformation if we replace the metric on the n’th copy of K by a metric n times larger, and require the morphisms to be bounded with respect to this metric.

Thinking of K as embedded in {1} × ℝ^{m} for some big m, and O(K) as the set of rays in ℝ^{m+1} from 0 going through K. The category C_{O(K)}(ℤ) is equivalent to the full subcategory where the objects are 0 except when the first coordinate is an integer, and the level of O(K) with first coordinate equal to n is a copy of K with a metric which is precisely n times the metric in K. It is now easy to see that the subcategory of C_{+}Cπ ; K̃(ℤ)^{∞} with morphisms bounded in the metric where the n’th copy of K has the metric multiplied by n, is isomorphic to the category C_{O(K)}(ℤ)^{∞}. The point is that we may divide out the π-action, and since morphisms near infinity are allowed to move very little there is a canonical choice of morphisms near infinity. We have thus shown that the torsion of the homeomorphism h lies in the image of a map from K_{2}(C_{O(K)}(ℤ)^{∞}) to K_{1}(ℤπ), but according to [7] we have

K_{2}(C_{O(K)}(ℤ)^{∞}) = H_{1}(K_{+}; AlgK(ℤ))

which by the Atiyah-Hirzebruch spectral sequence is

H_{1}(K, K_{0}(ℤ)) ⊕ H_{0}(K; K_{1}(ℤ)) = H_{1}(K, ℤ) ⊕ H_{0}(K; ℤ_{2}) = π/[π, π] ⊕ ±1.

But this is precisely what we divide out from K_{1}(ℤπ) to get the Whitehead group, so the image is trivial in Wh(π) as claimed.

Remark 5.3. We could obviously give a different proof by representing the torsion by just one sufficiently small automorphism, and then use the squeezing results of the previous sections to produce the element in K_{2}. Since a homeomorphism is squeezed as much as possible it seems more natural to utilize that directly.

6. Pedestrian Algebraic K-theory

The proof of topological invariance of Whitehead torsion in the previous section does depend on the computation in [7] of K_{2}(C_{O(K_{+})}(R)), and thus of the whole machinery of higher algebraic K-theory. This however is not necessary, it is perfectly possible to compute this group by elementary (pedestrian) methods which we shall proceed to indicate in this section.

Suppose M is a metric space, M_{1} and M_{2} metric subspaces. We shall consider the category C_{M}(R) and various subcategories. Abusing notation we denote the full subcategories on objects with support in a bounded neighborhood of M_{i} by C_{M_{i}}(R). This is not too bad because these subcategories are obviously equivalent to C_{M_{i}}(R). We shall denote the full subcategory on objects with support in a bounded neighborhood of M_{1} intersected with a bounded neighborhood of M_{2} by C_{M_{1},M_{2}}(R). This could be quite different from C_{M_{1}∩M_{2}}(R), in particular M_{1}∩M_{2} could be empty, but it does represent the intersection in the “bounded” sense, and in many favorable cases it is true that this category is equivalent to C_{M_{1}∩M_{2}}(R) in particular this will be the case if we take finite complexes K = K_{1} ∪K_{2} and put M_{i} = O(K_{i}) which is the case we have on mind.

Theorem 6.1. With assumptions as above, denoting the maps induced by inclusions by i_{i} and j_{i}, there is an exact sequence

K_{2}(C_{M_{1},M_{2}}(R)) ⟶^{(i_{1},i_{2})} K_{2}(C_{M_{1}}(R)) ⊕ K_{2}(C_{M_{2}}(R)) ⟶^{j_{1}−j_{2}} K_{2}(C_{M}(R)) ⟶^{∂}

K_{1}(C_{M_{1},M_{2}}(R)) ⟶^{(i_{1},i_{2})} K_{1}(C_{M_{1}}(R)) ⊕ K_{1}(C_{M_{2}}(R)) ⟶^{j_{1}−j_{1}} K_{1}(C_{M}(R)) ⟶^{∂}

K_{0}(C_{M_{1},M_{2}}(R)) ⟶^{(i_{1},i_{2})} K_{0}(C_{M_{1}}(R)) ⊕ K_{0}(C_{M_{2}}(R)) ⟶^{j_{1}−j_{2}} K_{0}(C_{M}(R)) ⟶^{∂}

K_{−1}(C_{M_{1},M_{2}}(R)) ⟶^{(i_{1},i_{2})} K_{−1}(C_{M_{1}}(R)) ⊕ K_{−1}(C_{M_{2}}(R)) ⟶^{j_{1}−j_{2}} K_{−1}(C_{M}(R)) ⟶^{∂} …

where K_{0} should be interpreted as K_{0} of the idempotent completion, or K_{1} of the category where M and M_{i} have been crossed with ℝ, K_{−1} similarly as K_{1} after crossing with ℝ^{2} etc.

Proof. This follows immediately from [7], see also [2] for an easier proof, but our aim here is to avoid higher Algebraic K-theory. The lower part of the sequence is proved by a slight generalization of the methods in [5] so we shall concentrate on the higher part. The boundary map is defined as follows: Given an element in K_{2}(C_{M}(R)) i. e. ∏_{i=1}^{k} e_{η_{i}} each bounded by some r. We cut the elementary matrices in a bounded neighborhood of M_{1} the following way: We write η_{i} = μ_{i} + ν_{i} where μ_{i} has support in a (k − i + 1) · r neighborhood of M_{1} and ν_{i} has support outside a (k − i) · r neighborhood of M_{1}. This means that, using Steinberg relations e_{ν_{i}} commutes with e_{μ_{j}} when i < j, so using Steinberg relations we get

e_{η_{1}} · . . . · e_{η_{k}} = e_{μ_{1}} · e_{ν_{1}} · . . . · e_{μ_{k}} · e_{ν_{k}} = e_{μ_{1}} · . . . · e_{μ_{k}} · e_{ν_{1}} · . . . · e_{μ_{k}}

The product ∏ e_{μ_{i}} will no longer be the identity, but it will be an automorphisms which is the identity except for a bounded neighborhood of M_{1} intersected with a bounded neighborhood of M_{2} thus defining an element in K_{1}(C_{M_{1},M_{2}}(R)). It is easy to see this is well defined. If we cut differently, the difference will be given by an elementary automorphism with support in a bounded neighborhood of M_{1} ∩ M_{2}. If the element is 0 in K_{1}(C_{M_{1},M_{2}}(R)) we may write it as a product of elementary matrices ∏ e_{λ_{j}} , and we see that we may write the original element in K_{2}

∏ e_{η_{i}} = ∏ e_{μ_{i}} · ∏ e_{ν_{i}} = ∏ e_{μ_{i}} · (∏ e_{λ_{j}})^{−1} · ∏ e_{λ_{j}} · ∏ e_{ν_{i}}

thus as the difference of an element ∏ e_{μ_{i}} ∏(e_{λ_{j}})^{−1} coming from K_{2}(C_{M_{1}}(R)) and an element (∏ e_{λ_{j}} · ∏ e_{ν_{i}})^{−1} coming from K_{2}(C_{M_{2}}(R)). While it is tedious to work out all the details the rest of the proof is elementary.

Writing ℝ^{k+2} as a union of two halfspaces along ℝ^{k+1} and using an Eilenberg swindle on the two half spaces it follows from this that K_{2}(C_{ℝ^{k+2}}(R)) ≅ K_{1}(C_{ℝ^{k+1}}(R)) which in turn is isomorphic to K_{−k}(R) by [5]. As remarked above when K is the union of two PLsubcomplexes K_{1} and K_{2} we have C_{O(K_{1}),O(K_{2})}(R) ≃ C_{O(K_{1}∩K_{2})}(R) and this thus provides sufficient information to compute K_{2}(C_{O(K)}(R)) by the usual Mayer-Vietoris arguments, attaching one cell at a time.

References

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

[2] M. Cárdenas and E. K. Pedersen, On the Karoubi filtration of a category, K-theory 12 (1997), 165–191.

[3] E. H. Connell and J. Hollingsworth, Geometric groups and Whitehead torsion, Trans. Amer. Math. Soc. 140 (1969), 161–181.

[4] E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser, Basel, 1990.

[5] E. K. Pedersen, On the K_{−i} functors, J. Algebra 90 (1984), 461–475.

[6] E. K. Pedersen and C. Weibel, A nonconnective delooping of algebraic K-theory, Algebraic and Geometric Topology (Rutgers, 1983), Lecture Notes in Mathematics, vol. 1126, Springer-Verlag, Berlin-New York, 1985, pp. 166–181.

[7] , K-theory homology of spaces, Algebraic Topology (Arcata, 1986), Lecture Notes in Mathematics, vol. 1370, Springer-Verlag, Berlin-New York, 1989, pp. 346–361.

[8] F. Quinn, Ends of maps, I, Ann. of Math. (2) 110 (1979), 275–331.

[9] , Ends of maps, II, Invent. Math. 68 (1982), 353–424.

[10] A. A. Ranicki and M. Yamasaki, Controlled K-theory, Topology Appl. 61 (1995), 1–59.

[11] J. Roe, Exotic cohomology and index theory, Bull. Amer. Math. Soc. (N.S.) 23 (1990), 447–454.

[12] J. Rosenberg, K and KK: Topology and Operator Algebras, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), Proc. Symp. Pure. Math., vol. 51, Amer. Math. Soc., Providence R. I., 1990, pp. 445–480.

[13] C. A. Weibel, K-theory of Azumaya Algebras, Proc. Amer. Math. Soc. 81 (1981), 1–7.

Department of Mathematical Sciences, SUNY at Binghamton, Binghamton, NY 13901