ALGEBRAIC K-THEORY WITH CONTINUOUS CONTROL AT INFINITY

DOUGLAS R. ANDERSON, FRANCIS X. CONNOLLY, STEVEN C. FERRY, AND ERIK K. PEDERSEN

Abstract. Let (\overline{E}, Σ) be a pair of spaces consisting of a compact Hausdorff space \overline{E} and a closed subspace Σ. Let 𝔄 be an additive category. This paper introduces the category B(\overline{E}, Σ; 𝔄) of geometric modules over E with coefficients in 𝔄 and with continuous control at infinity. One of the main results is to show that the functor that sends a CW comples X to the algebraic K-theory of B(cX, X; 𝔄) is a homology theory. Here cX is the closed cone on X and X is its base.

The categories B(\overline{E}, Σ; 𝔄) are generalizations of the categories C(Z; 𝔄) of geometric modules and bounded morphisms introduced by Pedersen and Weibel [8]. Here (Z, ρ) is a complete metric space. If X is a finite CW complex and O(X) is the metric space open cone on X considered in [9], then there is an inclusion of categories C(O(X); 𝔄) → B(cX, X; 𝔄). A second main result is that this inclusion induces an isomorphism on K-theory.

One advantage of the present approach is that B(\overline{E}, Σ; 𝔄) depends only on the topology of (\overline{E}, Σ) and not on any metric properties. This should make application of these ideas to problems involving stratified spaces, for example, more direct and natural.

0. Introduction

Let Z be a metric space with complete metric ρ, 𝔄 be an additive category and C(Z; 𝔄) be the category of geometric modules on Z with coefficients in 𝔄 and bounded morphisms introduced and studied in [8] and [9]. We recall that an object in C(Z, 𝔄) is a collection A = {A_{x} ∈ 𝔄|x ∈ Z} with the property that for every bounded set K ⊂ Z, {x ∈ K|A_{x} ≠ 0} is finite. A morphism f : A → B is a collection f = {f_{y}^{x}|f_{y}^{x} : A_{x} → B_{y} is a morphism in 𝔄} for which there is a number d such that f_{y}^{x} = 0 if ρ(x, y) > d.

The categories C(Z; 𝔄) have been of interest and use both in algebra and topology. In [8], the categories C(ℝ^{n+1}; 𝔄) are used to construct a nonconnective spectrum K(𝔄) delooping the usual K-theory spectrym for 𝔄. In [9], the categories C(O(X); 𝔄), where O(X) is the open cone on X, are used to construct a homology theory on the category of finite CW complexes. Finally in [4], an equivariant version of these categories is used in the study of group actions.

The first three named authors were partially supported by the National Science Foundation under grants numbered DMS88-03149, DMS90-01729, and DMS90-03746, respectively.

2 DOUGLAS R. ANDERSON, FRANCIS X. CONNOLLY, STEVEN C. FERRY, AND ERIK K. PEDERSEN

In this paper we describe a new approach to these ideas that depends only on the topology of the space Z and not on its metric properties. The advantage of this approach is that it can be applied more naturally and more directly to some topological problems than the “bounded” approach. This is especially true of problems that arise in stratified settings. An application that exploits the greater flexibility this gives will be given in [1].

Let E be the category whose objects are pairs (\overline{E}; Σ) where \overline{E} is a compact Hausdorff space and Σ is a closed subspace. Let E = \overline{E} − Σ. (Althought we do not require it, we often think of E as being dense in \overline{E}. We also think of Σ as the “space at infinity” of E.) A morphism in E is a (not necessarily continuous) function f : \overline{E}_{1} → \overline{E}_{2} having F^{−1}(Σ_{2}) = Σ_{1}, for which f|E_{1} : E_{1} → E_{2} is proper in the sense that if K ⊂ E_{2} is compact, then f^{−1}(K) has compact closure in E_{1}, and for which f is continuous at every point z ∈ Σ_{1}.

Let (\overline{E}, Σ) ∈ E and 𝔄 be an additive category. We define a new additive category B(\overline{E}, Σ; 𝔄) as follows: An object of B(\overline{E}, Σ; 𝔄) is a collection A = \{A_{x} ∈ 𝔄|x ∈ E\} of objects of 𝔄 with the property that for each compact set K ⊂ E, {x ∈ K|A_{x} ≠ 0} is finite. A morphism f : A → A′ is a collection f = {f_{y}^{x}|f_{y}^{x} : A_{x} → A′_{y} is a morphism in 𝔄 and x, y ∈ E} that has the properties that for all x, {y|f_{y}^{x} ≠ 0} is finite and that for every point z ∈ Σ and every neighborhood U of z in \overline{E}, there is a neighborhood V of z such that f_{y}^{x} = 0 whenever x ∈ V and y ∉ U. (In particular, if y ∈ E and U = \overline{E} − \{y\}, the compactness of Σ implies that {x|f_{y}^{x} ≠ 0} is finite.) We describe this condition by saying that f is continuously controlled at infinity. The additive structure of B(\overline{E}, Σ; 𝔄) is given by setting A ⊕ A′ = {A_{x} ⊕ A′_{x}}.

We may regard B(−, −; 𝔄) as a functor from E to additive categories as follows: Let h : (\overline{E}_{1}, Σ_{1}) → (\overline{E}_{2} → Σ_{2}) be a morphism in E. We define a functor h_{∗} : B(\overline{E}_{1}, Σ_{1}; 𝔄) → B(\overline{E}_{2}, Σ_{2}; 𝔄) as follows: If A = \{A_{x}|x ∈ E_{1}\} is an object of B(\overline{E}_{1}, Σ_{1}; 𝔄), let h_{∗}(A) be the object \{B_{z}\} of drawboxed() B(\overline{E}_{2}, Σ_{2}; 𝔄) with B_{z} = ΣA_{x} where the sum runs over x ∈ h^{−1}(z). Since the set h^{−1}(z) has compact closure in E_{1}, there are only finitely many x in h^{−1}(z) with A_{x} ≠ 0. Hence the sum is finite. Similarly for any compact set K ⊂ E_{2}, \{z ∈ K|B_{z} ≠ 0\} is finite. Hence \{B_{z}\} is an object of B(\overline{E}_{2}, Σ_{2}; 𝔄). Let f : A → A′ be the morphism f = {f_{y}^{x}|x.y ∈ E_{1}}. Then h_{∗}(f) = {g_{w}^{z}} : h_{∗}(A) → h_{∗}(A”) is the morphism with g_{w}^{z} = ∑ f_{y}^{x} where the sum runs over (x, y) ∈ h^{−1}(z) × h^{−1}(w). It is easily checked that h_{∗}(f) is a morphism in B(\overline{E}_{2}, Σ_{2}; 𝔄) and that h_{∗} is an additive functor.

For any additive category 𝔄, let 𝕂𝔄 denote the classifying space BA^{−1}A where A = Iso 𝔄 is the category of isomorphisms in 𝔄. The algebraic K-theory of 𝔄 is given by K_{m}(𝔄) = π_{m}(𝕂𝔄) for m ≥ 0. The algebraic K-theory of the additive category B(\overline{E}, Σ; 𝔄) is denoted by K_{∗}B(\overline{E}, Σ; 𝔄) and is called the algebraic K-theory of \overline{E} with continuous control at infinity. If h : (\overline{E}_{1}, Σ_{1} → (\overline{E}_{2}, Σ_{2}) is a morphism in E, h induces a homomorphism

h_{∗} : K_{∗}B(\overline{E}_{1}, Σ_{1}; 𝔄) → K_{∗}(\overline{E}_{2}, Σ_{2}; 𝔄)

and we may may regard K_{∗}B(−; 𝔄) as a functor from E to the category of graded abelian groups.

The closed-cone construction can be used to obtain many examples of pairs (\overline{E}, Σ) in E. In particular if X is a non-empty compact Hausdorff space, we let cX be the closed cone on X (i. e. the quotient space obtained from X × [0, 1] by identifying X × {0} to a point c) and (cX, X) be the pair in which X is the image of X × {1} in cX. In X = ∅, then we let (cX, X) = (c, ∅). In either case, (cX, X) is in E. In fact, the closed cone construction defines a functor c : F → E where F is the category of compact Hausdorff spaces. The image of X × [0, 1) in cX is denoted oX and is called the small open cone on X.

If 𝔄 is an additive category, we let 𝔄̂ be the idempotent completion [3] of 𝔄; that is, 𝔄̂ is the category with objects (A, p) where A ∈ 𝔄 and p : A → A has p^{2} = p anf with morphisms f : (A, p) → (B, q) those morphisms f : A → B in 𝔄 for which f = qfp.

Let X be a compact metrizable space and S^{n}X be its nth suspension. We will prove that the collection of spaces K̃(X; 𝔄) = {𝕂B̂(cS^{n}X, S^{n}X; 𝔄)|n = 0, 1, 2, …} is a spectrum. This spectrum is nonconnecteive and its stable homotopy is denoted by K̃_{∗}(X; 𝔄). It is studied in Section 4.

Theorem I. For any additive category 𝔄, the functor X ↦ K̃_{∗}(X; 𝔄) is a reduced homology theory on the category of compact metrizable spaces.

The reader should see Theorem 5.2 for a more complete statement of this theorem.

The main steps in the proof of Theorem I are showing that if X = vY is the closed cone on Y, the K-theory classifying space of X is contractible and establishing a Mayer-Vietoris property. Thes steps are done in Corollary 2.3 and Theorem 4.4 respectivley. Here we have written X = vY instead of X = cY since we must form cX to get the homology of X and it is convenient to dictinguish the two cone points c and v.

For any pair of compact metrizable spaces (X, A), let K_{∗}(X, A; 𝔄) = K̃_{∗}(X ∪_{A} vA; 𝔄) where vA is the cone on A with vertex v and we take vA to be a point if A = ∅. To simplify notation, throughout this paper we shall write X ∪_{v} A) instead of the more precise X ∪_{A} vA.

Theorem II. The functor K_{∗}(−; 𝔄) is a homology theory on the category of pairs of compact metrizable spaces. If we restrict this homology theory to the category of pairs of finite CW complexes, its representing spectrum K(pt; 𝔄) = K̃(S^{0}; 𝔄) has ΩK(pt; 𝔄) = K(𝔄).

The reader should see Theorem 5.3 for a more complete statement of this theorem. In this theorem K(𝔄) is the nonconnective spectrum for the K-theory of 𝔄 constructed in [8]. It follows from this theorem and the main result of [9] that K_{∗}(X, 𝔄) = H_{∗−1}(X, K(𝔄)) for every compact polyhedron X where the right-hand side is the homology of X with coefficients in the spectrum K(𝔄).

In proving the last part of Theorem II, the idea is to show that K̃(S^{0}; 𝔄) and the spectrum K̃′(pt; 𝔄) of [9] have isomorphic homotopy groups. In positive dimensions, this is accomplished in Lemma 5.5 and in nonpositive dimensions, in Lemma 5.6. The proof of Lemma 5.6 is based on the results in section 2 and the construction of a “Bass-Heller-Swan” homomorphism motivated by the discussion in [7, Section 2].

Let X be a finite CW complex and O(X) be the large open cone on X (i. e. the space obtained from X × [0, ∞) by identifying X × {0} to a point c). If we identify O(X) with oX under the homeomorphism that sends [ξ, t] to [ξ, 1 − t/(1 + t)], there is an inclusion of categories i : C(O(X); 𝔄) → B(cX, X; 𝔄) obtained by noticing that any morphism in C(O(X); 𝔄) is continuously controlled at infinity under the above identification. Let

i_{∗} : K_{∗}^{b}(X; 𝔄) → K_{∗}(X; 𝔄)

be the induced homomorphism on homology where K_{∗}^{b}(X; 𝔄) is the homology of X using bounded morphisms as described in [9]. The next result follows easily from Theorem II.

Corollary III. For any finite CW complex X, i_{∗} : K_{∗}^{b}(X; 𝔄) → K_{∗}(X; 𝔄) is an isomorphism.

In a recent paper [12], Vogell describes another approach to proving Theorems I and II for finite CW complexes. Although his approach in that case is simpler than the one taken here, it does not work in the full generality of this paper.

The first-named author would like to than the Department of Mathematics at the University of Notre Dame for provising a stimulating atmosphere for him during the period in which this paper was written. All four author would like to thank th ereferee for his detailed comments on the paper. They have greatly improved the exposition.

1. Some elementary properties of the categories B(\overline{E}, Σ; 𝔄)

We collect here some elementary properties of the categories B(\overline{E}, Σ; 𝔄) that will be useful later in this paper and illustrate the topological flavor of this category.

Let (\overline{E}, Σ) ∈ E, 𝔄 be an additive category and A = \{A_{x}|x ∈ E\} ∈ B(\overline{E}, Σ; 𝔄). The set supp_{∞}(A) = cl{x ∈ E|A_{x} ≠ 0} ∩ Σ is called the support of A at infinity. Notice that z ∈ supp_{∞}(A) if and only if every neighborhood U of z contains a point x ∈ E with A_{x} ≠ 0.

Lemma 1.1. Let A, B ∈ B(\overline{E}, Σ; 𝔄). If A is isomorphic to B, the supp_{∞}(A) = supp_{∞}(B).

Proof. Let f = {f_{y}^{x} : A_{x} → B_{y}} be an isomorphism from A to B. Let z ∈ supp_{∞}(A) and U be a neighborhood of z in \overline{E}. Then there is a smaller neighborhood V of z so that if x ∈ V and f_{y}^{x} ≠ 0, then y ∈ U. Since z ∈ supp_{∞}(A), there is an x ∈ V with A_{x} ≠ 0. Since f is an isomorphism, there is a y with f_{y}^{x} ≠ 0. Hence B_{y} ≠ 0 for some y in U, z ∈ supp_{∞}(B), and supp_{∞}(A) ⊂ supp_{∞}(B). Lemma 1.1 follows easily. □

Let (\overline{E}, Σ)×I = (\overline{E} ×I, Σ×I). Then lemma 1.1 shows that there are objects in B(\overline{E}, Σ)× I, 𝔄) that are not isomorphic to objects in the subcategory B((\overline{E}, Σ) × \{0\}, 𝔄). Hence the inclusion

i : B((\overline{E}, Σ) × \{0\}, 𝔄) → B((\overline{E}, Σ) × I, 𝔄)

is not an equivalence of categories.

Lemma 1.2. Let (\overline{E}, Σ) ∈ E, K ⊂ \overline{E} be closedand W ⊂ Σ be disjoint from K. Let A, B ∈ B(\overline{E}, Σ; 𝔄) and f : A → B. Then there exists a neighborhood U of W in \overline{E} − K so that if either (x, y) or (y, x) is in U × K, then f_{y}^{x} = 0.

Proof. Suppose first that W is a single point w. Then there are disjoint open sets U′ and V′ with w ∈ U′ and K ⊂ V′. For each z ∈ K ∩ Σ, there is a neighborhood V_{z} so that if f_{y}^{x} ≠ 0 and x ∈ V_{z}, then y ∈ V′. Let V = ⋃{V_{z}|z ∈ K ∩ Σ}. For all x ∈ V, if f_{y}^{x} ≠ 0, then y ∈ V′. Since C = K ∩ (\overline{E} − V) is compact and contained in E, the sets \{x ∈ C|A_{x} ≠ 0\} and {y ∈ E|f_{y}^{x} ≠ 0, x ∈ C} are finite. Let U = U′ − {y ∈ E|f_{y}^{x} ≠ 0, x ∈ K}. Then U is an open neighborhood of w and for all (x, y) ∈ K × U, f_{y}^{x} = 0. Since f is continuous at w, we may replace U by a smaller neughborhood of w, if necessary to insure that if (x, y) ∈ U × K, then f_{y}^{x} = o. The lemma follows in case W is a single point.

In the general case choose a neighborhood U_{w} for each w ∈ W as in the preceeding paragraph. Then U = ⋃{U_{w}|w ∈ W } works. □

Let f : (\overline{E}_{1}, Σ_{1}) → (\overline{E}_{2}, Σ_{2}) be a map in E for which f|_{Σ_{1}} : Σ_{1} → Σ_{2} is a homeomorphism. We say thatf is an equivalence at infinity if for every p ∈ Σ_{1} and every neighborhood U of p in \overline{E}_{1}, there exists a neighborhood V of f(p) in \overline{E}_{2} with f^{−1}(V) ⊂ U. This term is explained in the following lemma:

Lemma 1.3. Let f : (\overline{E}, Σ) → (\overline{E}, Σ be a map in E for which f|_{Σ} = 1. Then there is a natural transformation η : I → f_{∗} where I is the identity functor of B(\overline{E}, Σ; 𝔄). If f is an equivalence at infinity, then η is a natural equivalence. Hence f induces the identity on K-theory.

Proof. Let A = \{A_{x}\} be an object of B(\overline{E}, Σ; 𝔄) and set B = f_{∗}(A). Let η(A) = \{η_{y}^{x}\} : A → B be the morphism for which η_{y}^{x} is the inclusion of the A_{x} summand into B_{y} if y = f(x) and is 0 otherwise. Since f is a m map in E, for every p ∈ Σ and every neighborhood U of p there is a neighborhood V so that f(V) ⊂ U. hence η(A) is a morphism in B(\overline{E}, Σ; 𝔄). It is easily checked that η = {η(A) : I → f_{∗} is a natural transformation.

Suppose that f is an equivalence at infinity. To see that η is a natural equivalence, it suffices to show that for each A, η(A) is an isomorphism. Let ρ(A) : {ρ_{y}^{x}} : B → A be the morphism for which ρ_{y}^{x} is the projection of B_{x} on the A_{y} summand if y ∈ f^{−1}(x) and is 0 otherwise. Since the set {y|ρ_{y}^{x} ≠ 0} = {x|η_{x}^{y} ≠ 0}, it is finite. Let p ∈ Σ and U be a neighborhood of p. By hypothesis there is a neighborhood V of p with f^{−1}(V) ⊂ U. hence ρ(A) is a morphism in B(\overline{E}, Σ; 𝔄). It is easily checked that ρ(A) is an inverse of η(A).

The following general observations show that f induces the identity on K-theory: Let F, G : 𝔄 → 𝔅 be additive functors between additive categories and η : F → G be a natural equivalence. Then F, G induce functors \overline{F} , \overline{G} : 𝔄^{−1}𝔄 → 𝔅^{−1}𝔅 and η induces a natural transformation \overline{η} : \overline{F} → \overline{G}. Hence F and G induce the same homomorphism on K-theory. □

Corollary 1.4. Let h_{1}, h_{2} : (\overline{E}, Σ) → (\overline{E}, Σ) be homeomorphisms with h_{1}|_{Σ} = h_{2}|_{Σ}. Then there is a natural equivalence η : h_{1∗} → h_{2∗}. Hence h_{1} and h_{2} induce the same map on K-theory.

Proof. Since f = (h_{1}^{−1})h_{2} satisfies the hypothesis of Lemma 1.3, there is a natural equivalence η′ : I → f_{∗} where I is the identity functor. Then η = h_{1∗}(η′) is the desired natural equivalence. □

A morphism i : (\overline{E}_{1}, Σ_{1}) → (\overline{E}_{2}, Σ_{2}) is an inclusion in E if i : \overline{E}_{1} → \overline{E}_{2} is an inclusion and Σ_{1} = \overline{E}_{1} ∩ Σ_{2}.

Corollary 1.5. Let (\overline{E}_{1}Σ_{1}) and (\overline{E}_{2}, Σ_{2}) be objects in E with Σ_{1} = Σ_{2}. Let i : (\overline{E}_{1}, Σ_{1}) → (\overline{E}_{2}, Σ_{2}) be an inclusion and r : (\overline{E}_{2}, Σ_{2}) → (\overline{E}_{1}, Σ_{1}) be a retraction in E. Suppose that r is an equivalence at infinity. Then i_{∗} : B(\overline{E}_{1}, Σ_{1}; 𝔄) → B(\overline{E}_{2}, Σ_{2}; 𝔄) is an equivalence of categories.

Proof. Since r is a retraction, ri = 1 and r_{∗}i_{∗} is the identity functor. Since i is clearly an equivalence at infinity and r is an equivalence at infinity by hypothesis, ir is an equivalence at infinity. Hence i_{∗}r_{∗} is naturally equivalent to the idnetity by Lemma 1.3. Corollary 1.5 follows □

We now introduce a technical concept that is used in the proof of theorem 3.1. Let (\overline{E}_{0}, Σ_{0}) → (\overline{E}, Σ) ∈ E be an inclusion in E. We say that (\overline{E}_{0}, Σ_{0}) is an eventual neighborhood retract in (\overline{E}, Σ) if there is a closed neighborhood N of Σ_{0} in \overline{E} and a function r : (N − Σ) ∪ \overline{E}_{0} → \overline{E}_{0} with r^{−1}(Σ_{0}) = Σ_{0} and r|_{\overline{E}_{0}} = id with the following property: For every open neighborhood W of Σ − Σ_{0} in \overline{E} − \overline{E}_{0}, r_{W} = r| : [(N − Σ − W) ∪ \overline{E}_{0}] → (\overline{E}_{0}, Σ_{0}) is an equivalence at infinity. This amounts to requiring that for every triple (p, U, W) consisting of a point p ∈ Σ_{0}, a neighborhood U of p in \overline{E}, and an open neighborhood W of Σ − Σ_{0} in \overline{E}, there is a neighborhood V of p in \overline{E}_{0} with r^{−1}(V) ⊂ U ∪ W. The reader will note that only the restrictions r_{W} of r, and not r itself need be morphisms in E and that r need to be continuous only at infinity. The function r is called an eventual retraction.

The diagram below illustrates the definition of an eventual neighborhood retraction r. In it, r is defined on (N − Σ) ∪ \overline{E}_{0}. It is “well behaved” on (N − Σ − W) ∪ \overline{E}_{0} for every W.

[figure: \overline{E}; \overline{E}_{0}; Σ; Σ_{0}; W; N; .............; _{.............}]

Proposition 1.6. Suppose that Σ_{0} ⊂ Σ is a neighborhood retract. Then (cΣ_{0}, Σ_{0}) is an eventual neighborhood retract in (cΣ, Σ).

Proof. By replacing Σ with an appropriate closed neighborhood of Σ_{0}, we may assume there is a continuous retraction ρ : Σ → Σ_{0}. Let N = cΣ and r : N → cΣ_{0} be the cone on ρ. Let W be an open neighborhood of Σ − Σ_{0} not meeting cΣ_{0}. Then r_{W} = r|_{(cΣ−W)} : (cΣ − W, Σ_{0}) → (cΣ_{0}, Σ_{0}) is a morphism in E and ri = 1 where i : (cΣ_{0}, Σ_{0}) → (cΣ − W, Σ_{0}) is the inclusion.

Let U be any open neighborhood of p ∈ Σ_{0} in cΣ. Then there is an open neighborhood V of p in cΣ_{0} with (r_{W})^{−1}(V) ⊂ U. If not, then for each neighborhood V of p, there is a point x_{V} ∈ cΣ − W − U with r(x_{V}) ∈ V. Then the net {r(x_{V})|V ∈ V} converges to p where V is the set of neighborhood of p. Let B be the set of neighborhoods of p that do not contain c. Then B is a base of the neighborhoods of p. hence {r(x_{V}) = (cρ)(x_{V})|V ∈ B} also converges to p. On the other hand, {x_{V} |V ∈ B} converges to some point q of cΣ. Since x_{V} ∈ cΣ − W − U which is compact, q ∈ \overline{E} − W − U ⊂ cΣ − Σ. In particular q and (cρ)(q) are not in Σ. Hoevere, since cρ is continuous, (cρ)(q) = p ∈ Σ_{0}. This is a contradiction. □

If X is a nonempty space, let SX be the unreduced suspension of X (i. e. space obtained from X × [−1, 1] by identidying X × {i} to {i} for i = ±1). If X = ∅, let SX = S∅ = S^{0} be the discrete two-point space.

Proposition 1.7. If Σ_{0} ⊂ Σ is a neighborhood retract, then so is SΣ_{0} ⊂ Σ.

Proof. Let r : N → Σ_{0} be a continuous retraction of a closed neighborhoos N of Σ_{0} in Σ onto Σ_{0}. Let vΣ be the cone on Σ with vertex v. We show first that there is a closed neighborhood N_{1} of vΣ_{0} and a continuous retraction r_{1} : N_{1} → vΣ_{0} with N ⊂ N_{1}, N_{1} ∩ Σ_{0} = N, and r_{1}|_{N} = r. To see this, let h : I × I → I × I be a function with the following properties:

(i) For all s ∈ I, h(s, 0) = (s, 0) and h(s, 1) = (s, 1).

(ii) If \frac{3}{4} ≤ s ≤ 1 and \frac{1}{2} ≤ t ≤ 1, then h(s, t) = (s, 1).

Such a function can be constructed using standard elementary techniques. Let ϕ : Σ → I be a continuous map for which ϕ(Σ_{0}) = 0 and ϕ(Σ − Int N) = 1. Define H : Σ × I → Σ × I by setting H(x, t) = (x, h(ϕ(x), t)). By (1), H(x, 0) = (x, 0) and H(x, I) = (x, 1) for all x. Hence H induces a continuous map H′ : vΣ → vΣ for which H′[x, 0] = [x, 0] for all x and H′(v) = v. Let N_{1} = q(N × I ∪ Σ × [\frac{1}{2}, 1]). Then N_{1} is a closed neighborhood of vΣ_{0} in vΣ. Since H(x, t) = (x, 1) for all (x, t) ∈ ϕ^{−1}[\frac{3}{4}, 1]×]\frac{1}{2}, 1] by (2), for such (x, t), H′[x, t] = v and H′(N_{1}) ⊂ vN. We now set r_{1} = (vr)H′ : N_{1} → vΣ_{0} and observe that r_{1} has the properties claimed.

Let SΣ = vΣ ∪ wΣ where the union is along Σ. By gluing two copies of the map r_{1} together along the common subspace N, we obtain a neighborhood N_{2} of SΣ_{0} in SΣ and continuous retraction r_{2} : N_{2} → SΣ_{0}. This completes the proof of Proposition 1.7 □

2. The structure of the category B(cX, X; 𝔄)

Let (X, d) be a compact metric space X together with a metric for which diam X ≤ 2. In this section we shal determine the structure of B(cX, X; 𝔄). We also use this result to show that K_{∗}B(cX, X; 𝔄) = 0 if X = vY where Y is a compact metrizable space.

Let O(X) be the large open cone on X (i. e. the space obtained from X × [0, ∞) by identifying X × {0} to a point c). It was observed in [11] (cf. also [2] that setting

(1)  ρ(x, y) = min{s, t}d(ξ, η) + |s − t|

for x = [ξ, t] and y = [η, s] in polar coordinates, gives a metric on O(X). We identify O(X) with o(X) under the homeomorphism that sends [ξ, t] to [ξ, 1 − t/(1 + t)] and write cX as O(X) ∪ X(∞) where X(∞) is a copy of X “at infinity”.

For any numbers r < s ≤ ∞, let < r, s >= {[ξ, t] ∈ O(X)|r ≤ t < s}. If ξ ∈ X and β > 0, let ang(ξ, β) = {[ζ, s]|0 < s and d(ζ, ξ) < β}. By an abuse of language if x = [ξ, t] ≠ c in O(X) ∪ X(∞) and β > 0, we shall write ang(x, β) for ang(ξ, β). Finally for f ∈ B(O(X) ∪ X(∞), X(∞); 𝔄) and x ∈ O(X), let S_{x}(f) = {y ∈ O(X)|f_{y}^{x} ≠ 0}.

Let \underline{r} = (r_{1}, r_{2}, r_{3}, …) be an increasing sequence of positive integers. Let B\underline{r} be the subcategory of B(O(X)∪X(∞), X(∞); 𝔄) containing all the objects of B(O(X)∪X(∞), X(∞); 𝔄) but only those morphisms f : A → B for which there are constants α, β and γ so that

For almost all k, if x ∈< r_{k}, r_{k+1} >, then

(2)  S_{x}(f) ⊂< r_{k−α}r_{k+1+α} > ∩ ang(x, \frac{β}{k − γ}).

Of course for (2) to make sense, we must have k > γ and k ≥ α. A simple calculation shows that if f′ and f′′ are in B\underline{r}, then so is f′′f′ and thus that B\underline{r} really is a subcategory of B(O(X) ∪ X(∞), X(∞); 𝔄).

The structure of B(O(X) ∪ X(∞), X(∞); 𝔄) is given by the following theorem:

Theorem 2.1. Let X be a compact metric space with diam X ≤ 2 and 𝔄 be any additive category. Then the collection \{B\underline{r} forms a direct system of subcategoies of B(O(X) ∪ X(∞), X(∞); 𝔄) ordered by inclusion. that is

(i) B(O(X) ∪ X(∞), X(∞); 𝔄) = ⋃ B\underline{r} (ii) if \underline{r}′ and \underline{r}′′ are two such sequences, then there is a sequence \underline{r} with B\underline{r}′ ⊂ B\underline{r} and

B\underline{r}′′ ⊂ B\underline{r}. Furthermore, for every increasing sequence \underline{r}, B\underline{r} is isomorphic to C(O(X); 𝔄).

Theorem 2.1 is a direct consequence of Lemma 2.4-2.8.

Corollary 2.2. Let X be a compact metric space with diam X ≤ 2 and 𝔄 be an additive category. Then K_{∗}B(cX, X; 𝔄) = \underrightarrow{colim}K_{∗}B\underline{r}.

The colimit here is over the family \{B\underline{r}\} of subcategories of B(O(X)∪X(∞), X(∞); 𝔄) and we use the identification of this category with B(cX, X; 𝔄) given above to simplify notation. The corollary is an immediate consequence of Theorem 2.1.

We recall that for any additive category 𝔄, its idempotent completion [3] is the category 𝔄̂ with objects (A, p) where A ∈ 𝔄 and p : A → A has p^{2} = p and with morphisms f : (A, p) → (B, q) those morphisms f : A → B in 𝔄 for which f = qfp.

Corollary 2.3. Suppose X = vY is the closed cone on the compact metrizable space Y. Then the spaces 𝕂B(cX, X; 𝔄) and 𝕂B̂(cX, X; 𝔄) are contractible.

Proof. Any compact, metrizable space has a metric d of diameter ≤ 2. For this metric B(cX, X; 𝔄) = ⋃ B\underline{r} and hence B̂(cX, X; 𝔄) = ⋃ B̂\underline{r}. By Theorem 2.1, B\underline{r} is isomorphic to Ĉ(O(X); 𝔄) which is flasque by [9]. hence K_{∗}B̂\underline{r} = K_{∗}Ĉ(O(X); 𝔄) = 0 and K_{∗}B̂(cX, X; 𝔄) = \underrightarrow{colim}K_{∗}B̂\underline{r} = 0. It follows that 𝕂B̂(cX, X; 𝔄) is contractible. The proof that 𝕂B(cX, X; 𝔄) is contractibel is similar. □

Lemma 2.4. Let f ∈ B(O(X) ∪ X(∞), X(∞); 𝔄) and suppose that r > 0 and β > 0 are given. then there is an R > 0 so that for all x ∈< R, ∞ >, S_{x}(f) ⊂< r, ∞ > ∩ ang(x, β).

Proof. Suppose no such R exists. Then there exist sequences x_{n} = [ξ_{n}, t_{n}] and y_{n} = [ζ_{n}, s_{n}] (n = 1, 2, 3, …) so that for all n, x_{n} ∈< n, ∞ >, y_{n} ∈ S_{x_{n}}(f) and either s_{n} ≤ r or d(ζ_{n}, ξ_{n}) ≥ β. By choosing a subsequence if necessary, we may assume x_{n} → x ∈ X(∞). Since f is continuously controlled at infinity, y_{n} → x. Hence s_{n} → ∞ and d(ζ_{n}, ξ_{n}) → 0. This contradicts the choices of x_{n} and y_{n}. □

Lemma 2.5. For any f ∈ B(O(X) ∪ X(∞), X(∞); 𝔄) there is a sequence \underline{r} with f ∈ B\underline{r}.

Proof. The proof is inspired by an argument of Hughes given in [5, Section 5]. We construct the sequence \underline{r} by induction. Use Lemma 2.4 to choose r_{1}, so that if x ∈< r_{1}, ∞ >, then S_{x}(f) ⊂ ang(x, 1). Suppose r_{1}, …, r_{n} constructed so that

(i) if x ∈< r_{k}, ∞ >, then S_{x}(f) ⊂< r_{k−1}, ∞ > ∩ ang(x, 1/k) for k ≤ n; and (ii) if x ∈< 0, r_{k} >, then S_{x}(f) ⊂< 0, r_{k+1} > for k < n.

Now use Lemma 2.4 to choose r_{n+1} so that if x ∈< r_{n}, ∞ >, then (i) holds for k = n + 1. Since there are only finitely many non-zero f_{y}^{x} with x ∈< 0, r_{n} >, we may also choose r_{n+1} large enough that (ii) also holds. this completes the construction of \underline{r}.

If x ∈< r_{n}, r_{n+1} >, then S_{x}(f) ⊂< r_{n−1}, r_{n+2} > ∩ ang(x, 1/n). By letting α = 1, β = 1 and γ = 0 in (2), this shows that f ∈ B\underline{r}. □

Lemma 2.6. Let \underline{r}_{0} = (1, 2, 3, …). Then B_{\underline{r}_{0}} = C(O(X); 𝔄).

Proof. Let f ∈ B_{\underline{r}_{0}}, x = [ξ, t], y = [η, s], and suppose that f_{y}^{x} ≠ 0. Then if x ∈< k, k + 1 >,

ρ(x, y) = min{s, t}d(ξ, η) + |s − t|

≤ min\{s, t\}\frac{β}{k − γ} + α + 1 ≤ (k + α + 1)\frac{β}{k − γ} + α + 1.

Since (k + 1 + α)(k − γ) → 1 as k → ∞, ρ(x, y) < 2β + α + 1 for all x = [ξ, t] with t sufficiently large. Since there are only finitely many f_{y} omitted by this process, there is a number B so that ρ(x, y) < B for all x, y with f_{y} ≠ 0. Thus B_{\underline{r}_{0}} ⊂ C(O(X); 𝔄).

Let x ∈< n, n + 1 > and suppose d > 0 is given. Simple estimates using (1) show that B(x, d) ⊂< n−d, n+d+1 > ∩ ang(x, d/(n−d)). Hence if f ∈ C(O(X); 𝔄) has the property that ρ(x, y) < d when f_{y} ≠ 0, then f ∈ B_{\underline{r}_{0}} with constants α = β = γ = d in (2). Thus C(O(X); 𝔄) ⊂ B_{\underline{r}_{0}}. This completes the proof of Lemma 2.6. □

Lemma 2.7. For every sequence \underline{r}, B\underline{r} is a subcategory of B(O(X) ∪ X(∞), X(∞); 𝔄) and there is an isomorphism of categories h\underline{r}^{∗} : B_{\underline{r}_{0}} → B\underline{r}.

Proof. Let ρ\underline{r} : [0, ∞) → [0, ∞) be the homeomorphism that maps [k, k + 1] linearly onto [r_{k}, r_{k+1} and set h\underline{r}[ξ, t] = [ξ, ρ\underline{r}(t)]. Then h\underline{r} induces an automorphism h\underline{r}^{∗} of the category B(O(X) ∪ X(∞), X(∞); 𝔄) onto itself whose inverse is (h−1 ; \underline{r})_{∗}. Straighfoprward calculation shows that h\underline{r}^{∗}(B_{\underline{r}_{0}}) ⊂ B\underline{r} and that (h−1 ; \underline{r})_{∗}(B\underline{r}) ⊂ B_{\underline{r}_{0}} from which Lemmma 2.7 follows. □

Lemma 2.8. Let \underline{r}′ = (r′_{1}, r′_{2}, r′_{3}, …) and \underline{r}′′ = (r′′_{1}, r′′_{2}, r′′_{3}, …) be increasing sequences of integers. Then there is a sequence \underline{r} so that B\underline{r}′ ⊂ B\underline{r} ⊃ B\underline{r}′′.

We say that a squence \underline{s} splices the sequences \underline{t} if t_{n} < s_{n} < t_{n+1} for all n. Notice that if \underline{s} splices \underline{t}, the B\underline{t} = B\underline{s}.

Proof. Clearly one can choose subsequences \underline{s}′ and \underline{s}′′ of \underline{r}′ and \underline{r}′′ respectively so that \underline{s}′ splices \underline{s}′′. Then B\underline{s}′ = B\underline{s}′′. Since \underline{s}′′ is a subsequence of \underline{r}′, we see that B\underline{r}′ ⊂ B\underline{s}′. Similarly B\underline{r}′′ ⊂ B\underline{s}′′. The proof is completed by letting \underline{r} = \underline{s}′′. □

3. The Mayer-Voetories property

This section establishes a Mayer-Vietories property. Although the result given here is technical, the consequences of it obtained in the next section are far less so and are sufficient for proving the main theorems of this paper. The approach taken here is a modification of that in [2]. The difference is that the categories in [2] were filtered; whereas the categories that arise here are not. The reader who is interested in only the main results of this paper can sjip this section and refer back to it only as needed.

We recall some definitions and notation.

If fb is a full, additive subcategory of the additive category 𝔄, then the idempotent semicompletion (or simply, semicompletion) of 𝔄 with respect to 𝔅 is the full, additive subcategory of hat𝔄 containing those objects (A, p) isomorphic to (B, q) ⊕ (C, 1) with (B, q) ∈ 𝔅̂ and C ∈ 𝔄. The semicompletion is denoted by 𝔄̃.

Theorem 3.1. Let (\overline{E}, Σ) be the pushout of the diagram (\overline{E}_{1}, Σ_{1}) ⟵^{i_{1}} (\overline{E}_{0}, Σ_{0}) ⟶^{i_{2}} (\overline{E}_{2}, Σ_{2}) of inclusions in E in which \overline{E}_{0} − Σ_{0} ≠ ∅. If (\overline{E}_{0}, Σ_{0}) is an eventual neighborhood retract in (\overline{E}, Σ), then the square of spaces

\xymatrix{𝕂B̂(\overline{E}_{0}, Σ_{0}; 𝔄) \ar[r]^{i_{1}} \ar[d]^{i_{2}} & 𝕂B̃(\overline{E}_{1}, Σ_{1}; 𝔄) \ar[d]^{j_{1}} \\ 𝕂B̃(\overline{E}_{2}, Σ_{2}; 𝔄) \ar[r]^{j_{2}} & 𝕂B̃(\overline{E}, Σ; 𝔄)}

is a pullback up to homotopy. Here the semicompletions are with respect to B(\overline{E}_{0}, Σ_{0}; 𝔄). Hence there is a long exact sequence

⋯ → π_{n+1}(𝕂B) → π(𝕂B_{0}) → π_{n}(𝕂B_{1}) ⊕ π_{n}(𝕂B_{2}) → π_{n}(𝕂B) → ⋯

(n ≥ 0) where 𝕂B_{i} = 𝕂B̃(\overline{E}_{i}, Σ_{i}; 𝔄)(i = 1, 2, ∅) and 𝕂B_{0} = 𝕂B̂(\overline{E}_{0}, Σ_{0}; 𝔄).

The reader will recall that the definition of Σ_{0} being an eventual retract in \overline{E} is given in section 1. Therem 3.1 is an immediate consequence of Lemmas 3.2 and 3.3.

Suppose given a commutative diagram (*) of additive categories and additive functors

(⋆)  \xymatrix{𝔄_{0} \ar[d]^{i_{2}} \ar[r]^{i_{1}} & 𝔄_{1} \ar[d]^{j_{1}} \\ 𝔄_{2} \ar[r]^{j_{2}} & 𝔄}

and a directed set Λ, We say that (⋆) has the Mayer-Vietories property relative to Λ if the following conditions hold:

(1) The functors i_{1}, i_{2}, j_{1} and j_{2} are full and faithful.

(2) For every pair of objects A, B ∈ 𝔄 there is a family of subgroups {F_{λ} Hom(A, B)|λ ∈ Λ} such that if μ < λ, then F_{μ} Hom(A, B) ⊂ F_{λ} Hom(A, B) and with Hom(A, B) = ⋃ F_{λ} Hom(A, B).

(3) For every A ∈ 𝔄, there is a λ(A) ∈ Λ so that for every λ ≥ λ(A) there is a preferred decomposition of A as A = A_{1λ}⊕\overline{A}_{0λ}⊕A_{2λ} with A_{iλ} ∈ 𝔄_{i}(i = 1, 2) and an isomorphism σ_{0λ} : \overline{A}_{0,λ} → A_{0λ} to an object A_{0λ} ∈ 𝔄_{)}. It is further required that 1_{A} and 1_{1λ} ⊕ σ_{0λ} ⊕ 1_{2λ} be in F_{λ} Hom(A, A′) and that 1_{1λ} ⊕ (σ_{0λ})^{−1} ⊕ 1_{2λ} ∈ F_{λ} Hom(A′, A) for all λ ≥ λ(A) where A′ = A_{1λ} ⊕ A_{0λ} ⊕ A_{2λ}.

(4) If λ ≥ λ(A) and f ∈ F_{λ} Hom(A, B), then f decomposes as

\xymatrix{A_{1λ} \ar[d] & ⊕ & \overline{A}_{0λ} \ar[dll] \ar[d] \ar[drr] & ⊕ & A_{2λ} \ar[d] \\ B_{1} & ⊕ & B_{0} & ⊕ & B_{2}}

relative to the preferred decomposition of A and any decomposition B = B_{1}⊕B_{0}⊕B_{2} with B_{i} ∈ 𝔄_{i} (i = 1, 2).

(5) If λ ≥ λ(B) and f ∈ F_{λ} Hom(A, B), then f decomposes as

\xymatrix{A_{1} \ar[d] & ⊕ & \overline{A}_{0} \ar[dll] \ar[d] \ar[drr] & ⊕ & A_{2} \ar[d] \\ B_{1λ} & ⊕ & B_{0λ} & ⊕ & B_{2λ}}

re;ative to any decomposition A = A_{1} ⊕ A_{0} ⊕ A_{2} with A_{i} ∈ 𝔄 (i = 0, 1, 2) and the preferred decomposition of B.

Lemma 3.2. If (*) has the Mayer-Vietories property with respect to Λ and 𝔄_{0} is idempotent complete then

\xymatrix{𝕂𝔄_{0} \ar[d]_{i_{2}} \ar[r]^{i_{1}} & 𝕂𝔄_{1} \ar[d]^{j_{1}} \\ 𝕂𝔄_{2} \ar[r]^{j_{2}} & 𝕂𝔄}

is a pullback up to homotopy. Hence for n ≥ 0, there is a long exact sequence

⋯ → π_{n+1}(𝕂𝔄) → π(𝕂𝔄_{0}) → π_{n}(𝕂𝔄_{1}) ⊕ π_{n}(𝕂𝔄_{2}) → π_{n}(𝕂𝔄) → ⋯

Proof. The proof is similar to the proofs of the corresponding rsults in [8],[9] and [2]. Let 𝔅 be Thomasson’s simplified double mapping cylinder ogf the diagram 𝔄_{1} ⟵^{𝔄}_{0}⟶^{𝔄}_{2} and Σ : 𝔅 → 𝔄 be the obvious functor. We show Σ is a homotopy equivalence on K-theory classifying spaces using Quillen’s Theorem A by showing that for each object A ∈ 𝔄 , the category A ↓ Σ is contractible.

For any λ ∈ Λ, let F_{λ}(A ↓ Σ) be the full subcategory of A ↓ Σ) with objects those morphisms α : A → B_{1}⊕B_{0}⊕B_{2} = B for which α ∈ F_{λ} Hom(A, B) and α^{−1} ∈ F_{λ} Hom(B, A). Let σ be the composite

σ = 1 ⊕ σ_{0λ} ⊕ 1 : A = A_{1λ} ⊕ \overline{A}_{0λ} ⊕ A_{2λ} → A_{1λ} ⊕ A_{0λ} ⊕ A_{2λ}.

The argument given in [8],[9] or [2] shows that if λ ≥ max{λ(A), λ(B)}, then σ is an initial object in F_{λ}(A ↓ Σ). Hence this category is contractible. Since Λ is a directed set, it follows that A ↓ Σ is the directed union of contractible subcategories. Hence A ↓ Σ is contractible and Lemma 3.2 follows. □

Let (\overline{E}, Σ) ∈ E. Let A be an object of B(\overline{E}, Σ; 𝔄). The support of A is the set supp A = \{x ∈ E|A_{x} ≠ 0\}. If X ⊂ \overline{E}, we write A|_{X} for the object \{B_{y}|y ∈ E\} where B_{y} = A_{x} if y = x ∈ X and B_{y} is zero if y ∉ X. If f = {f_{y}^{x} : A_{x} → B_{y}|x, y ∈ E} : A → B is a morphism in B(\overline{E}, Σ; 𝔄), we let f|_{X} : A|_{X} → B|_{X} be the morphism \{f_{y}^{x}|x, y ∈ X\}.

Lemma 3.3. Let (\overline{E}, Σ) be the pushout of a diagram (\overline{E}_{1}, Σ_{1}) ⟵^{i_{1}} (\overline{E}_{0}, Σ_{0}) ⟶^{i_{2}} (\overline{E}_{2}, Σ_{2}) of inclusions in E in which \overline{E}_{0} − Σ_{0} ≠ ∅. Let Λ = \{U|U ⊂ \overline{E} − \overline{E}_{0} is a neighborhoos of Σ − Σ_{0} with U ∩ Σ_{0} ≠ ∅ and set U ≥ V if U ⊂ V. If (\overline{E}_{0}, Σ_{0}) is an eventual neighborhood retract in (\overline{E}, Σ), then the following diagram of additive categories and functors has the Mayer-Vietories property relative to Λ

(⋆⋆)  \xymatrix{B̂(\overline{E}_{0}, Σ_{0}; 𝔄) \ar[r]^{i_{1}} \ar[d]^{i_{2}} & B̃(\overline{E}_{1}, Σ_{1}; 𝔄) \ar[d]^{j_{1}} \\ B̃(\overline{E}_{2}, Σ_{2}; 𝔄) \ar[r]^{j_{2}} & B̃(\overline{E}, Σ; 𝔄)}

where the semicompletions are taken with respect to B̂(\overline{E}_{0}, Σ_{0}; 𝔄)

Proof. The semicompletion B̂(\overline{E}_{k}, Σ_{k}; 𝔄)(k = 1, 2, ∅ can be described more concretely as the full subcategory of B̂(\overline{E}, Σ; 𝔄) with objects those (A, p) for which there is a neighborhood U_{k} of Σ_{k} − Σ_{0} in \overline{E}_{k} − \overline{E}_{0} so that

(i) supp A ⊂ \overline{E}_{k}; (ii) p_{y}^{x} = 0 for (x, y) ∈ U_{k} × (E_{k} − U_{k}) ∪ (E_{k} − U_{k}) × U_{k} ∪ (U_{k} × U_{k} − Δ); and (iii) p_{x}^{x} = 1 if x ∈ U_{k}.

Here p = {p_{y}^{x} : A_{x} → A_{y}|x, y ∈ E} and Δ is the diagonal in U_{k} × U_{k}. Notice that (i)- (iii) implyu that p|_{U_{k}} = 1, that p|_{E_{k}−U_{k}} is a projection, and that (A, p) = (A|_{U_{k}}, 1) ⊕ (A|_{E_{k}−U_{k}}, p|_{E_{k}−U_{k}}).

We now verify that (⋆⋆) has the Mayer-Vietories property relative to Λ by verifying conditions (1)-(5) above.

It is clear from the description of B̃(\overline{E}_{k}, Σ_{k}; 𝔄)(k = 1, 2, ∅ that the functors i_{1}, i_{2}, j_{1} and j_{2} are full and faithfull, so (1) holds.

Let (A, p) ∈ B̂(\overline{E}_{k}, Σ_{k}; 𝔄), choose U ⊂ \overline{E} − \overline{E}_{0} satisfying (i)-(iii) above for k = ∅ (i. e. for \overline{E}), and set λ(A, p) = U. Suppose V = λ ≥ λ(A, p) = U. Then V is a neighborhood of \overline{Σ} − \overline{Σ}_{0} contained in U. We set V_{k} = V ∩ \overline{E}_{k} and let A_{kλ} = A|_{V_{k}} (k = 1, 2), \overline{A}_{0λ} = A|\overline{E} − V , p_{kλ} = p|_{V_{k}}(k = 1, 2), and \overline{p}_{0λ} = p|_{\overline{E}−V}. Then p_{kλ} = 1(k = 1, 2) and the preferres decomposition of (A, p) is given by setting

(A, p) = (A_{1λ}, 1) ⊕ (\overline{A}_{0λ}, \overline{p}_{0λ} ⊕ (A_{2λ}, 1).

Clearly (A_{kλ}, 1) ∈ B̃(\overline{E}_{k}, Σ_{k}; 𝔄)(k = 1, 2) and (\overline{A}_{0λ}, \overline{p}_{0λ}) ∈ B̂(\overline{E} − V, Σ_{0}; 𝔄). Since \overline{E}_{0}, Σ_{0}) is an eventual neighborhood retract in (\overline{E}, Σ), there is a closed neighborhood N of Σ_{0} in \overline{E} and function f : (N − Σ) ∪ \overline{E}_{0} → \overline{E}_{0} with r^{−1}(Σ_{0}) and r|_{Σ_{0}} = id which has r_{W} = r|; ((N − Σ − W) ∪ \overline{E}_{0}, Σ_{0}) → (\overline{E}_{0}, Σ_{0}) a morphism in E with r_{W} |_{Σ_{0}} for any neighborhood W of Σ − Σ_{0} in \overline{E} − \overline{E}_{0}. Let V be one such neighborhood. We may extend r_{V} to a morphism ρ : (\overline{E} − V, Σ_{0}) → (\overline{E}_{0}, Σ_{0}), also in E, by setting ρ(y) = x for all y ∈ E − (N ∪ V ∪ E_{0}) for some x ∈ E_{0}. Let (A_{0λ}, p_{0λ}) = ρ_{∗}(\overline{A}_{0λ}, ρ_{∗}\overline{A}_{0λ}, ρ_{∗}\overline{p}_{0λ}) ∈ B̂(\overline{E}_{0}, Σ_{0}; 𝔄). Since Corollary 1.5 shows that ρ_{∗} : B̂(\overline{E} − V, Σ_{0}; 𝔄) → B̂(\overline{E}_{0}, Σ_{0}; 𝔄) is an equivalence of categories with inverse the inclusion B̂(\overline{E}_{0}, Σ_{0}; 𝔄) → B̂(\overline{E} − V, Σ_{0}; 𝔄), (A_{0λ}, p_{0λ} is isomorphic to (\overline{A}_{0λ}\overline{p}_{0λ}).

Suppose f : A → B is a morphism in B(\overline{E}, Σ; 𝔄) and that U = λ ∈ Λ. We define a subgroup F_{λ} Hom(A, B) of Hom(A, B) by letting f ∈ F_{λ} Hom(A, B) if

(i) f decomposes as

\xymatrix{A_{1λ} \ar[d] & ⊕ & \overline{A}_{0λ} \ar[dll] \ar[d] \ar[drr] & ⊕ & A_{2λ} \ar[d] \\ B_{1} & ⊕ & B_{0} & ⊕ & B_{2}}

relative to the decomposition of A given above and any decomposition B = B_{1} ⊕ B_{0} ⊕ B_{2} with B_{k} ∈ B(\overline{E}_{k}Σ_{k}; 𝔄)(k = 0, 1, 2); and

(ii) f decomposes as

\xymatrix{A_{1} \ar[d] & ⊕ & \overline{A}_{0} \ar[dll] \ar[d] \ar[drr] & ⊕ & A_{2} \ar[d] \\ B_{1λ} & ⊕ & B_{0λ} & ⊕ & B_{2λ}}

relative to any decomposition A = A_{1} ⊕ A_{0} ⊕ A_{2} with A_{k} ∈ B(\overline{E}_{k}, Σ_{k}; 𝔄)(k = 0, 1, 2) and the decomposition given above.

For (A, p), (B, q) ∈ ˜(\overline{E}, Σ; 𝔄) and λ ∈ Λ, we now let F_{λ} Hom((A, p), (B, q)) = \{f|f = qgp for some g ∈ F_{λ} Hom(A, B)\}. Suppose f ∈ Hom((A, p), (B, q)) in B̃(\overline{E}, Σ; 𝔄). Then f : A → B in B(\overline{E}, Σ; 𝔄) and f = qfp. By Lemma 1.2 for k = 1, 2, there is a neighborhood U_{k} of Σ_{k}−Σ_{0} with U_{k} ⊂ \overline{E}−\overline{E}_{3−k} so that if either (x, y) or (y, x) is in U_{k}×\overline{E}_{3−k}, then f_{y}^{x} = 0. Let λ = U_{1} ∪ U_{2}. Clearly f ∈ F_{λ} Hom(A, B) and since f = qfp, f ∈ F_{λ} Hom((A, p), (B, q)). Thus (⋆⋆) satisfies (2).

That (⋆⋆) satisfies (3), (4) and (5) follows immediately from the definition of F_{λ} Hom((A, p), (B, q)) and the choice of the preferred decompositions of (A, p) and (B, q) given above.

This completes the proof of Lemma 3.3 □

4. Some consequences of the Mayer-Vietories property

Some consequences of the Mayer-Vietories property are obtained in this section. They are used in the next section to prove the main results of this paper. For the remainder of this paper, we let CM be the category of compact metrizable spaces and continuous maps.

Theorem 4.1. Let X be the pushout of the diagram X_{1} ⟵^{i_{1}} X_{0} ⟶^{i_{2}} X_{2} of inclusions of spaces in CM and suppose that X_{0} is a neighborhood retract in X_{i}(i = 1, 2). Then the square

\xymatrix{𝕂B̂(cX_{0}, X_{0}; 𝔄) \ar[r]^{i_{1}} \ar[d]^{i_{2}} & 𝕂B̃(cX_{1}, X_{1}; 𝔄) \ar[d]^{j_{1}} \\ 𝕂B̃(cX_{2}, X_{2}; 𝔄) \ar[r]^{j_{2}} & 𝕂B̃(cX, X; 𝔄)}

ALGEBRAIC K-THEORY WITH CONTINUOUS CONTROL AT INFINITY 15

is a pullback up to homotopy. hence there is a long exact sequence

⋯ → π_{n+1}(𝕂B) → π(𝕂B_{0}) → π_{n}(𝕂B_{1}) ⊕ π_{n}(𝕂B_{2}) → π_{n}(𝕂B) → ⋯

(n ≥ 0) where 𝕂B_{i} = 𝕂B̃(cX_{i}, X_{i}; 𝔄)(i = 1, 2, ∅) and 𝕂B_{0} = 𝕂B̂(cX_{0}, X_{0}; 𝔄).

We recall that if X_{0} = ∅, then (cX_{0}, X_{0}) = (c, ∅). Proof. Take (\overline{E}_{i}, Σ_{i}) = (cX_{i}, X_{i})(i = 0, 1, 2, ∅) in Theorem 3.1. Since X_{0} is a neighborhood retract in X_{i}(i = 1, 2), it is also a neighborhood retract in X. Hence (cX_{0}, X_{0}) is an eventual neighborhood retract of (cX, X) by Proposition 1.6 and Theorem 4.1 follows directly from Theorem 3.1 □ Corollary 4.2. For any compact metrizable space X, there are homotopy equivalences

𝕂B̂(cX, X; 𝔄) ⟵^{f} Ω𝕂B̃(cSX, SX; 𝔄) ⟶^{Ω_{i}} Ω𝕂B̂(cSX, SX; 𝔄)

where SX is the suspension of X and i is induced by the inclusion B̃(cSX, SX; 𝔄) → B̂(cSX, SX; 𝔄). Proof. The homotopy equivalence f is obtained by applying Theorem 4.1 to the pushout diagram vX ← X → wX of spaces in CM and using the fact that 𝕂˜(cuX, uX; 𝔄)(u = v, w) is contractible. Here we think of SX as the join {v, w}X. Since B̃(cSX, SX; 𝔄) is cofinal in B̂(cSX, SX; 𝔄), 𝕂B̂(cSX, SX; 𝔄) is a union of components of 𝕂B̃(cSX, SX; 𝔄) and Ωi is actually a homeomorphism. □ Corollary 4.3. For any additive category 𝔄, there are homotopy equivalences

𝕂(𝔄̂) ⟵^{f} Ω𝕂B̂(D^{1}, S^{0}; 𝔄) ⟶^{Ωi} Ω𝕂B̂(D^{1}, S^{0}; 𝔄)

Proof. Let X = ∅. Then cX is a point, B̂(cX, X; 𝔄) = 𝔄, SX is the two-point space and (cSX, X) = (D^{1}, S^{0}). Thus Corollary 4.3 is a special case of Corollary 4.2. □ Theorem 4.4. Let X be the pushout of the diagram X_{1} ⟵^{i_{1}} X_{0} ⟶^{i_{2}} X_{2} of inclusions of spaces in CM and suppose that X_{0} is a neighborhood retract in X_{i}(i = 1, 2). Then the square

\xymatrix{𝕂B̂(cX_{0}, X_{0}; 𝔄) \ar[r]^{i_{1}} \ar[d]^{i_{2}} & 𝕂B̂(cX_{1}, X_{1}; 𝔄) \ar[d]^{j_{1}} \\ 𝕂B̂(cX_{2}, X_{2}; 𝔄) \ar[r]^{j_{2}} & 𝕂B̂(cX, X; 𝔄)}

is a pullback up to homotopy. hence there is a long exact sequence

⋯ → π_{n+1}(𝕂B) → π(𝕂B_{0}) → π_{n}(𝕂B_{1}) ⊕ π_{n}(𝕂B_{2}) → π_{n}(𝕂B) → ⋯

(n ≥ 0) where 𝕂B_{i} = 𝕂B̂(cX_{i}, X_{i}; 𝔄)(i = 0, 1, 2, ∅).

The proof is given at the end of this section.

Corollary 4.5. Let X be a compact metrizable space and j_{0} : X → X × I be the inclusion at level 0. Then

𝕂B̂(cX, X; 𝔄) ⟶^{j_{0∗}} 𝕂B̂(c(X × I), X × I; 𝔄).

is a homotopy equivalence. hence the correspondence X ↦ 𝕂B̂(cX, X; 𝔄) is a homotopy functor.

Proof. Consider the inclusions ← X → X × I. Since X is a neigborhoos retract in each of these spaces, we may apply Theorem 4.4 to this diagram. Since the pushout of vX ← X → X × I is homeomorphic to vX, the two spaces at the bottom of the resulting diagram are contractible by Corollary 2.3. Corllary 4.5 now follows from theorem 4.4. The last sentence is a well-known consequence of the first part. □

Corollary 4.6. Let ; : A → X be an inclusion of spaces in CM. Then there is a fibration up to homotopy

𝕂B̂(cA, A; 𝔄) → 𝕂B̂(cX, X; 𝔄) → 𝕂B̂(c(X ∪ vA), X ∪ vA; 𝔄)

Proof. Suppose first that A is a neighborhood retract in X and consider the inclusions vA ← A → X. Since A is a neighborhood retract in each of these spaces, we may apply Theorem 4.4 to this diagram. Since 𝕂B̂(cvA, vA; 𝔄) is contractible by Corollary 2.3, this case of Corollary 4.6 follows from Theorem 4.4.

Now consider the general case. Let (X, A) be a pair of spaces in CM and consider the space X ∪ vA. Recall that vA = A × [0, 1]/A × {1}. Let v_{1/2}A be the image of A × [1/2, 1] in vA, Y = Cl[X ∪ vA − v_{1/2}A) and consider the inclusion of A into Y that sends a to (a, 1/2). We denote the image of this inclusion by A_{1/2}. Since A_{1/2} is a neighborhood retract in Y, the preceding paragraph shows there is a fibration

𝕂B̂(cA_{1/2}; 𝔄) → 𝕂B̂(cY, Y; 𝔄) → 𝕂B̂(c(X ∪ vA), X ∪ vA); 𝔄.

Since the inclusion X → Y is a homotopy equivalence, by Corollary 4.5 so is 𝕂B̂(cX, X𝔄) → 𝕂B̂(cY, Y; 𝔄). Thus we may replace 𝕂B̂(cY, Y; 𝔄) with 𝕂B̂(cX, X𝔄) to obtain the desired fibration up to homotopy. This completes the proof of Corollary 4.6. □

Proof of Theorem 4.4. Consider the pushout diagram SX_{1} ← SX_{0} → SX_{2} of inclusions of spaces in CM. Since X_{0} is a neighborhood retract in X_{i} (i = 1, 2), SX_{0} is a neighborhood retract in SX_{i}(i = 1, 2) by Proposition 1.7. By taking the loops on the spaces in the diagram obtained by applying Theorem 4.1 to these inclusions, we see that the diagram

(3)  \xymatrix{Ω𝕂B̂(cSX_{0}, SX_{0}; 𝔄) \ar[r] \ar[d] & Ω𝕂B̃(cSX_{1}, SX_{1}; 𝔄 \ar[d] \\ Ω𝕂B̃(cSX_{2}, SX_{2}; 𝔄 \ar[r] & Ω𝕂B̃(cSX, SX; 𝔄)}

is a pullback up to homotopy. On the other hand, if j_{1} : 𝕂B̂(cSX_{i}, SX_{i}; 𝔄) → 𝕂B̂(cSX_{i}, SX_{i}; 𝔄)(i = 0, 1, 2, ∅) is induced by the inclusion, then Ωj_{i} is a homeomorphism since 𝕂B̃(cSX_{i}, SX_{i}; 𝔄) is a union of components of 𝕂B̂(cSX_{i}, SX_{i}; 𝔄). Hence (3) is homeomorphic to the diagram

(4)  \xymatrix{Ω𝕂B̂(cSX_{0}, SX_{0}; 𝔄) \ar[r] \ar[d] & Ω𝕂B̂(cSX_{1}, SX_{1}; 𝔄 \ar[d] \\ Ω𝕂B̂(cSX_{2}, SX_{2}; 𝔄 \ar[r] & Ω𝕂B̂(cSX, SX; 𝔄)}

Let g_{i} : Ω𝕂B̂(cSX_{i}, SX_{i}; 𝔄) → 𝕂B̂(cX_{i}, X_{i}; 𝔄) (i = 0, 1, 2, ∅) be the inverse of the homotopy equivalence obtained by applying Corollary 4.2 to the space X_{i}. Consider the commutative cube obatined by using the maps g_{i} to map (4) into the diagram

(5)  \xymatrix{Ω𝕂B̂(cX_{0}, X_{0}; 𝔄) \ar[r] \ar[d] & Ω𝕂B̂(cX_{1}, X_{1}; 𝔄 \ar[d] \\ Ω𝕂B̂(cX_{2}, X_{2}; 𝔄 \ar[r] & Ω𝕂B̂(cX, X; 𝔄)}

We regard (4) and (5) as being the front and back faces of this cube, respectively. A straighforward argument using Corollary 4.2 shows that each of the other faces commutes up to a pointed homotopy. Since (4) is a pullback up to homotopy, the same is true for (5). This completes the proof of Theorem 4.4 □

5. The proofs of the main theorems

Let 𝔄 be a fixed additive category. To simplify notation, we suppress mention of 𝔄 when it is clear from the context. Let Ω − SPEC be the category in which an object is a Ω-spectrum {A_{n}; ε} → {B_{n}; η_{n}} is a sequence of maps {f_{n} : A_{n} → B_{n}|n ≥ 0} for which η_{n}f_{n} is pointed homotopic to (Ωf_{n+1})ε_{n}.

Let X ∈ CM and define the Ω-spectrum K̃(X; 𝔄) = {K̃(X; 𝔄)_{n}, ε_{n}} as follows: Let

(6)  K̃(X; 𝔄)_{n} = 𝕂B̂(cS^{n}X, S^{n}X; 𝔄) (n ≥ 0)

where S^{n}X is the nth suspension of X and S^{0}X = X. then Corollary 4.2 shows that there are homotopy equivalences

𝕂B̂(cY, Y; 𝔄) ⟵^{f} Ω𝕂B̃(cSY, SY; 𝔄) ⟶^{Ω_{j}} Ω𝕂B̂(cSY, SY; 𝔄)

form which we obtain a homotopy equivalence

(7)  ε_{n} = (Ωj)f^{1} : 𝕂B̂(cS^{n}X, S^{n}X; 𝔄) → Ω𝕂B̂(cS^{n+1}X, S^{n+1}X; 𝔄)

by taking Y = S^{n}X and choosing a homotopy inverse f^{−1} for f. Thus {K̃(X; 𝔄)_{n}, ε_{n}} is an Ω-spectrum and it is readily checked that the correspondance X ↦ K̃(X, 𝔄) denfines a functor

K̃(−; 𝔄) : CM → Ω − SPEC.

Proposition 5.1. The functor K̃(−; 𝔄) has the following properties: it is a homotopy functor; K̃(pt; 𝔄) is contractible; and there is a natural isomorphism of Ω-spectra ΣK̃(X; 𝔄) → K̃(SX; 𝔄).

Proof. To see K̃(−; 𝔄) is a homotopy functor, let j_{0} : X → X × I be the inclusion at level 0 and recall that

𝕂B̂(cX, X; 𝔄) ⟶^{j_{0∗}} 𝕂B̂(c(X × I), X × I; 𝔄)

is a homotopy equivalence by Corollary 4.5. A standard argument (cf. [2, p.575]) now easily adapts to prove that j_{0} induces a homotopy equivalence of spectra and that K̃(−; 𝔄) is a homotopy functor. The second statement follows from the definition of K̃(pt; 𝔄) and Corollary 2.3. The last statement is immediate from the definition of K̃(X). □

Let K̃_{∗}(−; 𝔄) = π_{∗}^{S}K̃(−; 𝔄) : CM → GA where π_{∗}^{S} is the stable homotopy functor aand GA is the category of graded abelian groups. Theorem I is a consequence of the following more complete result:

Theorem 5.2. The functor K̃_{∗}(−; 𝔄) = π_{∗}^{S}K̃(−; 𝔄) : CM → GA is a reduced homology theory on CM; that is,

(i) K̃_{∗}(−; 𝔄) is a homotopy functor.

(ii) There is a natural isomorphism Σ : K̃_{∗}(X; 𝔄) → K̃_{∗+1}(SX; 𝔄).

(iii) For any pair (X, A) of spaces CM, there is an exact sequence

⋯ → K̃_{∗}(A; 𝔄) → K̃_{∗}(X; 𝔄) → K̃_{∗}(X ∪ vA; 𝔄) → K̃_{∗−1}(A; 𝔄) → · · · .

(iv) widetildeK_{∗}(pt; 𝔄) = 0.

Proof. Part (i) follows directly from Proposition 5.1;(ii) follows from the definition of K̃(X; 𝔄); while (iv) follows from proposition 5.1. Finally for every n ≥ 0, there is a fibration up to homotopy

𝕂B̂(cS^{n}A, S^{n}; 𝔄) → 𝕂B̂(cS^{n}X, S^{n}X; 𝔄) → 𝕂B̂(cS^{n}(X ∪ vA), S^{n}(X ∪ vA); 𝔄)

by Corollary 4.6. The associated homotopy exact sequence may now be spliced together using (ii) to obtain the exact sequence of (iii). □

Fro any pair (X, A) of spaces in CM, let K_{∗}(X, A; 𝔄) = K̃_{∗}(X ∪ vA) where the union is over A and vA is a point if A = ∅. In particular, X ∪v∅ = X^{+} is X with a disjoint basepoint attaches and K_{∗}(X, ∅; 𝔄 = K̃_{∗}(X^{+}). The following result is Theorem II:

Theorem 5.3. The functor K_{∗}(X, A; 𝔄) is a homology theory on the category of pairs of spaces in CM; that is,

(i) K_{∗}(−, −; 𝔄) is a homotopy functor.

(ii) For any pair (X, A) of spaces in CM, there is an exact sequence

⋯ toK_{∗}(A^{+}; 𝔄) → K_{∗}(X^{+}; 𝔄) → K_{∗}(X, A; 𝔄; 𝔄) → K_{∗−1}(A^{+}; 𝔄) → · · · .

(iii) If U is an open set with Cl(U) ⊂ ∫ (A), then the inclusion map (X − U, A − U) → (X, A) induces an isomorphism K_{∗}(X − U, A − U) → K_{∗}(X, A).

Furthermore, if we restrict this homology theory to the category of finite CW complexes, the representing spectrum K(pt; 𝔄) = K̃(S^{0}; 𝔄) has Ω(K(pt; 𝔄) = K(𝔄).

In this theorem K(𝔄) is the non-connective spectrum that [8] associates with the additive category 𝔄. The reader will recall that K(𝔄) = K(𝔄̃).

Proof. That K_{∗} satisfies the homotopy and exactness condition axioms follows directly from the corresponding statements for K̃_{∗}. That K_{∗} satisfies the excision axiom follows from the arguments given in [6, p. 19]. □

It remains to identify the representing spectrum K(pt; 𝔄) for the restriction of K_{∗} to the category of finite CW complexes. Let K̃′(S^{0}; 𝔄) = {K̃′(S^{0}; 𝔄)_{n}, ε′_{n}} be the spectrum with

K̃′(S^{0}; 𝔄)_{n} = 𝕂Ĉ(ℝ^{n+1}; 𝔄) ⟶^{ε′_{n}} Ω𝕂Ĉ(ℝ^{n+2}; 𝔄) = ΩK̃′(S^{0}; 𝔄)_{n+1}

constructed from the homotopy equivalences of [8]

𝕂Ĉ(ℝ^{n+1}; 𝔄) ⟵^{f′} Ω𝕂C̃(ℝ^{n+2}; 𝔄) ⟶^{Ω′_{j}} Ω𝕂Ĉ(ℝ^{n+2}; 𝔄)

by choosing a homotopy inverse for f′. Recall that these are obtained by a Mayer-Vietories argument similar to the proof of Corollary 4.2 that uses the categories C(−; 𝔄) instead of B(−; 𝔄). Then K̃′(S^{0}; 𝔄) is the spectrum that [8] assocites with S^{0}.

We now identify ℝ^{n+1} = O(S^{n}) with oS^{n} = ∫ D^{n+1} = {x ∈ ℝ^{n+1}| ∥ x ∥< t} as above and write cS^{n} = D^{n+1}. Since every bounded map of ℝ^{n+1} extends via the identity on S^{n} to a map of D^{n+1}, there is an inclusion of additive categories

i : Ĉ(ℝ^{n+1}; 𝔄) → B̂(D^{n+1}, S^{n}; 𝔄).

It is easily checked that these inclusions for n ≥ 0 induce a map of spectra i_{∗} : K̃′(S^{0}; 𝔄) → K̃(S^{0}; 𝔄). The proof of Theorem 5.3 is completed by proving the following lemma:

Lemma 5.4. The map i_{∗} : K̃′(S^{0}; 𝔄) → K̃(S^{0}; 𝔄) is a homotopy equivalence of spectra. In particular, ΩK̃(S^{0}; 𝔄) = K(𝔄).

Proof. Since [8] and [9] show that ΩK̃′(S^{0}; 𝔄) = K(𝔄), the second sentence follows from the first. To show the first part, it suffices to show that i_{∗} : K̃′_{∗}(S^{0}; 𝔄) → K̃_{∗}(S^{0}; 𝔄) is an isomorphism. Notice that by the definition of K̃(S^{0}; 𝔄) and Corollary 4.2,

K̃_{n}(S^{0}; 𝔄) = K_{n}B̂(D^{1}, S^{0}; 𝔄) if n ≥ 0.

Similarly

K̃′_{n}(S^{0}; 𝔄) = K_{n}Ĉ(ℝ^{1}; 𝔄) if n ≥ 1,

and

K̃′_{−n}(S^{0}; 𝔄) = K_{1}Ĉ(ℝ^{n+2}; 𝔄) if n ≥ 0.

We examine

i_{∗} : K_{n}Ĉ(ℝ^{1}; 𝔄) → K_{n}B̂(D^{1}, S^{0}; 𝔄)

for n ≥ 1 in Lemma 5.5 and

i_{∗} : K_{1}Ĉ(ℝ^{n+2}; 𝔄) → K_{0}B̂(D^{n+2}, S^{n+1}; 𝔄)

for n ≥ 0 in Lemma 5.6. □

Lemma 5.5. For any additive category 𝔄, there is a homotopy commutative diagram of homotopy equivalences

\xymatrix{ & 𝕂(Â) \ar[dl]_{ε′} \ar[dr]^{ε} \\ Ω𝕂Ĉ(ℝ^{1}; 𝔄) \ar[rr]^{Ω_{i}} &  & Ω𝕂B̂(D^{1}, S^{0}; 𝔄)}

Hence for all n ≥ 1, i_{∗} : K_{n}Ĉ(ℝ^{1}; 𝔄) → K_{n}B̂(D^{1}, S^{0}; 𝔄) is an isomorphism.

Proof. We claim that there is a homotopy commutative diagram

\xymatrix{𝕂𝔄̂ \ar[d] & Ω𝕂C̃(ℝ^{1}; 𝔄) \ar[l]_{f′} \ar[d]^{Ωi} \ar[r]^{Ωj′} & Ω𝕂Ĉ(ℝ^{1}; 𝔄) \ar[d]^{Ωi} \\ 𝕂𝔄̂ & Ω𝕂B̃(D^{1}, S^{0}; 𝔄) \ar[l]_{f} \ar[r]^{Ωj} & Ω𝕂B̂(D^{1}, S^{0}; 𝔄)}

in which all the horizontal maps are homotopy equivalences. Letting ε′ = j′f^{′−1} and ε = jf^{−1}. where f^{′−1} and f^{−1} are homotopy inverses for f′ and f will complete the proof. To obtainn the diagram, recall the homotopy equivalences f was constructed in Corollary 4.3 using a Mayer-Voetories argument. A corresponding Mayer-Vietories argument using C, etc. instead of B, etc. gives the homotopy equivalence f′. Since the Mayer-Vietories diagram using C maps to the corresponding diagram using B, the left-hand square homotopy commutes. The right-hand square commutes since the underlying diagram of categories commutes. □

Lemma 5.6. Let n ≥ 0. Then i_{∗} : K_{1}Ĉ(ℝ^{n+2}, 𝔄)toK_{1}B̂(D^{n+2}, S^{n+1}; 𝔄) is an isomorphism.

Proof. Since the map Ω_{i} : Ω𝕂C(ℝ^{n+2}; 𝔄) → Ω𝕂Ĉ(ℝ^{n+2}; 𝔄) induced by the inclusion is a homeomorphism, we may replace Ĉ with C in this lemma. Similarly, we may replace B̂ with B.

To see that i_{∗} is onto, let x = [A, f] ∈ K_{1}B(D^{n+2}, S^{n+1}; 𝔄) where f : A → A is an isomorphism in B(D^{n+1}, S^{n}; 𝔄). By Theorem 2.1(i) and Lemma 2.7, there is a homeomorphism h of D^{n+2} = ℝ^{n+2} ∪ S^{n+1}(∞) onto itself with h|_{S^{n+1}}(∞) = 1 for which h_{∗}(f) is bounded. Since h_{∗} is the identity by Lemma 1.3, x = [A, f] = [h_{∗}(A), h_{∗}(f)] and i_{∗} is onto.

For any additive category 𝔄, let 𝔄[t, t^{−1} be the “polynomial extension” categoryu of 𝔄 introduced by Ranicki [10]. This category has two descriptions. In one description, the objects of 𝔄[t, t^{−1} are the same as the objects of 𝔄 and Hom(A, B) is the set of all formal sums ∑_{−∞}^{∞} t^{n}f_{n} with f_{n} = 0 for all n with |n| large enough. In the other description, the objects of 𝔄[t, t^{−1} are the objects {A_{n}|n ∈ ℤ} of C(ℤ, 𝔄) for which A_{n} = A_{0} for all n and the morphisms f : A → B are the ℤ-equivariant morphisms f = {f_{m}^{n} : A_{n} → B_{m}|n, m ∈ ℤ} in C(ℤ; 𝔄). In either case, we let A[t, t^{−1}] denote the object of 𝔄[t, t^{−1}] determined by A ∈ 𝔄.

The arguments given in Section 2 of [7] extend easily to define a “Bass-Heller-Swan” homomorphism λ_{C} : K_{1}C(ℝ^{n+1}; 𝔄) → K_{1}C(ℝ^{n}; 𝔄[t, t^{−1}]) for any n ≥ 0 and to show that λ_{C} is injective. Since K_{1}𝔄̂ = K_{1}𝔄 for nay additive category, the facxt that i_{∗} is one to one follows fron the next lemma by an obvious inductive argument. This completes the proof of Lemma 5.6. □

Lemma 5.7. Let n ≥ 0. there is a homomorphism λ_{B} making the following diagram commute:

\xymatrix{K_{1}C(ℝ^{n+1}; 𝔄) \ar[d]^{i_{∗}} \ar[r]^{λ_{C}} & K_{1}C(ℝ^{n}; 𝔄[t, t^{−1}]) \ar[d]^{i_{∗}} \\ K_{1}B(D^{n+1}, S^{n}; 𝔄) \ar[r]^{λ_{B}} & K_{1}B(D^{n}, S^{n−1}; 𝔄[t, t^{−1}])}

Proof. The construction of λ_{B} will follow Pedersen’s construction [7] of λ_{C} quite closely.

Let D(𝔄) be either C(ℝ^{n+1}; 𝔄) or B(D^{n+1}, S^{n}; 𝔄), A = {A_{x}|x ∈ X} ∈ D(𝔄), and A[t, t^{−1}] = {A_{x}[t, t^{−1}]|x ∈ X} be the object of D(𝔄[t, t^{−1}]) determined by A. Here X is either ℝ^{n+1} or D^{n}n + 1. Let p_{−} : A → A be the projection that is the identity on A_{x} if x = (x_{1}, …, x_{n+1}) has x_{n+1} < 0 and is zero otherwise. Let p_{t} be the morphism of A[t, t^{−1}] = {A_{x}[t, t^{−1}]|x ∈ X} given by p_{t} = tp_{−} + (1 − p_{−}) and notice that p_{t} is an automorphims with inverse t^{−1}p_{−} + (1 − p_{−}). Finally, if a : A → A is an automorphims, we let a_{t} = a[t, t^{−1}] be the automorphims of A[t, t^{−1}] induced from a.

Let [A, a] represent an element of K_{1}D(𝔄) and consider the commutator [a_{t}, p_{t}]. If D(𝔄) = C(ℝ^{n+1}; 𝔄) and a has bound d, [7, p. 469] shows that [a_{t}, p_{t}|ℝ^{n+1} − B = id where B is the band {x = (x_{1}, …, x_{n+1}||x_{n+1} ≤ 2d}. It also shows that [a_{t}, p_{t}] restricts to an automorphims of A[t, t^{−1}]|_{B} and lets λ_{C}([A, a]) be represented by [\overline{A}[t, t^{−1}], [a_{t}, p_{t}]]. here if y = (y_{1}, …, y_{n}), \overline{A}[t, t^{−1}] ∈ C(ℝ^{n}; 𝔄[t, t^{−1}]) has \overline{A}[t, t^{−1}]_{y} = ∑\{A_{x}[t, t^{−1}]|x = (y_{1}, …, y_{n}, x_{n+1}) with ||x_{n+1} ≤ 2d}.

If D(𝔄) = B(D^{n+1}, S^{n}; 𝔄) and a is continuously controlled at infinity, then by appl;ying Lemma 1.2 with K = H_{±} = {x = (x_{a}, …, x_{n+1}| ± x_{n+1} ≥ 0} and W = ∫ D_{∓} where D_{∓} = H_{±} ∩S^{n}, there are neighborhoods U_{±} of ∫ D_{±} so that if a^{−1} = b = {b_{y}^{x}|x, y ∈ ∫ D^{n+1}}, then b_{y}^{x} = 0 for (x, y) ∈ (U_{±} × H_{∓}) ∪ (H_{±} × U_{∓}). It now follows that c = [a_{t}, p_{t}] has

c|_{U+±} = id : A[t, t^{−1}]|_{U_{±}} → A[t, t^{−1}]|_{U_{±}}.

Consider f = c| : A[t, t^{−1}]|_{M} → A[t, t^{−1}]| where M = D^{n+1} − (U_{+} ∪ U_{−}). Applying Lemma 1.2 a second time with K = M and W = ∫ D_{+} ∪ D_{[}, there is a neighborhood V of W with f_{y}^{x} = 0 if (x, y) ∈ V × K ∪ K × V. Let W_{±} = U_{±} ∩ V. It is thenb easy to check that c|_{W_{±}} = id : A[t, t^{−1}]_{M_{±}} → A[t, t^{−1}|_{W_{±}} and that the rstriction of c maps A[t, t^{−1}]|_{D^{n+1}−(W_{+}∪W_{−})} isomorphically onto itself. Let λ_{B}([A, a]) be represented by [\overline{A}[t, t^{−1}], [a_{t}, p_{t}]] where if y = (y_{1}, …, y_{n}) then \overline{A}[t, t^{−1}] ∈ B(D^{n}, S^{n−1}; 𝔄[t, t^{−1}]) had \overline{A}[t, t^{−1}]_{y} = ∑ A_{x}[t, t^{−1}] where the sum runs over those x ∈ im{z × ℝ} ∩ {D^{n+1} − (W_{+} ∪ W_{−})}. Here z = (1− ∥ y ∥^{−1} y and im{z × ℝ} is the image of this line in ℝ^{n+1} under the identification of ℝ^{n+1} with ∫ D^{n+1} given above. The proof that λ_{C} is well defined given in [7, Theorem 2.3] carries over, almost without change, to show that λ_{B} is a well-defined homomorphism.

Since it follows immediately from the definitions of λ_{C} and λ_{B} that the diagram above commutes, this completes teh proof of Lemma 5.7. □

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ALGEBRAIC K-THEORY WITH CONTINUOUS CONTROL AT INFINITY 23

Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA Department of Mathematics, Suny at Binghamton, Binghamton, NY 13901, USA Department of Mathematics, SUNY at Binghamton, Binghamton, NY 13901, USA