A COMPARISON OF CONTINUOUSLY CONTROLLED AND CONTROLLED K-THEORY

DOUGLAS R. ANDERSON FRANCIS X. CONNOLLY HANS J. MUNKHOLM

Abstract. We define an unreduced version of the ϵ-controlled lower K-theoretic groups of Ranicki and Yamasaki, [?], and Quinn, [?]. We show that the reduced versions of our groups coincide (in the inverse limit and its first derived, lim^{1}) with those of [?]. We also relate the controlled groups to the continuously controlled groups of [?], and to the Quinn homology groups of [?].

Douglas R. Anderson was partially supported by the NSF (USA) under grant number DMS-9202598; Hans J. Munkholm was partially supported by the SNF (Denmark) under grants number 11-7792 and number 11-0062-1 PD.

Contents

1. Introduction.

1.1. The main results. The main results are the following two theorems which appear (the first one in a slightly more general form) in Sections ?? and ??, respectively. In them, the natural numbers do not include 0, i.e., ℕ = {1, 2, ⋯ }.

Theorem 1.1. Let p : H → B be a map from a topological space H into a compact metric space B, and let j ∈ ℕ. There is a natural, short exact sequence

0 → lim^{1}K_{2−j}(B; p)_{ϵ} → K_{2−j}^{cc}(c(B_{+}), B_{+}; c^{∘}(p_{+})) → K_{1−j}(B; p)_{c} → 0.

Assume further that each path component in H and in B is open. Then there is a reduced version of this seqeunce, i.e., a natural, short exact sequence

0 → lim^{1}K̃_{2−j}(B; p)_{ϵ} → K̃_{2−j}^{cc}(c(B_{+}), B_{+}; c^{∘}(p_{+})) → K̃_{1−j}(B; p)_{c} → 0.

Theorem 1.2. Let (\overline{Y} , C) be a pair of compact, metrizable spaces with C tame in \overline{Y}. Let

Y = \overline{Y} − C ⟵^{q} Holink(C, \overline{Y}) ⟶^{p} C

be the correponding holink diagram. There are natural chain complexes of abelian groups

K_{2}(Y) ⟶^{i_{2}} K_{2}^{cc}(\overline{Y} , C; id_{Y}) ⟶^{Δ_{2}} K_{1}(C; p)_{c} ⟶^{ℓ_{1}} K_{1}(Y) ⟶^{i_{1}} K_{1}^{cc}(\overline{Y} , C; id_{Y}) ⟶^{Δ_{1}} ⋯

and

W h(C; p)_{c} ⟶^{ℓ_{1}} W h(Y) ⟶^{i_{1}} W h^{cc}(\overline{Y} , C; id_{Y}) ⟶^{Δ_{1}} K̃_{0}(C; p)_{c} ⟶^{ℓ_{0}} K̃_{0}(Y) ⟶^{i_{0}} · · · .

For each j ∈ ℕ, these complexes are exact at K_{1−j}(C; p)_{c} and K_{1−j}(Y), respectively at K̃_{1−j}(Y; p)_{c} and K̃_{1−j}(Y); the homology groups at K_{2−j}^{cc}(\overline{Y} , C; id_{Y}) and at K̃_{2−j}^{cc}(\overline{Y} , C; id_{Y}) are naturally isomorphic to lim^{1}K_{2−j}(C; p)_{ϵ}.

For the reduced version there is the added hypothesis that each path component in each space in the holink diagram must be open.

Remarks: 1) In Section ??, we recall and comment upon a result of Ferry and Pedersen concerning the vanishing of the lim^{1} groups above.

2) We strongly believe that the theorems hold also for j = 0, but we emphasize that the present paper does not prove this to be true.

3) The extra conditions on path components assumed for the reduced version are discussed in Section ??. □

Let p : E → B = |K| be a map which admits an iterated mapping cylinder structure (also called a homotopy colimit structure, see [?]) over the finite simplicial complex K, and let S : T OP → SPEC be the functor which associates to a space X its non-connective K-theory spectrum as defined, e.g., in [?]. By Theorem 0.1 of [?], it is known that Quinn’s homology groups H_{∗}(ΣK, v_{+}; S(Σp)), derived from this functor S as in the appendix of [?], are isomorphic to the boundedly controlled groups K_{∗}(p) defined in [?]. On the other hand, in [?], the latter groups are shown to coincide with the cc K-theory groups occurring in Theorem 1.1. Thus, we have the following corollary.

Corollary 1.3. For each p as above, and each j ∈ ℕ, there is a short exact sequence

0 → lim^{1}K_{2−j}(|K|; p) → H_{2−j}(Σ|K|, v_{+}; S(Σp)) → K_{1−j}(|K|; p)_{c} → 0.

The groups that appear in the above theorems can briefly be described as follows. For each ϵ > 0, and each map p : H → B, ϵ-controlled, reduced K-groups W h(B; p)_{ϵ} (= K̃_{1}(B; p)_{ϵ}) and K̃_{0}(B; p)_{ϵ} have been defined in a geometric way by Chapman, [?], and in an algebraic one by Quinn, [?], as well as [?]. Our ϵ decorated groups are unreduced versions of the groups defined in [?] and [?]. They form inverse systems under connecting homomorphisms K_{2−j}(B; p)_{ϵ} → K_{2−j}(B; p)_{δ} defined for ϵ ≤ δ, and K_{2−j}(B; p)_{c} is the inverse limit group limK_{2−j}(B; p)_{ϵ} (and similarly for reduced groups). Our definition is given in Section ?? in terms of homotopy classes of path matrix morphisms between geometric modules in the space H and with “ϵ-control on maps and homotopies” in the space B via the map p.

The decoration ^{cc} means continuously controlled. Let (\overline{Y} , C) be a pair as in Theorem 1.2, and let R be any (reasonable) ring (which we shall usually suppress from the notation). In [?], the authors study the K-theory, K_{∗}^{cc}(\overline{Y} , C; R), of geometric Rmodules (or, more generally, objects in any additive category) in the space Y = \overline{Y} −C with morphisms which are “continuously controlled” at C. Although the ideas from [?] have had important applications (see e.g. [?] and [?]), from a geometric viewpoint it seems more natural to allow (homotopy classes of) path matrix morphisms between geometric modules in a space H which comes with a map p : H → Y = \overline{Y} − C. This map is then used to introduce control at C. This viewpoint is introduced in the continuously controlled setting by two of the present authors in [?].

In Theorem 1.1, the subscript + refers to the addition of a disjoint base point. Also, c^{∘}(p_{+}) is the induced map from the open cone c^{∘}(H ⨿ X) into c^{∘}(B_{+}) = c(B_{+}) − B_{+}. Thus, the cc groups appearing there are derived from the additive category of geometric modules in the open cone of H ⨿ X with morphisms that are cc homotopy classes of cc path matrix morphisms.

In Theorem 1.2, id_{Y} is the identity map on Y = \overline{Y} − C, so the cc groups appearing there date back to geometric modules in Y with path matrix morphisms which are cc at C (up to homotopies which are also cc at C).

We proceed to describe the background for our interest in these results.

1.2. Background. Both Quinn and Chapman were motivated by the controlled boundary problem for a manifold M, equipped with a map p : M → B with B as above. The problem is to decide under which conditions M admits a boundary in such a way that p extends over M ∪ ∂M. Under suitable a priori conditions on the dimension and the behaviour of M near ∞, Chapman shows that the problem is completely governed by a primary obstruction and a secondary obstruction which lie in groups that are constructed via inverse limits and lim^{1} from suitable groups of the above ϵ-controlled type. Quinn, in [?], also gave an obstruction theory solution to the controlled boundary problem, but the obstruction group is constructed (in Sections 5 and 8 in [?]) in a less transparent way. In his Notre Dame CBMS-lectures in 1984, Quinn outlined an algebraic definition of groups W h(B; p)_{ϵ} and K̃_{0}(B; p)_{ϵ} in terms of geometric modules in M and (ϵ-small homotopy classes of) path matrix morphisms in M with ϵ-short image paths in B. These groups can be directly related to the cellular chains on M (assuming a “controlled” cellular structure), and this introductory part of Quinn’s lecture series was apparently geared towards a direct description (in terms of chain complexes) of the obstructions obtained in [?]. Unfortunately, this approach was never carried to fruition, although part of the projected CBMS volume appeared later as [?]. In fact, to the best of the present authors’ knowledge,

(A) no one has ever established a strict link between the obstruction groups that

appear in [?] and the groups defined in [?]. In particular, even though Quinn’s obstruction groups do form a homology theory (as described in Section 8 of [?]),

(B) there is no documentation in the literature to support the folklore belief that in some great generality the inverse limit groups W h(B; p)_{c}, K̃_{0}(B; p)_{c}, ⋯ are the groups of a homology theory.

Also,

(C) no solid foundation for an interpretation of Quinn’s obstructions in terms of

chain complexes in some category of geometric modules seems to exist. Along a similar vein, apparently

(D) nobody has taken time out to construct the expected isomorphism between Chapman’s geometrically defined and Quinn’s algebraically defined ϵ-controlled groups, and to study its properties.

The present paper is related to all of these problems. Starting from behind, (D) should actually be very accessible now that the algebraic version of the groups involved is equally well understood (thanks to [?] and the present paper) as is the original, geometric version in [?].

Since Chapman’s obstructions are quite explicitly (and classically) given in [?], the precise form of a solution to (D) might actually be of value in trying to understand (C). Since the primary obstruction is in a group of type K̃_{0}(B; p)_{c} and the secondary in one of type lim^{1}W h(B; p)_{ϵ}, Theorem 1.1 suggests a total obstruction in a certain cc Whitehead group W h^{cc}(\overline{M}, B; id_{M}); see Remark 4) following ??.

Recently, in [?], Ranicki and Yamasaki have given a chain complex version of Quinn’s ϵ-controlled groups from [?]. Their main goal is to give an algebraic proof of the result of Chapman [?], that Whitehead torsion is a topological invariant. As such it represents the culmination of the program initiated by Connell and Hollingsworth, in [?], with the invention of the notion of a geometric group. They also reprove other results, including the Borsuk conjecture (originally proved by West, in [?]). Their work includes “stably exact” Mayer-Vietoris sequences which are, of course, very much related to our problem (B), but they do not attack (B) directly. In [?], it is shown that the cc groups appearing in Theorem 1.1 do form a homology theory, defined on a category of maps like p (or, more generally, on a category of diagrams of holink type). Thus, the controlled groups K_{2−j}(B; p)_{c} have homology theory behavior, provided the lim^{1}-terms in Theorem 1.1 vanish. We discuss a vanishing result (due essentially to Ferry and Pedersen) briefly in Section ??.

1.3. Historical remarks and acknowledgements. The exact sequences in Theorem 1.1 were conjectured to be true (with a boundedly controlled group instead of the cc group) a long time ago by two of the authors, DRA and HJM. Actually, HJM’s lecture at the Oberwolfach Topologie Tagung in 1987, contained an explicit version of this conjecture. One ingredient in the proof (injectivity of lim^{1}K_{1}(B; p)_{ϵ} → K_{1}^{cc}(c(B_{+}), B_{+}; c^{∘}(p_{+}))) eluded them and the project was essentially shelved until HJM came to University of Notre Dame on a sabbatical in 1992/93. There, discussions between him and the third author, FXC, about the groups K^{cc}(\overline{Y} , C; id_{Y}) appearing in Theorem 1.2, renewed the interest and the missing part of the proof was finally found.

DRA wants to thank the Mathematics Department at the University of Notre Dame and the Mathematical Sciences Research Institute, Berkely, for their kind hospitality during different periods when this paper was in preparation.

HJM wants to thank the Mathematics Department of University of Notre Dame for a pleasant atmosphere (mathematical and otherwise) during the above mentioned sabbatical. Special thanks go to Bruce Williams for his help in arranging the sabbatical and for his never ending willingness to discuss mathematics and to search through his extensive files of (p)reprints to find the one you want, or even the one you ought to want. He also thanks the Mathematical Sciences Research Institute, Berkeley, for its hospitality during the summer of 1993 when much of the actual writing was done. The delay in the appearance of this preprint arises from the authors’ desire to have [?] in final form before using results from it.

2. Geometric modules and path matrix morphisms.

In this section we set up the general framework (a priori involving no control) for geometric modules, path matrix morphisms, and path matrix homotopies in any topological space E.

We recall that a Moore path in E is a pair (ω, l_{ω}) where ω : [0, ∞) → E is continuous, and constant on the subray [l_{ω}, ∞). A homotopy between Moore paths is a pair (H, l_{H}) where l_{H} : [0, 1] → [0, ∞) and H : [0, ∞) × [0, 1] → E are continuous, and H(−, y) is constant on [l_{H}(y), ∞) for each y ∈ [0, 1]. As usual, (H, l_{H}) is a homotopy from δ_{1}(H, l_{H}) = (H(−, 0), l_{H}(0)) to δ_{0}(H, l_{H}) = (H(−, 1), l_{H}(1)). Often l_{ω} and l_{H} will be suppressed from the notation, ω(l_{ω}) will be denoted ω(∞), and H(l_{H}(y), y) will be denoted H(∞, y). The set, M(E), of Moore paths in E will be viewed as a category whose objects are the points of E and where the morphisms from x to y are the Moore paths ω with y = ω(0) and x = ω(∞) (so that the usual concatenation of paths defines the composition). Also, H(E) denotes the set of end point fixing homotopies, so δ_{i} is a map H(E) → M(E), for i = 0, 1.

A geometric module on E is a pair (S, σ) with S a discrete set and σ : S → E a function. We think of each s ∈ S as a basis element which “sits at” the point σ(s) ∈ E. A (path matrix) morphism φ : (T, τ) → (S, σ) is a function^{1} φ : S ×T → RM(E) into the free R-module generated by M(E). There are two requirements on φ(s, t) = Σ_{ω}r_{st}(ω)ω, viz.

r_{st}(ω) ≠ 0 ⇒ [ω(0) = σ(s) and ω(∞) = τ(t)], (1)

and

∀t ∈ T : {(s, ω) ∈ S × M(E) | r_{st}(ω) ≠ 0} is finite. (2)

The former condition says that each Moore path is to be considered a morphism from its end point to its beginning point; general morphisms are then R-linear combinations of such “atomic” morphisms. The second condition guarantees that matrix multiplication and the composition in M(E) give a well defined composition of morphisms. The resulting category will be denoted GM(E). It is additive, with addition of morphisms coming from the abelian group structure in RM(E) and with (S_{1}, σ_{1}) ⊕ (S_{2}, σ_{2}) given by the disjoint union (S_{1} ∐ S_{2}, σ_{1} ∐ σ_{2}). The inclusions and projections

(S_{i}, σ_{i}) ⟶^{ι_{i}} (S_{1}, σ_{1}) ⊕ (S_{2}, σ_{2}) ⟶^{π_{i}} (S_{i}, σ_{i}), i = 1, 2,

involve only constant paths γ_{x}, x ∈ E. For example, ι_{2}(s, s_{2}) = δ_{s,s_{2}}γ_{σ_{2}(s_{2})} for (s, s_{2}) ∈ (S_{1} ∐ S_{2}) × S_{2}, where δ is the Kronecker delta. Incidentally, if S_{1} = ∅, this defines the identity morphism on (S_{2}, σ_{2}).

1The indexing corresponds to classical matrix notation, i.e., the row index s corresponds to the range, and the column index t to the domain.

We call a morphism φ : M = (T, τ) → N = (S, σ) geometric^{2} if it has the form

φ(t, s) = δ_{t,α(s)}u_{s}ω_{s}, (3)

where α : S → T is a map; u_{s} ∈ R; and ω_{s} ∈ M(E) has ω_{s}(∞) = τα(s), ω_{s}(0) = σ(s)^{3}. We note that the above inclusions ι_{i} and projections π_{i} are examples of such geometric morphisms.

If α is a bijection, each u_{s} is a unit in R, and each ω_{s} is constant, then the corresponding φ is an isomorphism in GM(E) with an inverse obtained by inverting α and each u_{s}. Clearly, the direct sum operation ⊕ is associative and commutative up to coherent, geometric isomorphisms of this sort (with all u_{∗} = 1). By abuse of set theory, we shall actually proceed as if ⊕ is strictly associative, i.e., as if ∐ is strictly associative in the category of sets.

A (path matrix) homotopy from (T, τ) to (S, σ) is a function Φ : S × T → RH(E), say Φ(s, t) = Σ_{H}r_{st}(H)H, which satisfies

r_{st}(H) ≠ 0 ⇒ [H(0, y) = σ(s) and H(∞, y) = τ(t), ∀y ∈ [0, 1]], (4)

and

∀t ∈ T : {(s, H) ∈ S × H(E) | r_{st}(H) ≠ 0} is finite. (5)

The end point maps δ_{i} : H(E) → M(E), i = 0, 1, extend by linearity to maps δ_{i} : RH(E) → RM(E), and Φ is understood to be a homotopy from δ_{1}Φ to δ_{0}Φ. We also write Φ : δ_{1}Φ ≃ δ_{0}Φ. This homotopy notion behaves well viz-a-viz composition, sum and direct sums so that one has a corresponding, additive, homotopy category GM(E).

In this generality, GM(E) and GM(E) are of little interest. To arrive at interesting constructions, one has to impose conditions on the objects, the morphisms, and the homotopies allowed: For objects one has some “finiteness” condition on σ : S → E. For morphisms φ and homotopies Φ the conditions are expressed in terms of the following families of subsets of E (where φ(s, t) = Σr_{st}(ω)ω and Φ(s, t) = Σr_{st}(H)H).

(6)  F(φ) = {ω([0, ∞)) ⊆ E | ∃(s, t) ∈ S × T : r_{st}(ω) ≠ 0},
(7)  F(Φ) = {H([0, ∞) × [0, 1]) ⊆ E | ∃(s, t) ∈ S × T : r_{st}(H) ≠ 0}.

In Section ?? we study one such set of conditions (“ϵ-control”) in some detail. Another set of conditions (“continuous control”) is studied by two of the present authors in [?], in part in order to obtain the cc results needed in the present paper. Some definitions and results from [?] are summarized in Section ??.

2In [?], such a morphism (or rather its homotopy class) is said to be “induced by a basis morphism”. We have borrowed the present term from [?] and [?]. However, we use the term “geometric”

in connection with morphisms only in the context defined here, in contradistinction to [?], where all path matrix morphisms are called “geometric morphisms”; but where “geometric” is also used in our, restricted sense.

3i.e., for given s, σ(s) is the beginning point of at most one path having non zero coefficient in φ; this path ends at τα(s) and has multiplicity u_{s}.

3. Controlled K-theory.

The category T OP_{c} of controlled spaces has as its objects all pairs (B; p) where p : E → B be a map from an arbitrary space E into a compact metric space B. A morphism (f, g) : (B; p) → (B′; p′) is defined to be a pair of continuous maps

(f : E → E′, g : B → B′) with p′f = gp. (8)

Let j ∈ ℕ and ϵ > 0. In this section we define and study the ϵ-controlled groups K_{2−j}(B; p)_{ϵ}, (which are not functorial on T OP_{c}) as well as the controlled groups K_{2−j}(B; p)_{c} (= lim K_{2−j}(B; p)_{ϵ}), and the lim^{1}-groups lim^{1}K_{2−j}(B; p)_{ϵ} (which are functorial on T OP_{c}).

The first subsection deals with the cases j = 1 and j = 2, i.e., with K_{1}− and K_{0}−groups. Reduced versions of these are defined in Section ??, and compared to the similar groups defined by Ranicki and Yamasaki ([?]) in Section ??. Finally, in Section ??, results from [?] and Section ?? are utilized to define the lower groups K_{2−j}(B; p)_{ϵ}, K_{2−j}(B; p)_{c} and lim^{1}K_{2−j}(B; p)_{ϵ} for j > 2. This uses a geometric form of Bass’s contracted functor viewpoint, cf. [?].

First, however, we must discuss the relevant conditions on modules, morphisms and homotopies. All geometric modules σ : S → E will have S finite. They will often be denoted by M, N, L, ⋯ instead of (S, σ), (T, τ), (R, ρ), · · ·. The families F(φ) and F(Φ) associated to a morphism φ or a homotopy Φ in GM(E) (see (??) and (??)) are end pointed in the sense that each F ∈ F (= F(φ) or F(Φ)) comes equipped with an ordered pair of points (b_{F} , e_{F}). Such an end pointed family is said to be bounded by ϵ ≥ 0, if

∀F ∈ F : p(F) ⊆ B(p(b_{F}), ϵ) ∩ B(p(e_{F}), ϵ), (9)

where B(x, ϵ) denotes the closed ball of radius ϵ and with center x ∈ B. An ϵmorphism φ : N → M between geometric modules on E is a morphism for which F(φ) is bounded by ϵ. Similarly, one defines an ϵ-homotopy between ϵ-morphisms, written φ ≃_{ϵ} ψ. The collection GM_{ϵ}(B; p) of all (finite) geometric modules on E and all ϵ-morphisms between them, is not a subcategory of GM(E). The best one can say is that the composition maps GM_{δ}(B; p)(M, L) × GM_{ϵ}(B; p)(N, M) into GM_{δ+ϵ}(B; p)(N, L), and it is compatible with the inclusions GM_{ϵ}(B; p) ⊆ GM_{ϵ′}(B; p) for ϵ ≤ ϵ′. On the other hand, ϵ-homotopy is an equivalence relation in the collection GM_{ϵ}(B; p), so one does have the corresponding quotient “non-categories” GM_{ϵ}(B; p) with compositions

GM_{δ}(B; p)(M, L) × GM_{ϵ}(B; p)(N, M) → GM_{δ+ϵ}(B; p)(N, L).

Also, ϵ-homotopies can be composed with δ-morphisms or δ-homotopies on either side to give (ϵ + δ)-homotopies. We leave it to the reader to keep track of the obvious formal properties of such compositions, cf. Section 2 of [?].

An ϵ-morphism ξ : N → N is called an ϵ-idempotent, if ξ^{2} ≃_{2ϵ} ξ. If ξ is an ϵidempotent, then so is 1_{N} − ξ. A geometric ϵ-idempotent is an ϵ-morphism which is geometric in the sense of (??), and for which α = 1_{T} : S = T → T and each u_{s} ∈ {0, 1}.

An ϵ-morphism φ : N → M is called an ϵ-isomorphism, if there exists an ϵmorphism φ′ : M → N such that φ′φ ≃_{2ϵ} 1_{N} and φφ′ ≃_{2ϵ} 1_{M}. We also call φ′ an ϵ-inverse for φ. We note that an ϵ-morphism, φ, which is geometric and for which the map α : S → T is a bijection and each u_{s} is a unit in R, is an ϵ-isomorphism, with a preferred, geometric ϵ-inverse, φ^{−1}, obtained by inverting α, reversing all the paths involved, and inverting all the units u_{s}. We shall refer to such a morphism as a geometric ϵ-isomorphism. Remark: From a purely algebraic viewpoint, it might be natural to allow each u_{s} to be an arbitrary idempotent in R (respectively, an arbitrary unit in R) in the above definition of a geometric idempotent (respectively, a geometric isomorphism). However, in geometry the above choices seem preferable, cf. Section ??. □

Clearly, any geometric module M = (S, σ) admits a geometric 0-isomorphism to a module (T, τ) with T ⊂ E × ℕ and τ = proj_{1}|T : T → E. In fact, one may take each path involved in such an isomorphism to be constant. Therefore, up to conjugation by geometric 0-isomorphisms, there is only a set of ϵ-automorphisms and a set of ϵ-idempotents in GM_{ϵ}. This is important for set theoretic reasons in the following section.

3.1. Definition of K_{i}(B; p)_{ϵ} for i = 0, 1. The group K_{0}(B; p)_{ϵ} will be defined in terms of the class of ϵ-idempotents in GM_{ϵ}(B; p). Thus, let ξ : M → M and η : N → N be ϵ-idempotents. We write ξ ∼_{ϵ} η, if there exist a geometric module L and ϵ-morphisms φ : M ⊕ L → N ⊕ L, ψ : N ⊕ L → M ⊕ L such that

ψφ ≃_{2ϵ} ξ ⊕ 1_{L} and φψ ≃_{2ϵ} η ⊕ 1_{L}; (10)

and

ψ ≃_{2ϵ} (ξ ⊕ 1_{L})ψ ≃_{2ϵ} ψ(η ⊕ 1_{L}) and φ ≃_{2ϵ} (η ⊕ 1_{L})φ ≃_{2ϵ} φ(ξ ⊕ 1_{L}). (11)

The idea here is that up to homotopy φ is an isomorphism (M ⊕ L, ξ ⊕ 1_{L}) → (N ⊕L, η ⊕1_{L}) between objects in the idempotent completion of GM_{ϵ}(B; p) and that ψ is its inverse. We note that ∼_{ϵ} is not an equivalence relation, but it is symmetric and reflexive, and it does have the following properties

[ξ ∼_{ϵ} η and η ∼_{ϵ} ζ] ⇒ ξ ∼_{5ϵ} ζ, (12)

(13) ∀ϵ-idempotent ξ : M → M : ξ ⊕ (1_{M} − ξ) ∼_{ϵ} 1_{M},

[ξ_{i} ∼_{ϵ} η_{i}, i = 1, 2] ⇒ [ξ_{1} ⊕ ξ_{2} ∼_{ϵ} η_{1} ⊕ η_{2}], (14)

and

[ξ ∼_{δ} η and δ < ϵ] ⇒ [ξ ∼_{ϵ} η]. (15)

Let ∼∼_{ϵ} denote the equivalence generated by ∼_{ϵ}. Since ϵ-idempotents which are conjugate by means of a geometric 0-isomorphism, are ∼_{ϵ}-equivalent, the ∼∼_{ϵ}-equivalence classes form a set. Since ⊕ is associative and commutative up to geometric 0isomorphisms, this set is an abelian semigroup under a sum induced by ⊕. We define K_{0}(B; p)_{ϵ} to be the corresponding abelian group. Thus, any x ∈ K_{0}(B; p)_{ϵ} has the form x = [ξ] − [η] with ξ and η ϵ-idempotents, and [ξ] − [η] = [ξ′] − [η′] if and only if there exist ϵ-idempotents ξ_{1}, ξ_{2}, ⋯, ξ_{m+1}, and ζ, such that

ξ ⊕ η′ ⊕ ζ = ξ_{1} ∼_{ϵ} ξ_{2} ∼_{ϵ} ξ_{3} ∼_{ϵ} ⋯ ∼_{ϵ} ξ_{m+1} = ξ′ ⊕ η ⊕ ζ. (16)

Since there is no bound on m, it seems that there is no control on the relation between ξ ⊕ η′ and ξ′ ⊕ η. However, by using (??)–(??), one can imitate Theorem 3.5 of [?] (see also Lemma 4.4 of [?]) to get the following equivalences (where ξ̄ is short for 1 − ξ)

ξ_{1} ⊕ 1 ⊕ ⋯ ⊕ 1 ⊕ 1 ∼_{ϵ}
ξ_{1} ⊕ ξ̄_{2} ⊕ ξ_{2} ⊕ ⋯ ⊕ ξ̄_{m−1} ⊕ ξ_{m−1} ⊕ ξ̄_{m} ⊕ ξ_{m} ∼_{ϵ}
ξ_{2} ⊕ ξ̄_{2} ⊕ ξ_{3} ⊕ ⋯ ⊕ ξ̄_{m−1} ⊕ ξ_{m} ⊕ ξ̄_{m} ⊕ ξ_{m+1} ∼_{ϵ}
1 ⊕ 1 ⊕ ⋯ ⊕ 1 ⊕ 1 ⊕ ξ_{m+1} ∼_{0} ξ_{m+1} ⊕ 1 ⊕ 1 ⊕ ⋯ ⊕ 1 ⊕ 1.

In view of (??) and (??), this proves the following proposition.

Proposition 3.1. There is a universal constant k ∈ ℕ such that whenever ξ, η, ξ′, and η′ are ϵ-idempotents with [ξ] − [η] = [ξ′] − [η′] ∈ K_{0}(B; p)_{ϵ}, then ξ ⊕ η′ ∼_{kϵ} ξ′ ⊕ η′.

The inclusions GM_{δ} ⊆ GM_{ϵ}, δ < ϵ, give rise to “relax control” homomorphisms K_{0}(B; p)_{δ} → K_{0}(B; p)_{ϵ}. We define the controlled group K_{0}(B; p)_{c} to be the (inverse) limit of the resulting inverse system indexed on all ϵ > 0. We also need the first derived, lim^{1}K_{0}(B; p)_{ϵ}.

We note that K_{0}(B; p)_{ϵ} is not functorial on the category T OP_{c}. However, since B is assumed compact, the inverse system {K_{0}(B; p)_{ϵ}}_{ϵ}, when viewed as an object in the usual pro category (cf. the Appendix in [?]), is functorial. Hence the controlled groups K_{0}(B; p)_{c} and the groups lim^{1}K_{0}(B; p)_{ϵ} are also functorial.

The groups K_{1}(B; p)_{ϵ} and K_{1}(B; p)_{c} are constructed in a similar way, starting from the class of ϵ-automorphisms in GM_{ϵ}(B; p). An ϵ-automorphism γ : N → N is called ϵ-elementary, if N admits a decomposition^{4} N = N_{1} ⊕ N_{2} so that γ is of the form

γ = ( \begin{matrix}1 & γ′ \\ 0 & 1 \end{matrix} ) : N_{1} ⊕ N_{2} → N_{1} ⊕ N_{2} (17)

for some ϵ-morphism γ′ : N_{2} → N_{1}. Such a γ has a preferred ϵ-inverse

γ^{−1} = ( \begin{matrix}1 & −γ′ \\ 0 & 1 \end{matrix} ) : N_{1} ⊕ N_{2} → N_{1} ⊕ N_{2}. (18)

which is itself ϵ-elementary and has γγ^{−1} = γ^{−1}γ = 1. Let φ : M → M and ψ : N → N be ϵ-automorphisms. We write φ ∼_{ϵ} ψ, if there exist geometric modules L′, L′′; a geometric 0-isomorphism ν : M ⊕ L′ → N ⊕ L′′; an ϵ-elementary automorphism γ : N ⊕ L′′ → N ⊕ L′′; and a 2ϵ-homotopy

ν(φ ⊕ 1_{L′})ν^{−1} ≃_{2ϵ} (ψ ⊕ 1_{L′′})γ. (19)

The relation thus defined is not an equivalence relation, but it does satisfy

φ ∼_{ϵ} ψ ⇒ ψ ∼_{3ϵ/2} φ, (20)

[φ_{i} ∼_{ϵ} ψ_{i}, i = 1, 2] ⇒ φ_{1} ⊕ φ_{2} ∼_{ϵ} ψ_{1} ⊕ ψ_{2}, (21)

and

[φ ∼_{δ} ψ and δ < ϵ] ⇒ φ ∼_{ϵ} ψ. (22)

Let ∼∼_{ϵ} be the equivalence relation generated by ∼_{ϵ}. The equivalence classes form (a set, and) an abelian semigroup under a sum induced by ⊕. The zero element is the class of 1_{M} for any M.

If φ has φ′ as an ϵ-inverse, then successive multiplication on the right by the following six ϵ-elementary automorphisms

( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ),( \begin{matrix}1 & −1 \\ 0 & 1 \end{matrix} ),( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ),( \begin{matrix}1 & 0 \\ −φ & 1 \end{matrix} ),( \begin{matrix}1 & φ′ \\ 0 & 1 \end{matrix} ),( \begin{matrix}1 & 0 \\ −φ & 1 \end{matrix} ) (23)

gives a string of six ∼_{ϵ}-equivalences

( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} )∼_{ϵ}···∼_{ϵ}( \begin{matrix}φ & 0 \\ 0 & φ′ \end{matrix} ). (24)

This establishes the existence of inverses, and the group so defined will be called K_{1}(B; p)_{ϵ}.

4Recall that if N = (T, τ), then this means that N_{i} = (T_{i}, τ|T_{i}) where T = T_{1} ∐ T_{2}.

The following proposition is essentially a version of Lemma 4.4 of [?] or Theorem 3.5 of [?].

Proposition 3.2. There is a universal constant k ∈ ℕ such that whenever two ϵautomorphisms φ : M → M and ψ : N → N represent the same element in K_{1}(B; p)_{ϵ}, then there exist a geometric module L and thirteen ϵ-elementary automorphisms Γ^{i} : M ⊕ L ⊕ N → M ⊕ L ⊕ N, i = 1, 2, ⋯, 13, such that

φ ⊕ 1_{L} ⊕ 1_{N} ≃_{kϵ} (1_{M} ⊕ 1_{L} ⊕ ψ)Γ^{1}Γ^{2} ⋯ Γ^{13}.

Proof: Since conjugation by geometric 0-isomorphisms preserves ϵ-elementary automorphisms, we do not have to worry about associativity and commutativity isomorphisms in the proof. Using (??), we see that there is an integer m and a sequence of 3ϵ-homotopies ν_{i}(φ_{i} ⊕ 1_{L′_{i}})ν_{i}^{−1} ≃_{3ϵ} (φ_{i+1} ⊕ 1_{L′′_{i}})γ^{i}, i = 1, 2, ⋯, m − 1, where each φ_{i} is an ϵ-automorphism, each ν_{i} a geometric 0-isomorphism, and each γ^{i} an ϵelementary^{5} automorphism. Moreover, φ_{1} = φ and φ_{m} = ψ. After replacing each φ_{i} by 1_{L′′_{1}⊕···⊕L′′_{i−1}} ⊕ φ_{i} ⊕ 1_{L′_{i}⊕···⊕L′_{m−1}} (and adjusting each ν_{i} and γ^{i} correspondingly), we may assume that L′_{i} = L′′_{i} = 0 for all i. This, of course, also changes φ and ψ, but the conclusion for the new version does imply that for the old one. Next, we can replace each (new) φ_{i} by ν_{[i]}φ_{1}ν_{[i]}^{−1}, where ν_{[i]} = ν_{m−1}ν_{m−2} ⋯ ν_{i}, and each ν_{i} by 1. Again, this changes φ and ψ, but the conclusion for the new pair implies that for the old. We now have φ_{i} ≃_{3ϵ} φ_{i+1}γ^{i}, i = 1, 2, ⋯ m − 1, from which we conclude that

φ_{1} ⊕ [⊕_{i=2}^{m−1}(φ′_{i} ⊕ φ_{i})] ≃_{3ϵ} [⊕_{i=2}^{m−1}(φ_{i} ⊕ φ′_{i}) ⊕ φ_{m}]δ, (25)

where φ′_{i} is an ϵ-inverse of φ_{i} and δ = ⊕_{i=1}^{m−2}(γ^{i} ⊕ 1) ⊕ γ^{m−1} is an ϵ-elementary automorphism. For any ϵ-morphism φ with ϵ-inverse φ′, the product of the six factors in (??) is 3ϵ-homotopic to φ ⊕ φ′. Therefore, there are twelve ϵ-elementary automorphisms δ^{i} such that

1_{M} ⊕ [⊕_{i=2}^{m−1}(φ′_{i} ⊕ φ_{i}] ≃_{3ϵ} ∏_{i=1}^{6} δ^{i}, and [⊕_{i=2}^{m−1}φ_{i} ⊕ φ′_{i}] ⊕ 1_{N} ≃_{3ϵ} ∏_{i=7}^{12} δ^{i}. (26)

The desired conclusion follows easily from (??) and (??). □

If δ < ϵ, then by (??), the inclusions GM_{δ} ⊆ GM_{ϵ} give rise to “relax control” homomorphisms K_{1}(B; p)_{δ} → K_{1}(B; p)_{ϵ}. We define the controlled group K_{1}(B; p)_{c} to be the (inverse) limit of the resulting inverse system indexed on all ϵ > 0. We also need the first derived, lim^{1} K_{1}(B; p)_{ϵ}.

As with K_{0}, the individual groups K_{1}(B; p)_{ϵ} are not functorial, but the inverse system {K_{1}(B; p)_{ϵ}}_{ϵ} and the groups K_{1}(B; p)_{c} and lim^{1}K_{1}(B; p)_{ϵ} are functorial on T OP_{c}.

5The wording of (??) would give the bound 3ϵ/2 only, but the easy proof of (??) does give ϵ here.

3.2. The reduced groups for i = 0, 1. We recall the notion of a geometric morphism, defined in (??), and denote by GK_{i}(B; p)_{ϵ} the subgroup of K_{i}(B; p)_{ϵ} generated by all [φ] where φ is a geometric ϵ-idempotent (if i = 0), or a geometric ϵ-automorphism (if i = 1), cf. Section ??. The reduced groups K̃_{i}(B; p)_{ϵ}, are defined to fit into the short exact sequences.

0 → GK_{i}(B; p)_{ϵ} → K_{i}(B; p)_{ϵ} → K̃_{i}(B; p)_{ϵ} → 0, (27)

i = 0, 1. For i = 1, we also write W h(B; p)_{ϵ} for the reduced group. In order to study the behavior of the reduced groups under lim and lim^{1}, we need two lemmas.

Lemma 3.3. If two geometric modules are ϵ-isomorphic, then there is a geometric ϵ-isomorphism between them.

Proof: One can easily adapt the proof of Theorem 2.6, pp. 140ff. of [?] to the present situation. □

Lemma 3.4. For any geometric ϵ-isomorphism ν : M → N and any δ > 0, there exist an m ∈ ℕ; geometric modules M_{1} = M, M_{2}, ⋯, M_{m} = N; and geometric δ-isomorphisms ν_{i} : M_{i} → M_{i+1}, i = 1, 2, ⋯, m − 1, such that ν = ν_{m−1} ⋯ ν_{2}ν_{1}.

Proof: One uses a straigthforward subdivision of paths (cf. Section 4 of [?]). It is, of course, essential that B is compact. □

Proposition 3.5. The inverse system {GK_{0}(B; p)_{ϵ}}_{ϵ} is stably isomorphic to the constant system H_{0}(E, ℤ), where H_{0} denotes singular homology. Consequently, one has the following natural isomorphisms and natural short exact sequence

lim^{1}GK_{0}(B; p)_{ϵ} = 0 , lim^{1}K_{0}(B; p)_{ϵ} ≅ lim^{1}K̃_{0}(B; p)_{ϵ},
0 → H_{0}(E; ℤ) → K_{0}(B; p)_{c} → K̃_{0}(B; p)_{c} → 0.

The inverse system {GK_{1}(B; p)_{ϵ}}_{ϵ} satisfies the Mittag-Leffler condition. Consequently, one has the following natural isomorphisms and natural short exact sequence.

lim^{1}GK_{1}(B; p)_{ϵ} = 0 , lim^{1}K_{1}(B; p)_{ϵ} ≅ lim^{1}W h(B; p)_{ϵ},
0 → lim GK_{1}(B; p)_{ϵ} → K_{1}(B; p)_{c} → W h(B; p)_{c} → 0.

Proof: Any geometric ϵ-idempotent ξ : M → M is ∼_{ϵ}-equivalent to a geometric ϵ-idempotent ξ′ : M′ → M′ which is an ϵ-isomorphism (i.e., each of the elements u_{s} occurring in the presentation of ξ′ equals 1). In fact, if M = (S, σ), one can take ξ′ = ξ|M′ where M′ = (S′, σ|S′) and S′ = {s ∈ S|u_{s} = 1}. Thus we need only consider geometric ϵ-idempotents ξ : M → M which have ϵ-inverses (denoted ξ′). Then ξ ≃_{3ϵ} ξξξ′ ≃_{3ϵ} ξξ′ ≃_{2ϵ} 1, so ξ ≃_{3ϵ} 1_{M}. Hence, if two ϵ-invertible ϵidempotents, ξ : M → M and η : N → N, have ξ ∼_{ϵ} η so that (??) and (??) hold, then ψ and φ are \frac{3ϵ}{2}-inverses of one another. Then by Lemma ??, there is a geometric \frac{3ϵ}{2}-isomorphism M ⊕ L → N ⊕ L. Therefore, there is a homomorphism ρ_{ϵ} : GK_{0}(B; p)_{ϵ} → H_{0}(E, ℤ) mapping [ξ] to Σn_{β}[e_{β}] where M = (S, σ), [e_{β}] is the canonical, zero dimensional homology class in the path component E_{β} ⊆ E, and n_{β} is the cardinality of σ^{−1}(E_{β}). Conversely, if we fix the points {e_{β}}, there is an obvious homomorphism ρ′_{ϵ} : H_{0}(E; ℤ) → GK_{0}(B; p)_{ϵ} with ρ′_{ϵ}([e_{β}]) = 1_{(T,τ)} where T is a one point set which maps to e_{β} under τ. Clearly ρ_{∗} = {ρ_{ϵ}}_{ϵ} and ρ′_{∗} = {ρ′_{ϵ}}_{ϵ} are maps of the inverse systems involved and ρ_{∗}ρ′_{∗} = 1. On the other hand, it is easily seen that ρ′_{ϵ}ρ_{ϵ}([1_{M}]) = [1_{M}] for any geometric module M. Since (as remarked above) any [ξ] ∈ GK_{0}(B; p)_{ϵ} maps to an element of the form [1_{M}] when control is relaxed to 3ϵ, it follows that ρ′_{ϵ}ρ_{ϵ} and 1 are coequalized by the map GK_{0}(B; p)_{ϵ} → GK_{0}(B; p)_{3ϵ}. Thus ρ_{∗} is a stable isomorphism of inverse systems. Therefore, {GK_{0}(B; p)_{ϵ}}_{ϵ} satisfies the Mittag-Leffler condition, so its lim^{1} vanishes. The long exact lim^{∗}-sequence induced by (??) proves the rest of the results claimed for K_{0}.

The second part of the proposition follows in the same way once we have established the Mittag-Leffler condition for the system {GK_{1}(B; p)_{ϵ}}. To do so, it suffices to show that [ν] ∈ Im[GK_{1}(B; p)_{δ} → GK_{1}(B; p)_{2ϵ}] for every geometric ϵ-automorphism ν : M → M, every ϵ > 0, and every positive δ < ϵ. By Lemma ??, we can write ν as a composition, ν = ν_{m−1} ⋯ ν_{2}ν_{1}, where ν_{i} is a geometric δ-isomorphism M_{i} → M_{i+1}, i = 1, 2, ⋯, m − 1, and M_{1} = M_{m} = M. It follows easily that each sub-composition ν_{j}ν_{j−1} ⋯ ν_{i} with 1 ≤ i ≤ j ≤ m − 1 is a geometric 2ϵ-isomorphism. Therefore, it suffices to prove the following lemma.

Lemma 3.6. Let φ_{i} : M_{i} → M_{i+1} be a δ-isomorphism for i = 1, 2, ⋯ m−1. Assume that ϵ ≥ δ and that each composition φ_{i,j} = φ_{j−1}φ_{j−2} ⋯ φ_{i} : M_{i} → M_{j} is a 2ϵisomorphism. Let φ = φ_{1,m} : M_{1} → M_{m} = M_{1}. Then the δ-automorphism Φ defined below has [φ] = [Φ] ∈ K_{1}(B; p)_{2ϵ}. In particular, [φ] ∈ Im[K_{1}(B; p)_{δ} → K_{1}(B; p)_{2ϵ}].

Proof: Let

Φ = ( \begin{matrix}0 & φ_{m−1} \\ d & 0 \end{matrix} ) : [M_{1} ⊕ ⋯] ⊕ M_{m−1} → M_{1} ⊕ [⋯ ⊕ M_{m−1}], (28)

where d is the diagonal morphism

diag(−φ_{1}, ⋯, −φ_{m−2}) : M_{1} ⊕ ⋯ ⊕ M_{m−2} → M_{2} ⊕ ⋯ ⊕ M_{m−1}.

If each φ_{i} were an honest isomorphism (in some category), then a trivial exercise would show that Φ could be reduced to the form

φ ⊕ 1 : M_{1} ⊕ [M_{2} ⊕ ⋯ ⊕ M_{m−1}] → M_{1} ⊕ [M_{2} ⊕ ⋯ ⊕ M_{m−1}],

by elementary column operations, i.e., by right multiplication by elementary matrices. To do the actual proof, one has to write out such a reduction; to observe that each of the elementary matrices will represent a 2ϵ-elementary automorphism; and to check that the homotopies which occur because we deal with 2ϵ-isomorphisms rather than real isomorphisms are 4ϵ-homotopies. We leave the details to the reader. □

3.3. Ranicki and Yamasaki’s ϵ-controlled K-theory. In [?], two inverse systems of groups, {K̃_{0}(B, p, 0, ϵ)}_{ϵ} and {W h(B, p, 1, ϵ)}_{ϵ}, are introduced. In this section we compare these systems to those defined by us. The following proposition is essential for our treatment of the lower groups in the next section.

Proposition 3.7. The inverse systems {K̃_{0}(B; p)_{ϵ}}_{ϵ} and {W h(B; p)_{ϵ}}_{ϵ} are stably isomorphic to {K̃_{0}(B, p, 0, ϵ)}_{ϵ} and {W h(B, p, 1, ϵ)}_{ϵ}, respectively.

Proof: We note that our geometric modules, ϵ-morphisms and ϵ-homotopies coincide with the geometric modules, the geometric morphisms of radius ϵ and the ϵ homotopies in [?]^{6}. In [?], an ϵ-projection ξ : M → M is required to have bound ϵ and satisfy ξ^{2} ≃_{ϵ} ξ while we have required an ϵ-idempotent to have bound ϵ and satisfy ξ^{2} ≃_{2ϵ} ξ. Thus, each ϵ-idempotent is a 2ϵ-projection. As such, it can be viewed as a 0-dimensional 2ϵ projective chain complex (M, ξ) in the sense of the Definition following Proposition 2.2 in [?]. Moreover, if ξ ∼_{ϵ} η so that (??) and (??) hold, then ψ and φ show that (M, ξ) and (N, η) are 0-stable 2ϵ chain equivalent (in the sense of the definitions preceeding Proposition 3.1 in [?]). Thus they represent the same element of K̃_{0}(B, p, 0, 2ϵ), and there is a well defined homomorphism ρ_{ϵ} : K_{0}(B; p)_{ϵ} → K̃_{0}(B, p, 0, 2ϵ) with ρ_{ϵ}[ξ] = [M, ξ]. The collection {ρ_{ϵ}}_{ϵ} is compatible with the relaxation of control, so one has a short exact sequence of inverse systems

0 → {Kerρ_{ϵ}}_{ϵ} → {K_{0}(B; p)_{ϵ}}_{ϵ} → {Imρ_{ϵ}}_{ϵ} → 0. (29)

If ξ : M → M is a geometric ϵ-idempotent, then (cf. the proof of Proposition ??) ξ can be assumed ≃_{3ϵ} 1_{M}, so (M, ξ) represents 0 in K̃_{0}(B, p, 0, 3ϵ). Therefore,

Im[GK_{0}(B; p)_{ϵ} ⊆ K_{0}(B; p)_{ϵ} → K_{0}(B; p)_{\frac{3ϵ}{2}} ] ⊆ Kerρ_{\frac{3ϵ}{2}} . (30)

Conversely, if ξ : M → M has [ξ] ∈ Kerρ_{ϵ}, then, by Proposition 3.1 of [?], (M, ξ) ⊕ (E, 1) is 3ϵ chain equivalent to (F, 1) for suitable modules E and F. As noticed in [?] (immediately before Proposition 2.3), this implies that (F, 1) and (M ⊕ E, ξ ⊕ 1) are 3ϵ isomorphic. But then [ξ] = [1_{F}] − [1_{E}] ∈ K_{0}(B; p)_{3ϵ}, so [ξ] ∈ GK_{0}(B; p)_{3ϵ}, and we have shown that

Im[Kerρ_{ϵ} ⊆ K_{0}(B; p)_{ϵ} → K_{0}(B; p)_{3ϵ}] ⊆ GK_{0}(B; p)_{3ϵ}. (31)

6Actually, here we are slightly inaccurate, as Ranicki and Yamasaki’s geometric morphisms have 3(s, ω, t) + 2(s, ω, t) ≠ 5(s, ω, t). Since we always pass to homotopy classes, the inaccuracy is harmless.

The equations (??) and (??) imply that {Kerρ_{ϵ}}_{ϵ} and {GK_{0}(B; p)_{ϵ}}_{ϵ} are stably isomorphic subsystems of {K_{0}(B; p)_{ϵ}}_{ϵ}. It follows that the inverse systems {K̃_{0}(B; p)_{ϵ}}_{ϵ} and {Imρ_{ϵ}}_{ϵ} are also stably isomorphic, and we can finish our study of the K_{0}-case by showing that {Imρ_{ϵ}}_{ϵ} is stably isomorphic to {K̃_{0}(B, p, 0, ϵ)}_{ϵ}. A map from one to the other is given by the inclusions Imρ_{ϵ} ⊆ K̃_{0}(B, p, 0, 2ϵ), and this is a stable isomorphism of inverse systems because of the inclusions Im[K̃_{0}(B, p, 0, 2ϵ) → K̃_{0}(B, p, 0, 4ϵ)] ⊆ Imρ_{2ϵ}. The latter inclusions, in turn, hold because any 2ϵ projection in the sense of [?] is a 2ϵ idempotent in our sense.

We next turn to the K_{1} case. Any ϵ-automorphism φ : M → M can be viewed as a 1-dimensional, free ϵ chain complex, to be denoted (φ : M → M), in the sense of the Definition following Proposition 2.2 in [?]. As such it is strongly^{7} ϵ chain contractible, so it represents an element [φ : M → M] ∈ W h(X, p, 1, ϵ), cf. the definition preceding Proposition 4.1 in [?]. If ψ : N → N is another ϵ-automorphism, and (??) holds, then the pair (γν, ν) describes a 1-stable 2ϵ-simple^{8} chain equivalence between (φ : M → M) and (ψ : N → N). Since the equivalence relation in the above mentioned definition in [?] is generated by 1-stable 40ϵ-simple chain equivalence, there results a homomorphism ρ_{ϵ} : K_{1}(B; p)_{ϵ} → W h(B, p, 1, ϵ) given by ρ_{ϵ}([φ]) = [φ : M → M], and a short exact sequence of inverse systems

0 → {Kerρ_{ϵ}}_{ϵ} → {K_{1}(B; p)_{ϵ}}_{ϵ} → {Imρ_{ϵ}}_{ϵ} → 0. (32)

If φ : M → M is an ϵ-automorphism with ρ_{ϵ}[φ] = 0, then by Proposition 4.1 of [?], the ϵ chain complex (φ : M → M) is 1-stable 86ϵ-simple chain equivalent to the 0 complex, i.e. there are free modules T and T′ and two 86ϵ deformations, say D_{1} = f_{m}, f_{m−1}, ⋯, f_{1} and D_{0} = g_{n}, g_{n−1}, ⋯, g_{1}, such that the compositions f = f_{m}f_{m−1} ⋯ f_{1} and g = g_{n}g_{n−1} ⋯ g_{1} and their inverses define 86ϵ chain maps (f, g) : (φ⊕1 : M ⊕T → M ⊕T) → (1 : T′ → T′), respectively, (f^{−1}, g^{−1}) : (1 : T′ → T′) → (φ ⊕ 1 : M ⊕ T → M ⊕ T). It follows that φ ⊕ 1_{T} ≃_{172ϵ} g_{1}^{−1} ⋯ g_{m}^{−1}f_{m} ⋯ f_{1}. Since each of the factors is either a 172ϵ-elementary automorphism or a geometric 172ϵ-isomorphism, it follows from Lemma ?? that

Im[Kerρ_{ϵ} ⊆ K_{1}(B; p)_{ϵ} → K_{1}(B; p)_{344ϵ}] ⊆ GK_{1}(B; p)_{344ϵ}, (33)

Conversely, it is easily seen that any geometric ϵ-isomorphism ξ : M → M when viewed as chain complex ξ : M → M is 1-stable ϵ-simple equivalent to the 0 chain complex. Therefore

GK_{1}(B; p)_{ϵ} ⊆ Kerρ_{ϵ} (34)

7Actually, “strongly” is vacuous for contractions of 1-dimensional complexes.

8We get 2ϵ rather than ϵ, because [?] insists that an ϵ chain map between free ϵ chain complexes commute with the boundary maps up to ϵ homotopy.

The equations (??) and (??) imply that {Kerρ_{ϵ}}_{ϵ} and {GK_{1}(B; p)}_{ϵ} are stably isomorphic subsystems of {K_{1}(B; p)}_{ϵ}.

As in the proof of the K_{0} case, it now suffices to show that the system of inclusions {Imρ_{ϵ}}_{ϵ} ⊆ {W h(B, p, 1, ϵ)}_{ϵ} is a stable isomorphism. To that end, let [δ : C_{1} → C_{0}] be a typical generator for W h(B, p, 1, ϵ), i.e., δ : C_{1} → C_{0} is a free ϵ chain complex which is ϵ chain contractible. Since an ϵ chain contraction is precisely the same as an ϵ-inverse for δ, Lemma ?? supplies a geometric ϵ-isomorphism ν : C_{1} → C_{0}. Since [δ : C_{1} → C_{0}] = [δν^{−1} : C_{0} → C_{0}] = ρ_{2ϵ}[δν^{−1}] ∈ W h(B, p, 1, 2ϵ), it follows that

Im[W h(B, p, 1, ϵ) → W h(B, p, 1, ϵ] ⊆ Imρ_{2ϵ},

and the proof is complete.

3.4. Lower controlled K-theory. In the Appendix of [?], Ranicki and Yamasaki introduce the lower, reduced K-theory groups K̃_{1−j}(B, p, n−1, ϵ) and W h_{2−j}(B, p, n, ϵ) for all ϵ > 0 and all j, n ∈ ℕ. They show that the inverse systems {W h_{1−j}(B, p, n, ϵ)}_{ϵ} and {K̃_{1−j}(B, p, n, ϵ)}_{ϵ} are stably isomorphic for all j, n ∈ ℕ. Moreover, they (essentially) give the following inductive characterization of the lower groups in terms of the maps induced by the standard inclusion i : (B; p) → (B × S^{1}; p × id_{S^{1}}).

Proposition 3.8. For each j ∈ ℕ and each n ∈ ℕ, there is a natural stable isomorphism of inverse systems

{Coker[i_{∗} : K̃_{2−j}(B, p, n, ϵ) → K̃_{2−j}(B × S^{1}, p × id, n, ϵ)]}_{ϵ} → {K̃_{1−j}(B, p, n, ϵ)}_{ϵ}.

Proof: (Sketch) Of course, K̃_{2−j} has to be interpreted as W h when j = 1. In that case, the result then follows from the stably exact sequence

0 → W h(B, p, n, ϵ) → W h(B × S^{1}, p × id, n, ϵ) → K̃_{0}(B, p, n, ϵ) → 0,

which is established immediately after Corollary 7.3 in [?] as a formal consequence of the Mayer-Vietoris sequence (Theorem 6.2 of [?]) for the triad B × (S^{1}; S_{+}^{1}, S_{−}^{1}) and Lemma 7.2 of [?]. In the Appendix of [?] it is shown that Theorem 6.2 generalizes to lower, reduced K-theory, and it is easy to check a generalized version of Lemma 7.2. Therefore, the whole proof goes through for all j ∈ ℕ. □

Definition 3.9. For any j ∈ ℕ, we define K_{−j}(B; p)_{ϵ} to be the group K̃_{−j}(B, p, 1, ϵ) of [?].

Since the Ranicki-Yamasaki groups are independent of the choice of n up to stable isomorphism (see [?], immediately before Corollary 3.5 and immediately before Proposition 4.8), and since the inclusion i : (B; p) → (B × S^{1}, p × id) is easily seen to induce a stable isomorphism on the inverse systems {GK_{0}(−; −)_{ϵ}}_{ϵ}, it is not hard to derive the following inductive characterization of the inverse systems of unreduced groups.

Proposition 3.10. For each j ∈ ℕ, there is a natural stable isomorphism of inverse systems

{Coker[i_{∗} : K_{1−j}(B; p)_{ϵ}) → K_{1−j}(B × S^{1}; p × id)_{ϵ})]}_{ϵ} → {K_{−j}(B; p)_{ϵ})}_{ϵ}.

The morphisms i_{∗} in the above propositions have natural splittings induced by the projection (B × S^{1}; p × id) → (B; p). Therefore, one gets the following corollary, where naturality refers to morphisms (B; p) → (B′; p′) in the sense of (??). Note that the first of the three sequences involves degrees^{9} 1 − j < 1 while the other two hold in degrees 2 − j ≤ 1. Also note that the third sequence coincides with its own reduced version because of Proposition ??.

Corollary 3.11. For any j ∈ ℕ, there are natural, naturally split, exact sequences

0 → K_{1−j}(B; p)_{c} →^{i_{∗}} K_{1−j}(B × S^{1}; p × 1)_{c} → K_{−j}(B; p)_{c} → 0,

0 → K̃_{2−j}(B; p)_{c} →^{i_{∗}} K̃_{2−j}(B × S^{1}; p × 1)_{c} → K̃_{1−j}(B; p)_{c} → 0,

and

0 → lim^{1}K_{2−j}(B; p)_{ϵ} →^{i_{∗}} lim^{1}K_{2−j}(B × S^{1}; p × 1)_{ϵ} → lim^{1}K_{1−j}(B; p)_{ϵ} → 0.

This corollary will be used in Section ?? in order to extend the basic exact sequence to lower K-theory.

4. K-theory with continuous control.

4.1. Recollections from [?]. Warning: The functor that we call K_{∗}^{cc} in the present paper is denoted K̃_{∗}^{cc}, or sometimes just K̃_{∗}, in [?]. Here we use the notation K̃_{∗}^{cc} for a quotient of K_{∗}^{cc} obtained by factoring out a certain subfunctor. In particular, for us K̃_{1}^{cc} is a just a different name for the cc Whitehead group functor, W h^{cc}. Further details are given below in (??). □

Continuously controlled K-theory “with constant coefficients” was introduced in [?]. A version with “variable coefficients” was introduced by two of the present authors in [?] as a functor K_{∗}^{cc} : T OP/CM^{∗} → AB_{∗} Here AB_{∗} is the category of ℤ-graded abelian groups while T OP/CM^{∗} is the category of diagrams of holink type. We recall that an object in this category is a pair of maps ξ = (B ⟵^{p} H ⟶^{q} X) with H and X Hausdorff, and B compact, metrizable. Also, a morphism is a triple of maps making an obvious diagram commutative.

If j > 0, the functor K_{j}^{cc} arises as a composition

T OP/CM^{∗} ⟶^{c} T OP^{cc}/LC ⟶^{GM^{cc}} ADDCAT ⟶^{K_{j}^{Q}} AB,

9Actually, one can analyze the map i_{∗} : GK_{1}(B; p) → GK_{1}(B × S^{1}; p × id) and prove exactness of the first sequence also for 1 − j = 1, but we do not need this fact here.

where the ingredients are described in detail in [?]. Here we recall the major facts.

We start with the category T OP^{cc}/LC whose objects are triples (\overline{X}, B; p) with \overline{X} a locally compact, Hausdorff space, B ⊆ \overline{X} a compact subspace, and p a map p : E → X = \overline{X} − B. Even though E is a very important part of such an object, it is part of the notation only insofar as any map determines its domain. Morphisms f : (\overline{X}, B; p) → (\overline{X′}, B′; p′) are maps f : E → E′ admitting a suitable extension \overline{f} : \overline{E} → \overline{E′} where \overline{E} = E ⨿ B with a teardrop topology.

The external cone functor c : T OP/CM^{∗} → T OP^{cc}/LC maps ξ = (B ⟵^{p} H ⟶^{q} X) to the triple (v(B), B; v^{∘}(p)) where v(B) is the standard cone on B, and v^{∘}(p) : cyl(q) → v^{∘}(B) = v(B) − B is the map from the open mapping cylinder of q (with a teardrop topology; see [?]) to the open cone on B, induced by p.

To each (\overline{X}, B; p) we associate an additive subcategory GM^{cc}(\overline{X}, B; p) ⊆ GM(E), and a corresponding additive homotopy category GM^{cc}(\overline{X}, B; p), by imposing the following conditions on objects, morphisms and homotopies in GM(E). Each object (S, σ) must have pσ : S → B proper. For each morphism φ, the family F(φ) defined in (??) must have p_{∗}F(φ) continuously controlled at B (briefly, cc) in the sense of [?]. Similarly, p_{∗}F(Φ) must be cc for each homotopy Φ. If φ is a morphism in GM^{cc}(\overline{X}, B; p), then its image in GM^{cc}(\overline{X}, B; p) will be denoted cls(φ).

For each additive category A and each j ∈ ℕ, we denote by K_{j}^{Q}(A) the Quillen K_{j}-group of A, as defined, e.g., in [?].

The above three functors suffice to define K_{j}^{cc} for j ∈ ℕ, but in order to define K_{1−j}^{cc} for j ∈ ℕ, we also need the suspension functor Σ : T OP/CM^{∗} → T OP/CM^{∗} (which is called S in [?]). It has

Σ(ξ) = (Σ(B) ⟵ Cyl(q) ∪_{H} Cyl(q) ⟶ X).

where once again we refer to [?] for further details if the reader needs such. Finally then, the complete definition as given in [?] can be summarized as follows.

Definition 4.1. Let j ∈ ℕ, and ξ ∈ T OP/CM^{∗}. If c and Σ denote the external cone and the suspension, respectively, then

K_{j}^{cc}(ξ) = K_{j}^{Q}(GM^{cc}(c(ξ))), and K_{1−j}^{cc}(ξ) = K_{1}^{cc}(Σ^{j}(ξ)).

Remarks: 1) We do not know whether there is a filtration on the additive category GM^{cc}(c(ξ)) such that Pedersen and Weibels lower K-theoretic groups of the filtered additive category GM^{cc}(c(ξ)), as defined in [?], coincide with the groups K_{1−j}^{cc}(ξ) above.

2) An equivalent definition takes K_{0}^{cc}(ξ) = K_{0}^{Q}(\begin{matrix} \\ GM & cc \end{matrix}(c(ξ))) where the symbol indicates idempotent completion. □

From the object ξ = (B ← H → X) as above, we construct two new objects

ξ × S^{1} = (B × S^{1} ⟵^{p×id} E × S^{1} ⟶^{q∘pr} X) and ξ_{+} = (B_{+} ⟵^{p⨿c_{X}} E ⨿ X ⟶^{q∨id} X),

where + denotes the addition of a base point (in [?], + was put as a superscript). Proposition 8.1 of [?] states that the lower K^{cc}-theory of ξ_{+} can be computed inductively as follows.

Proposition 4.2. For each j ∈ ℤ and each ξ ∈ T OP/CM^{∗}, there is a natural, naturally split, short exact sequence of abelian groups

0 → K_{j}^{cc}(ξ_{+}) →^{i_{∗}} K_{j}^{cc}((ξ × S^{1})_{+}) → K_{j−1}^{cc}(ξ_{+}) → 0,

where i_{∗} is induced by the obvious inclusion i : ξ → ξ × S^{1} given by a base point in S^{1}.

If X is connected, then we shall write K_{∗}(X) for the K-theory (including the lower groups) of the ring Rπ_{1}(X). More generally, K_{∗}(X) will denote the direct sum of such groups, one for each path component of X. By Corollary 9.4 of [?], these groups are naturally isomorphic to K_{∗}^{cc}(∅_{ξ}), so the second exact sequence obtained in Section 8 of [?] takes the following form.

Proposition 4.3. There is a natural exact sequence

⋯ → K_{j}(X) → K_{j}^{cc}(ξ) → K_{j}^{cc}(ξ_{+}) → K_{j−1}(X) → ⋯, j ∈ ℤ,

for any ξ ∈ T OP/CM^{∗}.

4.2. The reduced functors W h^{cc} and K̃_{2−j}^{cc}, j ∈ ℕ. We shall also consider reduced groups K̃_{2−j}^{cc}(ξ), j ∈ ℕ. For these, the ground ring is supposed to be ℤ. Moreover, in order to identify - in Proposition ?? below - the difference between the reduced and the unreduced groups, we shall assume that each path component in each of the spaces that appear in ξ = (B ← H → X) is open. We do not know whether this extra assumption is really needed.

By definition,

K̃_{2−j}^{cc}(ξ) = K_{2−j}^{cc}(ξ)/F_{2−j}(ξ), (35)

where the subgroup F_{2−j}(ξ) is given as follows. For j = 1, F_{1}(ξ) is is the subgroup generated by all [cls(φ)] where the automorphism cls(φ) : (S, σ) → (S, σ) in GM^{cc}(c(ξ)) has a representative φ : (S, σ) → (S, σ) in GM^{cc}(c(ξ)) which is a geometric in the sense of (??). For j = 2, F_{0}(ξ) is the subgroup generated by all elements of the form [S, σ], with (S, σ) ∈ GM^{cc}(c(ξ)) ⊆ \begin{matrix} \\ GM & cc \end{matrix}(c(ξ)), cf. Remark 2) following Definition ??. Finally, for j > 2, F_{2−j}(ξ) = 0, i.e., in strictly negative degrees K_{∗}^{cc} and K̃_{∗}^{cc} coincide.

We have the following partial computation of F_{1}(ξ) and F_{0}(ξ). In it, for any set S, ℤS is the free abelian group generated by S, and [S^{1}, X] is the set of free homotopy classes of maps S^{1} → X.

Proposition 4.4. Let ξ = (B ⟵^{p} H ⟶^{q} X) ∈ T OP/CM^{∗}. With the above assumption on path components, there is a natural exact sequence

Coker(q_{∗} : ℤ[S^{1}, H] → ℤ[S^{1}, X]) ⟶^{α_{1}} F_{1}(ξ) ⟶^{β} ℤπ_{0}(H) ⟶^{q_{∗}} ℤπ_{0}(X) ⟶^{α_{0}} F_{0}(ξ) ⟶ 0.

An outline of a proof is given below. First, however, here are a couple of applications. By specializing to objects of the form ξ_{+}, we get the following.

Corollary 4.5. For any ξ ∈ T OP/CM^{∗}, F_{0}(ξ_{+}) = 0 for j > 1, and there is a natural isomorphism F_{1}(ξ_{+}) → ℤπ_{0}(H).

A straightforward application of Propositions ??, ?? and ?? immediately reveal that the exact sequences of the latter two propositions have the following reduced versions.

Proposition 4.6. Let ξ ∈ T OP/CM^{∗} and j ∈ ℕ. There are a natural, naturally split, short exact sequence

0 → K̃_{2−j}^{cc}(ξ_{+}) → K̃_{2−j}^{cc}(ξ_{+}) → K̃_{1−j}^{cc}(ξ_{+}) → 0

and a natural exact sequence

W h(X) → W h^{cc}(ξ) → W h^{cc}(ξ_{+}) → K̃_{0}(X) → K̃_{0}^{cc}(ξ) → · · · .

Proof of Proposition ?? - Outline:

We start by defining the maps α_{0}, β, and α_{1}. For any point y ∈ cyl(q) = X ⨿ H × (0, 1), we denote by M_{y} the geometric module having one generator which sits at y, i.e., M_{y} = ({1}, μ_{y}) where μ_{y}(1) = y. If ω is path in cyl(q) from y_{1} to y_{0}, then the corresponding geometric morphism is denoted cls(ω) : M_{y_{0}} → M_{y_{1}}. With this notation, for [x] ∈ π_{0}X, we let

α_{0}([x]) = [M_{x}] ∈ F_{0}(ξ). (36)

Let φ : (S, σ) → (S, σ) be a geometric isomorphism representing a typical element z = [cls(φ)] ∈ F_{1}(ξ). Thus φ consists of a permutation π : S → S, a collection of paths {ω_{s}|s ∈ S} where ω_{s} joins σ(s) to σ(π(s)), and a collection of units {u_{s} = ±1| s ∈ S}, cf. (??). Let A be a π-orbit in S. To A there corresponds a composable sequence of paths. Let the composite path be ω_{A}. If A is finite, this is an oriented loop ω_{A} : S^{1} → cyl(q). If A is infinite, it is an oriented, infinite path ω_{A} : ℝ → cyl(q). In both cases, it is defined only up to an oriented reparametrization. We may decompose S as S = S_{fX} ⨿ S_{iX} ⨿ S_{H} where S_{fX} is the union of all finite orbits A, for which ω_{A} meets X, S_{iX} is the union of those infinite orbits A for which ω_{A} meets X, and S_{H} is the union of those (finite or infinite) orbits for which ω_{A} avoids X. The local finitenes conditions imposed on morphisms guarantee that S_{iX} and S_{fX} consist of finitely many orbits. For later use, we note that z splits similarly

z = z_{fX} + z_{iX} + z_{H} ∈ F_{1}(ξ). (37)

Let A be an orbit in S_{iX}. The oriented path ω_{A} must stay away from X near ±∞. Therefore, it determines two path components of H which we shall denote [ω_{A}(±∞)]. In terms of these, we let

β(z) = ∑_{A}([ω_{A}(∞)] − [ω_{A}(−∞)]) ∈ ℤπ_{0}H, (38)

where the sum extends over all orbits in S_{iX}.

Ultimately, generators [a] in the domain of α_{1} are represented by maps a : S^{1} → X. We let 1 ∈ S^{1} be the canonical base point. Then a is a path from a(1) to itself, so it determines a geometric automorphism cls(a) : M_{a(1)} → M_{a(1)}, and we let

α_{1}([a]) = [cls(a)] ∈ F_{1}(ξ). (39)

It is very easy to show that α_{0} is well defined. To see that it is onto, we note that any M = (S, σ) ∈ GM^{cc}(c(ξ)) splits as M_{X} ⊕ M′_{X} where M_{X} = (σ^{−1}(X), σ|σ^{−1}(X)) and M′_{X} is the complement. Clearly, [M_{X}] is in the image of α_{0}, so we may assume that M = (S, σ) = M′_{X}. Then each σ(s) has the form σ(s) = (w_{s}, t_{s}), with w_{s} ∈ H and 0 < t_{s} < 1, and we may define another object N = (S×ℕ, Σ) by letting Σ(s, k) = (w_{s}, 1 − 2^{−k}(1 − t_{s})). The obvious paths from σ(s) to Σ(s, 1), and from Σ(s, k) to Σ(s, k + 1), define an isomorphism N → N ⊕ M. This proves that [M] = 0 ∈ F_{0}(ξ), and finishes the proof.

Next, we turn to exactness at ℤπ_{0}(X). If w ∈ H, then the geometric module N_{w} = (ℕ, ν_{w}) with ν_{w}(k) = (w, 1 − 2^{−k}) ∈ H × (0, 1) admit an obvious geometric (and cc) isomorphism M_{q(w)} ⊕ N_{w} → N_{w} consisting of paths in the (0, 1) direction. Therefore, Imq_{∗} ⊆ Kerα_{0}. To get the opposite inclusion, we need the following lemma, a proof of which can be obtained by adaptation from that of Theorem 2.6, pp. 140ff of [?].

Lemma 4.7. If two geometric modules are isomorphic in GM^{cc}(c(ξ)), then there is a cc geometric isomorphism between them.

A typical element of ℤ[π_{0}(X)] is a difference a − b with a and b of the form a = ∑_{i=1}^{n}[x_{i}], b = ∑_{j=1}^{m}[y_{j}] for certain points x_{i} and y_{j} in X. We have to show that a−b ∈ Im(q_{∗}) under the assumption that α_{0}(a) = α_{0}(b). We let S = {s_{1}, s_{2}, ⋯, s_{n}}, T = {t_{1}, t_{2}, ⋯, t_{m}} and define σ and τ to have σ(s_{i}) = x_{i} and τ(t_{j}) = y_{j}. The assumption implies the existence of a geometric module (R, ρ) and an isomorphism in GM^{cc}(c(ξ)) cls(φ) : (S, σ) ⊕ (R, ρ) ⟶ (T, τ) ⊕ (R, ρ). Let R′ = ρ^{−1}(X) and (R, ρ) = (R′, ρ′) ⊕ (R′′, ρ′′). Also, let (S′, σ′) = (S, σ) ⊕ (R′, ρ′), and define (T′, τ′) similarly. If R′ = {r_{1}, r_{2}, ⋯, r_{p}} and ρ′(r_{k}) = z_{k} ∈ X, and we set c = ∑_{k=1}^{p}[z_{k}], then we may replace a and b with a′ = a + c and b′ = b + c, respectively. In other words, we may assume that R′ is empty, so that ρ maps into H × (0, 1). The lemma lets us assume that the morphism φ : (S, σ) ⊕ (R, ρ) → (T, τ) ⊕ (R, ρ) which represents the given isomorphism is geometric, i.e., it establishes a bijection β : S ⨿ R → T ⨿ R and supplies a path between corresponding elements (or rather their images in cyl(q) under σ ∨ ρ or τ ∨ ρ). Now we choose subsets R_{1}, R_{2} ⊆ R, so that T ∪ β(S) = T ⨿ R_{2} and S ∪ β^{−1}(T) = S ⨿ R_{1}. Then β restricts to give a bijection β′ : S ⨿ R_{1} → T ⨿ R_{2}. By restricting attention to the paths corresponding to S ⨿ R_{1} and T ⨿ R_{2}, we finally arrive at a geometric isomorphism between (S, σ)⊕(R_{1}, ρ_{1}) and (T, τ)⊕(R_{2}, ρ_{2}), and a moment’s reflection shows that this suffices to show that a − b ∈ Im(q_{∗}) as desired.

We go on to show that β is well defined. There is a morphism

η = (η_{B}, η_{H}, η_{X}) : ξ → π_{0}(ξ) = (π_{0}(B) ⟵ π_{0}(H) ⟶ π_{0}(X)),

where η_{?} is the map from a space to its (by our assumption, discrete) quotient space of path components. By naturality, it suffices to deal with the case of π_{0}(ξ), i.e., we may assume that X, H and B are discrete. Then the groups involved are direct sums of similar groups indexed on the points of X, so we can further assume that X is just one point. Also, since B is finite, the cc control is unchanged, if we replace B by one point. If H is infinite, the groups are direct limits of similar groups corresponding to the finite subsets of H so we can take H to be finite. After that is done, once again we can observe that the cc condition is unchanged if we replace H → {∗} by 1 : H → H. In summary, we have to show that β is well defined when ξ = (H ←^{1} H → {∗}) with H finite and discrete. If H = ∅, there is nothing to prove. If H ≠ ∅, Theorem 9.1 of [?] allows us to use bc K-theory instead of K^{cc}, and Theorem II of [?] shows that F_{1}(ξ) is free abelian of rank |H| − 1 , generated by infinite shifts, and the rest of the proof is easy.

Exactness at ℤπ_{0}H is shown as follows. For each orbit A in S_{iX}, the path ω_{A} meets X, so q_{∗}[ω_{A}(∞)] = q_{∗}[ω_{A}(−∞)]. Therefore, Imβ ⊆ Kerq_{∗}. To get the opposite inclusion, we note that the kernel of q_{∗} is generated by elements of the form [w_{+}]−[w_{−}] where w_{±} ∈ H and there is a path ω from ω(−1) = q(w_{−}) to ω(1) = q(w_{+}) in X. Let L = (ℤ, λ) where λ(k) = (w_{+}, 1 − 2^{−k}) for k > 1, λ(k) = (w_{−}, 1 − 2^{k}) for k < −1, and λ(k) = ω(k) for k = −1, 0, 1. An obvious collection of paths in the (0, 1) direction combine with ω to define a geometric automorphism of L which maps to [w_{+}] − [w_{−}] under β.

Clearly [a] ↦ [cls(a)] ∈ F_{1}(ξ), for a : S^{1} → X, defines a homomorphism ℤ[S^{1}, X] → F_{1}(ξ), and to see that α_{1} is well defined, we only have to show that [cls(q ∘ b)] = 0 for any b : S^{1} → H. The proof of this proceeds as follows. For n ∈ ℕ, let ζ_{n} be a primitive 2^{n}^{th} root of unity, and define the geometric module M_{n} to be M_{n} = ({0, 1, 2, ⋯, 2^{n} − 1}, σ_{n}), where σ_{n}(i) = [1 − 2^{−n}, b(ζ_{n}^{i})] ∈ cyl(q). The restrictions of b to the arcs from ζ_{n}^{i} to ζ_{n}^{i+1} for i = 0, 1, 2, ⋯ 2^{n} − 1, define a geometric isomorphism μ_{n} : M_{n} → M_{n}. Let μ_{0} : M_{0} → M_{0} be cls(q ∘ b) : M_{q∘b(0)} → M_{q∘b(0)}. It is easily seen that the infinite direct sums μ_{even} = ⨁_{i≥0} μ_{2i}, μ′_{even} = ⨁_{i>0} μ_{2i}, and μ_{odd} = ⨁_{i≥0} μ_{2i+1}, make sense. Moreover an easy subdivision and conjugation argument along the lines of the proof of Lemma 4.3 in [?] gives the following identity

[cls(μ_{even})] = [cls(μ_{odd})] = [cls(μ′_{even})] ∈ F_{1}(ξ),

which implies that [cls(μ_{0})] = 0 ∈ F_{1}(ξ), as desired.

The proof of exactness at F_{1}(ξ) relies on a closer analysis of the splitting in (??). The necessary ideas can be dug out of [?], so we shall be content with these hints: In the splitting, the elements z_{fX} are precisely those in the image of α_{1}; this uses a version of the subdivision procedure (also used above) defined in Section 4 of [?]. Also, the elements z_{H} vanish; this uses an Eilenberg swindle combined with subdivison and radial push defined as in Section 5 of [?]. Finally, by the definition of β above, the elements z_{iX} precisely account for the image of β.

4.3. cc K-theory of compact, metrizable, tame pairs. Let (\overline{Y} , C) be a pair of spaces. We recall from [?] that C is tame in \overline{Y}, if there is a neighborhood N of C in \overline{Y} and a nearly strict deformation retraction of N into C, i.e., a strict map

(N × I, C × I ∪ N × \{0\}) → (\overline{Y} , C)

which is the inclusion on N × {1} and the projection on C × I. We denote by CM_{tame}^{2} the category of compact pairs (\overline{Y} , C) of metrizable spaces with C tame in \overline{Y}. The morphisms are strict maps f : (\overline{Y} , C) → (\overline{Y′}, C′), i.e. maps of pairs with f^{−1}(C′) = C.

The holink diagram of Section 2 of [?] defines a functor χ : CM_{tame}^{2} → T OP/CM^{∗} with

χ(\overline{Y} , C) = (Y ⟵^{q} H = Holink(C, \overline{Y}) ⟶^{p} C).

Here H is the space of strict maps ([0, 1], \{0\}) → (\overline{Y} , C) while p and q are the evaluation maps at 0 and 1, respectively.

There is also an inclusion functor ι : CM_{tame}^{2} → T OP^{cc}/LC with ι(\overline{Y} , C) = (\overline{Y} , C; id_{Y}), where Y = \overline{Y} − C. For a morphism g : (\overline{Y} , C) → (\overline{Y′}, C′), ι(g) = g|Y : Y → Y′. Since \overline{Y} is compact, the necessary conditions for ι(g) to be a morphism in T OP^{cc}/LC as defined in [?], are easily verified. We also note that when we start with (\overline{Y} , C) ∈ CM_{tame}^{2}, take Y = \overline{Y} − C, and give Y ⨿ C the teardrop topology relative to the function id_{Y} ⨿ id_{C}, as in Section 1 of [?], then we do recover the space \overline{Y}, i.e., there is no conflict between our present use of the notation \overline{Y} and that in [?].

Proposition 4.8. Let j ∈ ℕ and (\overline{Y} , C) ∈ CM_{tame}^{2}. There is a natural isomorphism

K_{j}^{Q}(GM^{cc}(ι(\overline{Y} , C))) → K_{j}^{cc}(χ(\overline{Y} , C)).

Proof: We shall introduce a homotopy notion in the category T OP^{cc}/LC, show that the composite functor K_{j}^{Q} ∘ GM^{cc} : T OP^{cc}/LC → AB is a homotopy functor, and prove that there is a natural homotopy equivalence of functors c ∘ χ ≃ ι : CM_{tame}^{2} → T OP^{cc}/LC. The desired conclusion then follows since K_{j}^{cc} ∘ χ = K_{j}^{Q} ∘ c ∘ χ for j ∈ ℕ.

The relevant homotopy notion in T OP^{cc}/LC is obtained by declaring the cylinder on (\overline{X}, B; p) to be

(\overline{X}, B; p) × I = ([(\overline{X} × I) ⨿ B]/∼, B; p × id),

where the equivalence relation ∼ identifies each {b}×I to the point b ∈ B, and where p × id : H × I → X × I ⊆ [(\overline{X} × I) ⨿ B]/∼. One then has two inclusions

i_{ν} : (\overline{X}, B; p) → (\overline{X}, B; p) × I

(ν = 0, 1), and as usual, two morphisms f_{ν} : (\overline{X}, B; p) → (\overline{Z}, D; r) (ν = 0, 1) are called homotopic if there is a morphism F : (\overline{X}, B; p)×I → (\overline{Z}, D; r) with F ∘i_{ν} = f_{ν} for ν = 0, 1.

In order to show that K_{j}^{Q} ∘ GM^{cc} is a homotopy functor for each j ∈ ℕ, it suffices to establish the following lemma.

Lemma 4.9. There is a natural equivalence of additive functors between additive categories

i_{0∗} ≅ i_{1∗} : GM^{cc}(\overline{X}, B; p) → GM^{cc}((\overline{X}, B; p) × I), (\overline{X}, B; p) ∈ T OP^{cc}/LC.

Proof: Let σ : S → H be an arbitrary object of GM^{cc}(\overline{X}, B; p). The image of (S, σ) under i_{ν∗} is (S, σ_{ν}) with σ_{ν} : S → H × I having σ_{ν}(s) = (σ(s), ν), (ν = 0, 1, s ∈ S). The obvious paths in the I direction from σ_{1}(s) to σ_{0}(s) define a morphism (S, σ_{0}) → (S, σ_{1}) in GM(H × I). It is not hard to check that this morphism is cc, and that its class in GM^{cc}((\overline{X}, B; p) × I) defines the desired natural equivalence. □

Finally, the desired natural homotopy equivalence, χ ∘ c ≃ ι, is essentially a special case of Lemma 2.4 of [?]. In fact, let P = cyl(p, q) be the double mapping cylinder of the diagram χ(\overline{Y} , C) with the teardrop topology described in [?] or in [?]. We apply the lemma quoted^{10} with ϵ = diam(\overline{Y}) to get a strict homotopy equivalence, relative to C, \overline{α} : (P, C) → (\overline{Y} , C). Let α : P − C → \overline{Y} − C be the homotopy equivalence obtained by restriction, and observe that P − C is the domain of v^{∘}(p) in the object c(χ(\overline{Y} , C) = (v(C), C; v^{∘}(p) while \overline{Y} − C = Y is the domain of id_{Y} in ι(\overline{Y} , C) = (\overline{Y} , C; id_{Y}). It is not hard to check that α is a morphism c ∘ χ(\overline{Y} , C) → ι(\overline{Y} , C), and in fact a homotopy equivalence, in the category T OP^{cc}/LC. Since the naturality is clear, this finishes the proof of Proposition ??. □

Motivated by Proposition ??, we introduce the following definition. Note that it does not say that K_{∗}^{cc}(\overline{Y} , C; id_{Y}) is obtained from the Pedersen Weibel delooping of GM^{cc}(\overline{Y} , C; id_{Y}).

10Actually, the map P → \overline{Y} given in [?] is usually not continuous, but this malady can be repaired by a reference to [?]. See also [?] and [?].

Definition 4.10. If (\overline{Y} , C) ∈ CM_{tame}^{2}, then we write K_{∗}^{cc}(\overline{Y} , C; id_{Y}) for the group K_{∗}^{cc}(χ(\overline{Y} , C)). A similar notation is used for W h^{cc} and K̃_{0}^{cc}.

5. The basic exact sequence.

This section is devoted to the proof of our first main theorem. In it, ξ = (X ←^{q} E →^{p} B) is an object of the category T OP/CM^{∗} defined in Section ??. Also, recall that

ξ_{+} = (X ⟵^{q∨id} E ⨿ X ⟶^{p⨿c} B_{+}).

Theorem 5.1. For each j ∈ ℕ, there is a natural short exact sequence

0 → lim^{1}K_{2−j}(B; p)_{ϵ} → K_{2−j}^{cc}(ξ_{+}) → K_{1−j}(B; p)_{c} → 0.

Remark: If we take X to be a point, this gives the unreduced version of Theorem 1.1 of the introduction. We leave it to the reader to formulate and prove the reduced version. □

Proof: We first note that it suffices to prove the result for j = 1. In fact, the inclusion i : {∗} → S^{1} induces a split monomorphism from the exact sequence for j = 1 and (B; p) to that for j = 1 and (B × S^{1}; p × id). In view of Corollary ?? and Proposition ??, the resulting exact sequence of cokernels gives the desired exact sequence for j = 2 and (B; p). One proceeds inductively to get the exact sequence for each j ∈ ℕ.

Next we observe that we may assume that X is a point in which case the object ξ is denoted simply p. In fact, it is known from [?] that the obvious morphism π : ξ → p induces an isomorphism π_{∗} : K_{2−j}^{cc}(ξ_{+}) → K_{2−j}^{cc}(p_{+}) for each j ∈ ℕ.

In [?], K_{1}^{cc}(p_{+}) is defined to be K_{1} of the additive category GM^{cc}(c(B_{+}), B_{+}; c^{∘}(p_{+})) where c^{∘}(ξ_{+}) : c^{∘}(E_{+}) → c^{∘}(B_{+}) is the map of open cones induced by p_{+}. In this proof it will be convenient to work with a different additive category which carries the same K_{1} group. In particular, we consider

p × id : E × (−∞, 0) → B × (−∞, 0),

and note that there is a morphism in T OP^{cc}/LC (cf. Section ?? and [?])

f : (B × (−∞, 0], B × 0; p × id) → (c(B_{+}), B_{+}; c^{∘} (p_{+}))

given as follows, for (e, t) ∈ E × (−∞, 0)

f(e, t) = \{ \begin{matrix}[e, t + 1] ∈ c^{∘}(E) ⊂ c^{∘}(E_{+}), & if − 1 < t < 0; \\ [+, 1 + t^{−1}] ∈ c^{∘}(+) ⊂ c^{∘}(E_{+}), & if − ∞ < t ≤ −1. \end{matrix}

Lemma 5.2. The morphism f induces a natural isomorphism

f_{∗} : K_{1}(GM^{cc}(B × (−∞, 0], B × 0; c^{∘}(p_{+}))) → K_{1}(GM^{cc}(c(B_{+}), B_{+}; c^{∘} (p_{+}))).

Proof: The commutative diagram of maps of topological spaces

E × (−∞, −1] ⟶^{incl} E × (−∞, 0)
f|↓  ↓^{f}
c^{∘}(+)  ⟶^{incl}  c^{∘}(E_{+})

is a diagram of morphisms in T OP^{cc}/LC, so it induces a commutative diagram of additive functors

GM^{cc}(B × (−∞, −1], ∅; p × id) ⟶^{incl_{∗}} GM^{cc}(B × (−∞, 0], B × 0; p × id)
f|_{∗}↓  ↓^{f_{∗}}
GM^{cc}(c(+), +; c^{∘}(id_{+}))  ⟶^{incl_{∗}}  GM^{cc}(c(B_{+}), B_{+}; c^{∘}(p_{+}))

An easy argument along the lines of the proof of Theorem 3.1 in [?] shows that the categories in the left hand column are flasque, so their Quillen K-theory vanishes. The horizontal functors are clearly full and faithful, and the categories in the right hand column are Karoubi filtered (see e.g., p. 355 in [?]) by those in the left hand column. Finally, since f|E × [δ, 0) is a homeomorphism onto its image for any δ ∈ (−1, 0), the two Karoubi quotient categories are equivalent under the induced functor. The desired conclusion now follows by considering the homotopy fibrations from Theorem 5.3 of [?], and the induced maps connecting them. [Actually, one has to pass to the idempotent completions in order to apply the theorem quoted, but that does not change K_{1}, nor does it disturb flasque-ness]. □

In the actual proof of the theorem for j = 1, we shall use the following abbreviations and notation.

GM_{ϵ} = GM_{ϵ}(B; p);  GM^{cc} = GM^{cc}(B × (−∞, 0], B × 0; p × id);
GM_{ϵ} = GM_{ϵ}(B; p); GM^{cc} = GM^{cc}(B × (−∞, 0], B × 0; p × id);
lim^{1}K_{1} = lim^{1}K_{1}(B; p)_{ϵ}; K_{1}^{cc} = K_{1}(GM^{cc}(B × (−∞, 0], B × 0; p × id));
limK_{0} = limK_{0}(B; p)_{ϵ}.

If M = (S, σ), where σ : S → E × (−∞, 0), is a geometric module on E × (−∞, 0), and U ⊆ B × (−∞, 0], we denote by j_{U} : M|U → M the inclusion of the direct summand M|U = (σ^{−1}(p × id)^{−1}(U), σ|) into M; by q_{U} : M → M|U the corresponding projection; and by π_{U} the idempotent π_{U} = j_{U}q_{U} : M → M. Note that j_{U}, q_{U}, π_{U} are considered morphisms in GM rather than GM.

The necessary connection between geometric modules on E × (−∞, 0) and E will be given by the homotopy preserving functors

Σ : GM(E × (−∞, 0)) → GM(E), (40) i_{n∗} : GM(E) → GM(E × (−∞, 0)), n ∈ ℤ. (41)

The (“summation”) functor Σ is induced by the projection E × (−∞, 0) → E, and i_{n} : E → E ×(−∞, 0) has i_{n}(e) = (e, −2^{−n}). In general, if φ : M → N in GM^{cc}, then Σ(φ) : Σ(M) → Σ(N) will not be in GM_{ϵ}, even for ϵ = ∞ (in fact, Σ(M) will not be finite). However, for judiciously chosen subsets U, V ⊆ B × (−∞, 0) and a suitable ϵ, one may have Σ(q_{V} φj_{U}) : Σ(M|U) → Σ(N|V) in GM_{ϵ}. On the other hand, i_{n∗} does map GM_{ϵ} into GM^{cc}, but we will often have to consider infinite direct sums ⊕_{n}i_{n∗} in which case one has to be careful that the sums make sense in GM^{cc}.

The composite functors satisfy

Σi_{n∗} = 1_{GM(E)} and i_{n∗}Σ ≃ 1_{GM(E×(−∞,0))}, (42)

where the natural homotopy consists of paths in the (−∞, 0)-direction. Such paths also define natural transformations

ν_{n} : i_{n∗} → i_{(n+1)∗} and μ_{n} : i_{(n+1)∗} → i_{n∗}, (43)

and the composite transformations are homotopy equivalent to the identity transformations, i.e.,

ν_{n}μ_{n} ≃ 1 and μ_{n}ν_{n} ≃ 1. (44)

Also in this case, when infinite direct sums of the transformations ν_{n} or μ_{n} are formed, attention has to be paid to the cc condition, and the homotopy equivalence from i_{n∗}Σ to the identity is in GM^{cc} only for suitable objects.

In our dealings with lim and lim^{1} we shall actually work over the cofinal subset {2^{−n}|n ∈ ℤ}, so that an element x ∈ lim^{1}K_{1} is represented by a sequence {[φ_{n}]}_{n} ∈ ∏_{n} K_{1}(B; p)_{2^{−n}} where φ_{n} : M_{n} → M_{n} in GM_{2^{−n}}, n ∈ ℤ, and there exist ψ_{n} : M_{n} → M_{n} in GM_{2^{−n}} and homotopies Φ_{n} : φ_{n}ψ_{n} ≃ 1, Ψ_{n} : ψ_{n}φ_{n} ≃ 1 in GM_{2^{1−n}}. It is easily seen that

φ_{⊕} = ⨁_{n∈ℤ}i_{n∗}(φ_{n}) : M_{⊕} = ⨁_{n∈ℤ}i_{n∗}(M_{n}) → M_{⊕} (45)

defines a morphism in GM^{cc}. Similarly, one has ψ_{⊕} and homotopies Φ_{⊕} : φ_{⊕}ψ_{⊕} ≃ 1, Ψ_{⊕} : ψ_{⊕}φ_{⊕} ≃ 1 in GM^{cc}. Thus, φ_{⊕} represents an automorphism cls(φ_{⊕}) ∈ GM^{cc}, and we claim that

(46) i(x) = [cls(φ_{⊕})] defines a monomorphism i : lim^{1} → K_{1}^{cc}.

First note that cls(φ_{⊕}) ∈ GM^{cc} is unchanged if the sequence φ_{n} is replaced by a sequence φ′_{n} with φ_{n} ≃ φ′_{n} in GM_{2^{−n}}; that cls(φ_{⊕}) is replaced by cls(φ_{⊕})cls(γ_{⊕}) with cls(γ_{⊕}) elementary in GM^{cc}, if the sequence φ_{n} is replaced by a sequence φ_{n}γ_{n} [∈ GM_{2^{1−n}}] with γ_{n} elementary in GM_{2^{−n}}; that a stabilization of each φ_{n} (to φ_{n}⊕1_{N_{n}}) leads to a stabilization of φ_{⊕} (to φ_{⊕} ⊕ 1_{N_{⊕}}); and that the operation preserves (direct) sums. Therefore, {[φ_{n}]}_{n} → [cls(φ_{⊕})] defines a homomorphism ∏_{n} K_{1}(B; p)_{2^{−n}} → K_{1}^{cc}.

We must show that this homomorphism factors over lim^{1}K_{1}, so we suppose that {[φ_{n}]}_{n} represents 0 in lim^{1}K_{1}. This means that we have an element {[σ_{n}]}_{n} ∈ ∏_{n} K_{1}(B; p)_{2^{−n}} so that [φ_{n}] = [σ_{n}] − [σ_{n+1}] ∈ K_{1}(B; p)_{2^{−n}}, n ∈ ℤ. Let σ_{n} : N_{n} → N_{n} have θ_{n} as a 2^{−n}-inverse. Then −[σ_{n+1}] = [θ_{n+1}], so we may assume that M_{n} = N_{n}⊕N_{n+1}, and φ_{n} = σ_{n}⊕θ_{n+1} : N_{n}⊕N_{n+1} → N_{n}⊕N_{n+1}, n ∈ ℤ. We note that M_{⊕} ≠ N_{⊕} ⊕ N_{⊕}, but the natural transformations ν_{n}, μ_{n} of (??) give rise to an isomorphism (in GM^{cc}) M_{⊕} → N_{⊕} ⊕ N_{⊕} which conjugates cls(φ_{⊕}) into cls(σ_{⊕}) ⊕ cls(θ_{⊕}). Since cls(σ_{⊕}) and cls(θ_{⊕}) are inverses in GM^{cc}, it follows that [cls(φ_{⊕})] = 0 ∈ K_{1}^{cc}, so we do have the desired factorization.

To show that i is monic, we need the following lemma, the straightforward proof of which will be left to the reader.

Lemma 5.3. Let F be a cc family of subsets of B×(−∞, 0] and let U_{0} = B×(−∞, 0]. There is a strictly increasing function λ : ℕ → ℤ such that any F ∈ F which meets U_{n} = B × [−2^{−λ(n)}, 0] is completely contained in U_{n−1}, and has diameter diam(F) ≤ 2^{−n}, n ∈ ℕ.

If this lemma is applied to F(φ) for some morphism φ : M → N, then for any n < m ∈ ℕ ∪ ∞, φ maps M|(U_{n} − U_{m}) into N|(U_{n−1} − U_{m+1}), and the resulting morphism Σ(M|(U_{n} −U_{m})) → Σ(N|(U_{n−1} −U_{m+1}), has bound 2^{−n}. A similar remark holds for homotopies.

Returning to the proof of the injectivity of i, we assume that x = {[φ_{n}]}_{n} has [cls(φ_{⊕})] = 0 ∈ K_{1}^{cc}. After a stabilization (which can easily be absorbed into the given morphisms φ_{n}), then in GM^{cc} we have thirteen^{11} elementary automorphisms γ^{i} : M_{⊕} → M_{⊕}, i = 1, 2, ⋯, 13 and a homotopy Γ : φ_{⊕} ≃ γ = γ^{1}γ^{2} ⋯ γ^{13}. We apply Lemma ?? to F = F(Γ) ∪ ⋃_{i} F(γ^{i}) to determine λ : ℕ → ℤ and the neighborhoods U_{n} of B, n ∈ ℕ.

For each pair (n_{1}, n_{2}) ∈ (ℕ ∪ {∞})^{2} with n_{1} < n_{2}, let n = [n_{1}, n_{2}) be the corresponding half open interval. Let M_{n} = M_{⊕}|(U_{n_{1}} − U_{n_{2}}) with complement M_{n}^{⊥} (= M_{⊕}|(U_{0} − U_{n_{1}}) ⊕ M_{⊕}|U_{n_{2}}); inclusion j_{n} : M_{n} → M_{⊕}; projection q_{n} : M_{⊕} → M_{n}; and idempotent π_{n} = j_{n}q_{n} : M_{⊕} → M_{⊕}. Also let φ_{n} = q_{n}φ_{⊕}j_{n} : M_{n} → M_{n} (so Σφ_{n} = ⊕_{k}φ_{k} with λ(n_{1}) ≤ k < λ(n_{2})). Finally, for the above n = [n_{1}, n_{2}), let

n′′ = [n_{1}, n_{2} + 1) ; n′ = [n_{1} − 1, n_{2}), if n_{1} > 1; and (47)

n + ke = [n_{1} − k, n_{2} + k), if k ∈ ℤ and 0 < n_{1} − k < n_{2} + k. (48)

For any n there is the elementary automorphism (of bound 2^{−n_{1}}) γ_{n}^{i} = 1 + (γ^{i} − 1)π_{n} : M_{⊕} → M_{⊕}. It maps M_{n+e} into itself, is the identity on M_{n+e}^{⊥}, and agrees with the restriction of γ^{i} on the submodule M_{n} ⊆ M_{⊕}. We think of γ_{n}^{i} as a “concentration of γ^{i} on M_{n}” although this is not literally true. If n_{2} > n_{1} +26, then the composition

11In [?] (see pp. 145-46), two of the present authors observed that 16 elementary automorphisms will do. The reduction to 13 factors can be found in [?], or by contemplating the proof of Proposition ??.

γ_{n} = γ_{n−e}^{1}γ_{n−2e}^{2} ⋯ γ_{n−13e}^{13} : M_{⊕} → M_{⊕} (49)

has bound 2^{−n_{1}−13}(1 + 2 + 4 + ⋯ + 2^{12}) < 2^{−n_{1}}. Moreover, each factor maps M_{n} into itself and is the identity on M_{n}^{⊥}. Hence q_{n}γ_{n}j_{n} : M_{n} → M_{n} is a product of the 13 elementary automorphisms q_{n}γ_{n−ie}^{i}j_{n} : M_{n} → M_{n}, so

[Σ(q_{n}γ_{n}j_{n})] = 0 ∈ K_{1}(B; p)_{2^{−n_{1}}}, if n_{2} > n_{1} + 26. (50)

We now assume that n_{2} − n_{1} ≥ 39, and write M_{n} = M_{l} ⊕ M_{m} ⊕ M_{r} where l = [n_{1}, n_{1} +13), m = n−13e, and r = [n_{2} −13, n_{2}). The indexing is supposed to convey the left hand, the middle, and the right hand summand of M_{n}, respectively. We note that M_{r} = 0 when n_{2} = ∞. Since the length of the middle interval m is at least 13, γ_{n} maps the left summand M_{l} into M_{l} ⊕ M_{m} and the right summand M_{r} into M_{m} ⊕ M_{r}. Also, γ_{n} coincides with γ on M_{m}, so

q_{n}γ_{n}j_{m} = q_{n}γj_{m} ≃ q_{n}φ_{⊕}j_{m} : M_{m} → M_{n}

(the homotopy is the reverse of q_{n}Γj_{m} and has bound 2^{−n_{1}}). In terms of matrices w.r.t. the decomposition M_{n} = M_{l} ⊕ M_{m} ⊕ M_{r} this gives us the equation

(51) q_{n}γ_{n}j_{n} = ( \begin{matrix}γ_{n}^{(l)} & q_{l}γ_{n}j_{m} & 0 \\ ∗ & q_{m}γ_{n}j_{m} & ∗ \\ 0 & q_{r}γ_{n}j_{m} & γ_{n}^{(r)} \end{matrix} ) ≃ ( \begin{matrix}γ_{n}^{(l)} & 0 & 0 \\ ∗ & φ_{m} & ∗ \\ 0 & 0 & γ_{n}^{(r)} \end{matrix} )

in GM(E × (−∞, 0); p × id)_{2^{−n_{1}}} (note that the center columns represent q_{n}γj_{m} and q_{n}φ_{⊕}j_{m}, respectively, and that the equation defines γ_{n}^{(l)} and γ_{n}^{(r)}).

Since the left hand side and φ_{m} are 2^{−n_{1}}-isomorphisms, it easily follows that γ_{n}^{(l)} and γ_{n}^{(r)} are 2^{2−n_{1}}-isomorphisms. Clearly, (??) and (??) imply that^{12}

(52) [Σγ_{n}^{(l)}] + [Σφ_{m}] + [Σγ_{n}^{(r)}] = 0 ∈ K_{1}(B; p)_{2^{3−n_{1}}}, if 39 ≤ n_{2} − n_{1} < ∞.

It is easily seen that the upper left hand corner of the above matrix is unchanged if one augments n_{2}, keeping n_{1} fixed, i.e., γ_{n}^{(l)} = γ_{n′′}^{(l)} (and, similarly, γ_{n}^{(r)} = γ_{n′}^{(r)}). Hence the definition b_{n_{1}−3} = [Σ(γ_{n}^{(l)})] ∈ K_{1}(B; p)_{2^{3−n_{1}}} gives an element which is independent of the choice of n_{2} ∈ [n_{1} + 39, ∞). For any n ∈ ℕ, we can apply (??) to n = [n + 4, n + 43) and n′ = [n + 3, n + 43) to get

12The loss of a factor 2 in control happens because one must get rid of the off diagonal terms by an elementary operation.

b_{n+1} + ∑_{k=λ(n+17)}^{λ(n+30)−1}[φ_{k}] + [Σ(γ_{n}^{(r)})] = 0 ∈ K_{1}(B; p)_{2^{−n−1}}, and

b_{n} + ∑_{k=λ(n+16)}^{λ(n+30)−1}[φ_{k}] + [Σ(γ_{n′}^{(r)})] = 0 ∈ K_{1}(B; p)_{2^{−n}}.

If we let c_{n} = −b_{n} + ∑_{k=n}^{λ(n+16)−1}[φ_{k}] ∈ K_{1}(B; p)_{2^{−n}}, then the above formulae immediately imply that c_{n} − r_{n,n+1}(c_{n+1}) = [φ_{n}] ∈ K_{1}(B; p)_{2^{−n}} for all n ∈ ℕ. This shows that x = {[φ_{k}]}_{k} = 0 ∈ lim^{1}K_{1} and finishes the proof of the injectivity of i.

We go on to define the map ∂ : K_{1}^{cc} → limK_{0}. Let y = [clsφ] ∈ K_{1}^{cc}. Thus φ : M → M in GM^{cc} and there exist ψ : M → M and homotopies Φ : φψ ≃ 1 and Ψ : ψφ ≃ 1 in GM^{cc}. We use Lemma ?? with F = F(φ) ∪ F(ψ) ∪ F(Φ) ∪ F(Ψ) and continue to denote the resulting neighborhoods by U_{n}. Also, the notation j_{n}, q_{n}, π_{n}, M_{n}, introduced in relation to M_{⊕} immediately before (??), will now be used for M. Thus

π_{n}φj_{k} = φj_{k} whenever n ⊇ k + e,

and analogous identities hold for ψ, Φ, and Ψ. For any n = [n_{1}, n_{2}) with n_{2} − n_{1} ≥ 3, we let

(53) p_{n} = p_{n}(φ, ψ) = q_{n}φπ_{n−e}ψj_{n} : M_{n} → M_{n}

Since j_{n}q_{n}φπ_{n−e} = φπ_{n−e}, the homotopy q_{n}φπ_{n−e}Ψπ_{n−e}ψj_{n} shows that p_{n} is a homotopy idempotent of bound 2^{−n_{1}} on the module M_{n}.

If we assume that n_{2} − n_{1} ≥ 6, then we get a decomposition into a left hand, a middle, and a right hand summand as before, M_{n} = M_{l} ⊕ M_{m} ⊕ M_{r}, but this time l = [n_{1}, n_{1} + 2), m = n − 2e, r = [n_{2} − 2, n_{2}). As before, the right hand summand vanishes if n_{2} = ∞. Since m has length at least two, p_{n} maps M_{l} into M_{l} ⊕ M_{m} and M_{r} into M_{m} ⊕ M_{r}. Also,

p_{n}|M_{m} = q_{n}φπ_{n−e}ψj_{m} = q_{n}φψj_{m} ≃ q_{n}j_{m} = incl : M_{m} → M_{n}

by the homotopy q_{n}Φj_{m} of bound 2^{−m_{1}} = 2^{−(n_{1}+2)}. In matrix language relative to the decomposition M_{n} = M_{l} ⊕ M_{m} ⊕ M_{r}, this means that we have a homotopy of bound 2^{−m_{1}}

(54) p_{n} = ( \begin{matrix}p_{n}^{(l)} & q_{l}φψj_{m} & 0 \\ ∗ & q_{m}φψj_{m} & ∗ \\ 0 & q_{r}φψj_{m} & p_{n}^{(r)} \end{matrix} ) ≃ ( \begin{matrix}p_{n}^{(l)} & 0 & 0 \\ ∗ & 1 & ∗ \\ 0 & 0 & p_{n}^{(r)} \end{matrix} )

in GM(E×(−∞, 0); p×id). It follows easily that the right hand matrix is a homotopy idempotent of bound 2^{1−n_{1}}. Hence p_{n}^{(l)} and p_{n}^{(r)} (which are defined by the equation) are homotopy idempotents of bound 2^{1−n_{1}}. Moreover, the left hand idempotent p_{n}^{(l)} is unchanged if one augments n_{2}, keeping n_{1} fixed (and dually for the right hand idempotent p_{n}^{(r)}), i.e., with notation from (??),

p_{n′}^{(r)} = p_{n}^{(r)} and p_{n′′}^{(l)} = p_{n}^{(l)}. (55)

We define ∂ : K_{1}^{cc} → limK_{0}(B; p)_{2^{−n}} by

∂[clsφ] = {∂_{n}(φ, ψ)}_{n}, (56)

where

∂_{n−1}(φ, ψ) = [Σp_{[n,n+k)}^{(l)}(φ, ψ)] − [Σp_{[n,n+k)}^{(l)}(1_{M}, 1_{M})], 6 ≤ k ≤ ∞. (57)

To check that this actually gives an element of limK_{0} which is independent of the various choices we need the following identity in K_{0}(B; p)_{2^{1−n_{1}}}

(58) [Σp_{n}(1_{M}, 1_{M})] = [Σp_{n}(φ, ψ)] = [Σp_{n}^{(l)}(φ, ψ)] + [1_{ΣM_{m}}] + [Σp_{n}^{(r)}(φ, ψ)].

The second part of (??) follows from (??). To prove the first part, let f = q_{n}φπ_{n−e}j_{n} : M_{n} → M_{n} and g = q_{n}π_{n−e}ψj_{n} : M_{n} → M_{n}. Clearly fg = p_{n}(φ, ψ), and, since π_{n}φπ_{n−e} = φπ_{n−e}, gf ≃ p_{n}(1_{M}, 1_{M}) with bound 2^{−1−n_{1}}.

The identity shows that ∂_{n−1}(φ, ψ) is also given by

∂_{n−1}(φ, ψ) = −[Σp_{[n,n+k)}^{(r)}(φ, ψ)] + [Σp_{[n,n+k)}^{(r)}(1_{M}, 1_{M})], 6 ≤ k < ∞. (59)

It follows from (??) and (??) that {∂_{n}(φ, ψ)}_{n} ∈ limK_{0}(B; p)_{2^{−n}}. It also follows that the choice of the neighborhoods U_{n} satisfying Lemma ?? is immaterial. Indeed, if {U′_{n}}_{n} is a different choice, then {U_{n} ∩ U′_{n}}_{n} is also a possible choice, so in a comparison we may assume that U′_{n} ⊆ U_{n}. But then for any n_{0}, the system {V_{n}^{(n_{0})}}_{n} which coincides with U_{n} for n < n_{0} and with U′_{n} for n ≥ n_{0} is allowed and one only has to use (??) and (??) to the system {V_{n}^{(n_{0})}}_{n} for a suitable choice of n_{0} and k to finish the proof. Similarly, a modification of φ (or ψ) by a homotopy or by right multiplication by an elementary γ can be decomposed into two consecutive modifications where one does not change anything above some arbitrarily chosen U_{n_{0}} while the other one leaves everything intact above the complement of U_{n_{0}+k} for suitable k. Therefore, the above argument easily adapts to show that, indeed, (??) and (??) give a well defined homomorphism.

To show that ∂ is epic, let z = {[α_{n}] − [β_{n}]}_{n} ∈ limK_{0}(B; p)_{2^{−n}}. This means that α_{n} : A_{n} → A_{n} and β_{n} : B_{n} → B_{n} are homotopy idempotents of bound 2^{−n}, and in GM_{2^{−n}} there exist geometric modules S_{n} and morphisms ψ_{n} : A_{n} ⊕ B_{n+1} ⊕ S_{n} → A_{n+1} ⊕ B_{n} ⊕ S_{n} and φ_{n} : A_{n+1} ⊕ B_{n} ⊕ S_{n} → A_{n} ⊕ B_{n+1} ⊕ S_{n} such that we have the following homotopies of bound 2^{1−n}

(60)  α_{n}^{2} ≃ α_{n},  and β_{n}^{2} ≃ β_{n};
(61)  φ_{n}ψ_{n} ≃ α_{n} ⊕ β_{n+1} ⊕ 1_{S_{n}}, and ψ_{n}φ_{n} ≃ α_{n+1} ⊕ β_{n} ⊕ 1_{S_{n}};
(62)  φ_{n} ≃ φ_{n}(α_{n+1} ⊕ β_{n} ⊕ 1_{S_{n}}) ≃ (α_{n} ⊕ β_{n+1} ⊕ 1_{S_{n}})φ_{n};
(63)  ψ_{n} ≃ ψ_{n}(α_{n} ⊕ β_{n+1} ⊕ 1_{S_{n}}) ≃ (α_{n+1} ⊕ β_{n} ⊕ 1_{S_{n}})ψ_{n}.

We now set T = ⊕_{n}i_{n∗}(A_{n} ⊕ B_{n} ⊕ S_{n}), T′ = ⊕_{n}i_{n∗}(A_{n+1} ⊕ B_{n} ⊕ S_{n}), and T′′ = ⊕_{n}i_{n∗}(A_{n} ⊕ B_{n+1} ⊕ S_{n}), where n ranges over ℤ. These three geometric modules in GM(E ×(−∞, 0)) are connected by natural transformations (cf. (??)) ν′ : T′ → T, μ′ : T → T′, ν′′ : T′′ → T, μ′′ : T → T′′ which are pairwise inverses up to natural, cc homotopies. Put Φ = ⊕_{n}i_{n∗}(φ_{n}) : T′ → T′′, Ψ = ⊕_{n}i_{n∗}(ψ_{n}) : T′′ → T′, and Γ = ⊕_{n}i_{n∗}((1 − α_{n}) ⊕ (1 − β_{n}) ⊕ 0) : T → T. Elementary computations using the homotopies from (??) – (??) show that

Φ̃Γ ≃ 0, ΓΦ̃ ≃ 0, Ψ̃Γ ≃ 0, ΓΨ̃ ≃ 0, Γ^{2} ≃ Γ, Ψ̃Φ̃ ≃ 1 − Γ, and Φ̃Ψ̃ ≃ 1 − Γ,

where Ψ̃ = ν′Ψμ′′ : T → T, and Φ̃ = ν′′Φμ′ : T → T. Moreover, all of these maps and homotopies are in GM^{cc}, so ψ = Ψ̃ + Γ : T → T and φ = Φ̃ + Γ : T → T are cc homotopy inverses. We claim that ∂[cls(φ)] = z.

To compute ∂_{n}(φ, ψ), we chose U_{n} = B × [−2^{−n}, 0] in Lemma ?? and k = 6 in (??). The homotopy idempotent Σp_{[n,n+6)}^{(l)}(φ, ψ) then lives on the module A_{n} ⊕ B_{n} ⊕ S_{n} ⊕ A_{n+1} ⊕ B_{n+1} ⊕ S_{n+1}, and a simple computation gives us the following matrix form up to 2^{1−n}-homotopies coming from (??) – (??). Here a superscript like ^{AB} denotes the component of a morphism which maps the relevant B-summand to the relevant A-summand.

( \begin{matrix}φ_{n}^{AA}ψ_{n}^{AA} & 0 & φ_{n}^{AA}ψ_{n}^{AS} & 0 & φ_{n}^{AA}ψ_{n}^{AB} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ φ_{n}^{SA}ψ_{n}^{AA} & 0 & φ_{n}^{SA}ψ_{n}^{AS} & 0 & φ_{n}^{SA}ψ_{n}^{AB} & 0 \\ 0 & 0 & 0 & 1_{A_{n+1}} & 0 & 0 \\ φ_{n}^{BA}ψ_{n}^{AA} & 0 & φ_{n}^{BA}ψ_{n}^{AS} & 0 & φ_{n}^{BA}ψ_{n}^{AB} + (1 − β_{n+1}) & 0 \\ 0 & 0 & 0 & 0 & 0 & 1_{S_{n+1}} \end{matrix} )

This is the sum of two homotopy orthogonal homotopy idempotents, namely the diagonal matrix diag(0, 0, 0, 1_{A_{n+1}}, 1 − β_{n+1}, 1_{S_{n+1}}) and the product X^{t}Y where X = (φ_{n}^{AA}, 0, φ_{n}^{SA}, 0, φ_{n}^{BA}, 0), Y = (ψ_{n}^{AA}, 0, ψ_{n}^{AS}, 0, ψ_{n}^{AB}, 0) and ^{t} denotes a transposition. These terms represent [1_{A_{n+1}}] + [1 − β_{n+1}] + [1_{S_{n+1}}], respectively, [X^{t}Y] = [Y X^{t}] = [(ψ_{n}φ_{n})^{AA}] = [α_{n+1}]. Since [p_{[n,n+6)}^{(l)}(1_{T} , 1_{T})] = [1_{A_{n+1}⊕B_{n+1}⊕S_{n+1}}], it follows that ∂_{n−1}(φ, ψ) = [α_{n+1}] − [β_{n+1}] ∈ K_{0}(B; p)_{2^{1−n}}, so ∂[clsφ] = z as claimed.

We leave to the reader the easy proof that ∂i = 0 and go on to prove that Ker(∂) ⊆ Im(i). We keep the notation from the last few lines preceding (??) and assume that y = [clsφ] ∈ Ker(∂), i.e., for each n ∈ ℕ, the element ∂_{n−1}(φ, ψ) defined in (??) vanishes in K_{0}(B; p)_{2^{1−n}}. The basic lemma is the following one.

Lemma 5.4. There exists a universal constant c ∈ ℕ for which the following holds. If ∂_{n−1}(φ, ψ) = 0 ∈ K_{0}(B; p)_{2^{1−n}}, then there exist a (finite) geometric module L_{n+1} in GM(B; p), seven 2^{c−n}-elementary automorphisms γ^{i} : M′ = M ⊕ i_{n+1∗}L_{n+1} → M ⊕ i_{n+1∗}L_{n+1}, and a morphism φ′ ≃ γ^{7}γ^{6} ⋯ γ^{1}φ : M′ → M′ such that φ′ is diagonal w.r.t. the decomposition M′ = M′_{(−∞,n+1)} ⊕ M′_{[n+1,∞)}. Moreover, each γ^{i} is the identity on (M′_{[n,n+3)})^{⊥}. Finally, the homotopy is bounded by 2^{c−n} measured in B and it is stationary on (M′_{[n,n+3)})^{⊥}.

Proof: Throughout the proof, the constant c will be increased a few times without our changing its name. We take k = ∞ in the definition (??) and write n = [n, ∞). Then the right hand summand in the decomposition used in (??) vanishes, so M = M_{n}^{⊥} ⊕ M_{l} ⊕ M_{m} where l = [n, n + 2) and m = [n + 2, ∞). Up to 2^{c−n}-homotopy, one has the following matrix presentation relative to this decomposition (compare (??) and (??)).

φπ_{n−e}ψ = ( \begin{matrix}0 & 0 & 0 \\ 0 & p_{n}^{(l)}(φ, ψ) & 0 \\ 0 & x & 1 \end{matrix} ) (64)

Since φπ_{n−e}ψ is a 2^{c−n}-idempotent, xp_{n}^{(l)}(φ, ψ) ≃ 0 (with bound 2^{c+1−n}). We let γ^{1} = e_{32}(x) where the classical index-notation for the elementary automorphism refers to the decomposition of M used in (??). The effect of replacing (φ, ψ) by (γ^{1}φ, ψ(γ^{1})^{−1}) then is to replace x by 0 in (??) (up to 2^{1+c−n}-homotopy). Thus we shall proceed with x = 0 in (??). Note that this first elementary γ^{1} is supported on M_{[n,n+3)} as claimed. The remaining six elementary automorphisms will actually be supported on M′_{l} (recall that l = [n, n + 2)).

Since p_{n}^{(l)}(1_{M}, 1_{M}) : M_{l} → M_{l} is the canonical projection onto M_{[n+1,n+2)}, Proposition ?? gives the following translation of the vanishing of ∂_{n−1}(φ, ψ). For some constant c ∈ ℕ we have a geometric module L′_{n} ∈ GM_{0}; two 2^{c−n}-morphisms

α_{n} : ΣM_{l} ⊕ L′_{n} → ΣM_{[n+1,n+2)} ⊕ L′_{n} and (65)

β_{n} : ΣM_{[n+1,n+2)} ⊕ L′_{n} → ΣM_{l} ⊕ L′_{n}; (66)

and four 2^{c+1−n}-homotopies (note that two of the conditions in (??) are vacuous, because the second idempotent involved here is the identity)

(67) β_{n}α_{n} ≃ Σp_{n}^{(l)}(φ, ψ) ⊕ 1_{L′_{n}} and α_{n}β_{n} ≃ 1_{ΣM_{[n+1,n+2)}⊕L′_{n}}; and

α_{n}(Σp_{n}^{(l)}(φ, ψ) ⊕ 1_{L′_{n}}) ≃ α_{n} and β_{n} ≃ (Σp_{n}^{(l)}(φ, ψ) ⊕ 1_{L′_{n}})β_{n}. (68)

If we stabilize φ and ψ by adding 1_{i_{n+1∗}L′_{n}}, the summands L′_{n} will appear in the original data, and no stabilization will be needed. Hence, we proceed under the assumption that L′_{n} = 0.

We now let L_{n+1} = ΣM_{[n+1,n+2)}. Then α_{n} : ΣM_{l} → L_{n+1}, β_{n} : L_{n+1} → ΣM_{l}, and we also have the inclusion j : L_{n+1} → ΣM_{l} and the projection q : ΣM_{l} → L_{n+1}. The following product of six 2^{c−n}-elementary automorphisms in the module ΣM_{l} ⊕ L_{n+1}

( \begin{matrix}1 & 0 \\ −q & 1 \end{matrix} )( \begin{matrix}1 & j \\ 0 & 1 \end{matrix} )( \begin{matrix}1 & 0 \\ −q & 1 \end{matrix} )( \begin{matrix}1 & 0 \\ α_{n} & 1 \end{matrix} )( \begin{matrix}1 & −β_{n} \\ 0 & 1 \end{matrix} )( \begin{matrix}1 & 0 \\ α_{n} & 1 \end{matrix} )

is easily seen to conjugate^{13} the diagonal matrix diag(β_{n}α_{n}, 0) into diag(jq, 0) (up to 2^{c−n}-homotopy). Now we get the elementaries γ^{i}, i = 2, 3, ⋯ 7, by applying i_{n+1∗} to each of the elementaries above, using (??) to identify i_{n+1∗}ΣM_{l} with M_{l}, and extending the resulting elementaries to be the identity on (M′_{l})^{⊥}. With the identification mentioned, i_{n+1∗}(jq) : M_{l} → M_{l} is the standard projection onto M_{[n+1,n+2)}, so the resulting φ′ = γ^{7}γ^{6} ⋯ γ^{2}φ : M′ → M′ conjugates π_{[n+1,∞)} : M′ → M′ into itself (up to a 2^{c−n}-homotopy supported on M′_{l}), and the proof is complete. □

We can now finish the proof of Theorem ?? as follows. Recall that we have assumed that y ∈ K_{1}^{cc} has ∂(y) = 0. Using the lemma, we can first write a representative of y as φ′ ⊕φ : M′ ⊕M → M′ ⊕M with M′ = M′_{[0,4)} and M = M_{[4,∞)}. A standard Eilenberg swindle on the ray (−∞, 0] shows that [clsφ′] = 0 ∈ K_{1}^{cc}. Repeated applications of the lemma let us split φ as an infinite direct sum

φ = ⨁_{k=1}^{∞} φ_{4k} : ⨁ M_{[4k,4k+4)} → ⨁ M_{[4k,4k+4)}

Indeed, the resulting (seven) infinite families of elementary automorphisms assemble to seven cc elementary automorphisms on the direct sum, because of the conditions in Lemma ??, and a similar remark holds for the infinite family of stabilizations, and the infinite family of homotopies. The family {Σφ_{4k}}_{k} defines an element z ∈ lim^{1}K_{1} and clearly i(z) = y.

6. Computations for tame pairs.

Let (\overline{Y} , C) be a compact, metric pair with C tame in \overline{Y} and with χ = (Y ⟵^{q} H ⟶^{p} C) the corresponding holink diagram. The groups K_{∗}(Y) and K_{∗}^{cc}(\overline{Y} , C; id_{Y}), as well

( \begin{matrix}1 & 0 \\ ξ & 1 \end{matrix} )( \begin{matrix}1 & −η \\ 0 & 1 \end{matrix} )( \begin{matrix}1 & 0 \\ ξ & 1 \end{matrix} )

13This is based on the following (well known) fact: If, in some additive category, we have morphisms ξ : X → Y and η : Y → X satisfying ξηξ = ξ and ηξη = η, then the product conjugates diag(ηξ, 0) into diag(0, ξη).

as K_{2−j}(C; p)_{c} and lim^{1}K_{2−j}(C; p)_{ϵ} for each j ∈ ℕ, are functorial in (\overline{Y} , C) ∈ CM_{tame}^{2}, and we shall fit them into a natural chain complex

K_{2}(Y) ⟶^{i_{2}} K_{2}^{cc}(\overline{Y} , C; id_{Y}) ⟶^{Δ_{2}} K_{1}(C; p)_{c} ⟶^{ℓ_{1}} K_{1}(Y) ⟶^{i_{1}} ⋯ (69)

and prove

Theorem 6.1. Let j ∈ ℕ. The above chain complex is exact at K_{1−j}(C; p)_{c} and K_{1−j}(Y), and the homology group at K_{2−j}^{cc}(\overline{Y} , C; id_{Y}) is naturally isomorphic to lim^{1}K_{2−j}(C; p)_{ϵ}.

Remarks: 1) This is the unreduced version of Theorem 1.2. All the ingredients in the following proof exist in reduced versions, so the proof of the other version is the same, mutatis mutandis.

2) We note that the theorem as stated contains no claims about the homology at K_{2}^{cc}(\overline{Y} , C; id_{Y}) or exactness at K_{1}(Y) and K_{1}(C; p)_{c}. In fact, two of the present authors (FXC and HJM) have worked out a definition of ϵ-controlled groups K_{2}(C; p)_{ϵ} which appear to fit into an exact sequence

0 → lim^{1}K_{2}(C; p)_{ϵ} → K_{2}^{cc}(χ_{+}) → K_{1}(C; p)_{c} → 0.

Assuming this, the following proof will also cover the three cases currently excluded.

3) If we introduce “bumpy groups” BK_{∗}^{cc} (cf. [?]) by letting BK_{∗}^{cc}(\overline{Y} , C; id_{Y}) = KerΔ_{∗}, then the theorem translates into two exact sequences for each j ∈ ℕ, viz.

0 → BK_{2−j}^{cc}(\overline{Y} , C; id_{Y}) → K_{2−j}^{cc}(\overline{Y} , C; id_{Y}) → K_{1−j}(C; p)_{c} → K_{1−j}(Y),

and

K_{1−j}(C; p)_{c} → K_{1−j}(Y) → BK_{1−j}^{cc}(\overline{Y} , C; id_{Y}) → lim^{1}K_{1−j}(C; p)_{ϵ} → 0.

They are analogous to the sequences in Siebenmann’s Theorems I and II in [?]. In fact, if Y is locally compact, metric with a metric one point compactification \overline{Y} = Y ∪\{∞\}, and if C = \{∞\} is tame in \overline{Y}, then the connection to Siebenmann’s exact sequences can be made more than an analogy. We shall not pursue this here, in part because we need the extension mentioned in 2) in order to recover Siebenmann’s Theorem II in the correct degree.

4) We recall that Chapman, see Theorems 13.1 and 13.2 of [?], studies the controlled boundary problem for a manifold M^{m} parametrized over a compact metric space B via a map p : M → B. Assuming m ≥ 6, and suitable “movability”and “tameness” conditions, he obtains a solution in terms of a primary obstruction σ_{∞}^{p}(M) ∈ limK_{0}(M − C_{i})_{1/i} and a secondary obstruction τ_{∞}^{p}(M) ∈ lim^{1}W h(M − C_{i})_{1/i}. Here {C_{i}|i = 1, 2, 3 ⋯ } is an expanding and exhaustive sequence of compacta in M. For any space Y with a map p : Y → B, the group W h^{p}(Y)_{ϵ} is defined geometrically in terms of ϵ-controlled (in B) strong deformation retraction pairs, and the group K_{0}^{p}(Y)_{ϵ} is given as the transfer invariant subgroup of W h^{p∘proj}(Y × S^{1})_{ϵ}. We may choose the sequence C_{i} to be of the form C_{i} = ρ^{−1}([0, 1 − 2^{−i}]) for some proper map ρ : M → [0, 1). Now ρ and p together define a map p̄ : M → c^{∘}(B), and we let \overline{M} = M ⨿ B with the teardrop topology. Urysohn’s Metrizability Theorem can be used to show that \overline{M} is metrizable, and presumaby Chapman’s movability and tameness conditions imply that B is tame in \overline{M}. Assuming this and the extension mentioned in 2), it is easy to produce natural maps

limK_{0}(M − C_{i})_{1/i} → K̃_{0}^{cc}(B; p̂)_{c}, (70)

and

lim^{1}W h(M − C_{i})_{1/i} → lim^{1}W h(B; p̂)_{ϵ}, (71)

where χ(\overline{M}, B) = (M ⟵^{q} H ⟶^{p̂} B). We conjecture that these maps are isomorphisms in some reasonable generality. If this is true, then it should be possible to give a direct definiton of a total obstruction ς(M; p) ∈ W h(\overline{M}, B; id_{M}) such that Δ_{1}(ς(M; p)) correponds to σ_{∞}^{p}(M) and such that the homology class represented by ς(M; p) corresponds to τ_{∞}^{p}(M) when σ_{∞}^{p}(M) vanishes. We plan to return to these questions in a later publication. □

Proof: By Proposition ?? and Definition ??, we have a long exact sequence

⋯ ⟶ K_{2−j}(Y) ⟶^{i_{2−j}} K_{2−j}^{cc}(\overline{Y} , C; id_{Y}) ⟶^{Δ̄_{2−j}} K_{2−j}^{cc}(χ_{+}) ⟶^{δ_{2−j}} K_{1−j}(Y) ⟶ · · · .

It is valid for all j ∈ ℤ, but right now we concentrate on j ∈ ℕ. Also, by Theorem ??, the groups K_{2−j}^{cc}(χ_{+}) fit into short exact sequences

0 → lim^{1}K_{2−j}(C; p)_{ϵ} ⟶^{γ′_{2−j}} K_{2−j}^{cc}(χ_{+}) ⟶^{γ′′_{2−j}} K_{1−j}(C; p)_{c} → 0

Since lim^{1}K_{2−j}({c}; c_{Y})_{ϵ} = 0, a naturality argument applied to the morphism

(c_{C}, q, id_{Y}) : χ ⟶ ({c} ⟵^{c_{Y}} Y ⟶^{id_{Y}} Y)

shows that the composition δ_{2−j} ∘ γ′_{2−j} = 0. Hence δ_{2−j} induces a homomorphism ℓ_{1−j} : K_{1−j}(C; p)_{c} → K_{1−j}(Y), and if we take Δ_{2−j} = γ′′_{2−j} ∘ Δ̄_{2−j}, an easy diagram chase establishes the desired chain complex (and the properties claimed), except for the first three terms. To get at these, on needs to show that the homomorphism Δ̄_{2} : K_{2}(χ_{+}) → K_{1}(Y) can be factored via K_{1}(C; p)_{c}. We leave this easy part of the extension mentioned in 2) above, to the reader.

7. Ferry and Pedersen’s vanishing result for lim^{1}K_{2−j}(B; p)_{ϵ}.

Let K be a finite, simplicial complex and let A be an additive category. Some time around 1987, Erik Kjær Pedersen, in a collaboration with Steve Ferry, gave a nice proof of the following result.

Theorem 7.1. (Ferry and Pedersen - unpublished) There exists an ϵ > 0 such that any element of K_{1}(|K|; A) which can be represented by an automorphism of bound ϵ (of some geometric A-object on |K|), can be “squeezed”, i.e. it can be similarly represented with an arbitrarily small bound δ > 0.

Unfortunately, the proof is still unpublished, although there does exist an incomplete preprint, [?].

When Ferry and Pedersen first presented the proof at Odense University, two of the present authors (DRA and HJM) went through it in a variable coefficient version and found that it carries over with little extra trouble. Since the main source is unpublished, here we prefer to present the result as a conjecture, although we do feel it is a very strong such.

Conjecture 7.2. Let |K| be a finite polyhedron, and p : E → B = |K| be a map which has an iterated mapping cylinder stucture w.r.t. to the triangulation K of B. Then the inverse systems {K_{1}(B; p)_{ϵ}}_{ϵ} and {W h(B; p)_{ϵ}}_{ϵ} satisfy the Mittag Leffler condition. Hence the groups lim^{1}K_{2−j}(B; p)_{ϵ} and lim^{1} K̃_{2−j}(B; p)_{ϵ} vanish for each j ∈ ℕ.

We hope that a final version of [?] will eventually be published.

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Douglas R. Anderson Dept. of Math., Syracuse University, Syracuse, NY 13210, USA E-mail address: anderson@euclid.syr.edu

Francis X. Connolly Dept. of Math., University of Notre Dame, Notre Dame, IN 46556, USA E-mail address: francis.x.connolly.1.@nd.edu

Hans J. Munkholm IMADA, Universitet, DK 5230 Odense M, Denmark E-mail address: hjm@imada.ou.dk