Douglas R. Anderson

Algebraic K-theory with continuous control at infinity

Let (E, Σ) be a pair of spaces consisting of a compact Hausdorff space Ē and a closed subspace Σ. Let U be an additive category. This paper introduces the category B(E, Σ; U of geometric modules over E with coefficients in U and with continuous control at infinity. One of the main results is to show that the functor that sends a CW complex X to the algebraic K-theory of B(cX, X; U) is a homology theory. Here cX is the closed cone on X and X is its base. The categories B(E, Σ; U) are generalizations of the categories C(Z; U) of geometric modules and bounded morphisms introduced by Pedersen and Weibel [8]. Here (Z, ϱ) is a complete metric space. If X is a finite CW complex and O(X) is the metric space open cone on X considered in [9], then there is an inclusion of categories C(O(X); U)→B(cX, X; U). A second main result is that this inclusion induces an isomorphism on K-theory. One advantage of the present approach is that B(E, Σ; U) depends only on the topology of (E, Σ) and not on any metric properties. This should make application of these ideas to problems involving stratified spaces, for example, more direct and natural.

A comparison of continuously controlled and controlled K-theory

Abstract We define an unreduced version of the e-controlled lower k -theoretic groups of Ranicki and Yamasaki (1995) and Quinn (1985). We show that the reduced versions of our groups coincide (in the inverse limit and its first derived, lim 1 ) with those of Ranicki and Yamasaki. We also relate the controlled groups to the continuously controlled groups of Anderson and Munkholm (1994), and to the Quinn homology groups of Quinn (1982).

Continuously controlled K-theory with variable coefficients

Abstract The purpose of this paper is to use geometric modules and path matrix morphisms of construct a continuously controlled K-theory with variable coefficients. The theory constructed here can be thought of as a “pushout” of the boundedly controlled K-theory with variable coefficients constructed in D.R. Anderson, H.J. Munkholm (Geometric modules and algebraic K -Homology theory, K -Theory 3 (1990) 561–602) and the continuously controlled K-theory with constant coefficients constructed in D.R. Anderson, F.X. Connolly, S. Ferry, E.K. Pedersen (Algebraic K -theory with continuous control at infinity, J. Pure Appl. Algebra 94 (1994) 25–47) over the boundedly controlled K-theory with constant coefficients constructed (E.K. Pedersen, C. Weibel, K -Theory Homology of Spaces, Lecture Notes in Mathematics, vol. 1370, Springer, Berlin, New York, 1989, pp. 346–361). This theory should be directly and easily applicable in the study of stratified spaces. This paper also relates the theory constructed here to the controlled K-theory constructed in D.R. Anderson, F.X. Connolly, H.J. Munkholm (A comparison of continuously controlled and controlled K -theory, Topology Appl. 71 (1996) 9–46) as an inverse limit of e -controlled K-groups and shows that, under suitable conditions, the controlled K-theory of D.R. Anderson, F.X. Connolly, H.J. Munkholm (A comparison of continuously controlled and controlled K -theory, Topology Appl. 71 (1996) 9–46) is a Quinn homology theory.

The Boundedly Controlled Whitehead Theorem

This note contains a version of the Whitehead Theorem for boundedly controlled maps of CW complexes that is often useful in applications and complements the Whitehead Theorem in our book Boundedly controlled topology (Lecture Notes in Math., vol. 1323, Springer-Verlag, 1988). We also include a version of the Whitehead Theorem valid for simply connected boundedly controlled CW complexes.

Steenrod's Equivariant Moore Space Problem for Cohomologically Trivial Modules

Throughout the paper G will denote a finite group and, unless otherwise stated, any module considered will be assumed to be a finitely generated Z G module. Let G be a group, A be a ZG module, and n>0. We say that the G-space X is an equivariant Moore space of type (A,n), i f Ho(X)=Z, Hn(X ) is ZG isomorphic to A, and HI(X )=0 for i4=0, n. The following question was posed by Steenrod: Given A, does there exist a finite G-CW complex (i.e. a CW complex on which G acts cellularly) of type (A, n) for some n? If such an X exists, A of course, must be finitely generated. Various authors have considered this problem. In [10], Swan gave examples showing that in general finite G-CW complexes of type (A, n) do not exist. In [1] and [2], Arnold showed that if G is cyclic of prime power order, finite dimensional, but infinite, G-CW complexes of type (A, n) always exist and he described an obstruction to obtaining a finite G-CW complex of type (A, n). Finally in [41, Carlsson showed that if G is a non-cyclic abelian group, there exist ZG modules A for which no space of type (A,n) exists. In this paper we investigate Steenrods problem for finitely generated cohomologically trivial modules; and, henceforth, we assume all modules are finitely generated ZG modules. We recall that Rim [-7; Theorem4.12] has shown that A is cohomologically trivial if and only if there is a short exact sequence 0 ~ P 1 -*Po-+A ~ 0 where P0 and P1 are projective. In particular, the category NG of projective modules is a subcategory of the category CgYG of cohomologically trivial modules and every object in (gYG is a quotient of a monomorphism in ~G. Furthermore, the inclusion of categories ~GcCgJ-G induces an isomorphism k,: KoZG--*GoCg3-G of Grothendieck groups. (The relations in these groups arise from short exact sequences.) We identify these groups under this isomorphism. If AcCgYG, then [A] will denote the class of A in either KoZG or I(oZG.