Hans J. Munkholm

shm Maps of differential algebras, I. A characterization up to homotopy

In this note we study the category DASH,, of differential graded algebras over a field with homotopy classes of shm (strongly homotopy multiplicative) maps as morphisms. Algebraic shm maps were first introduced by Clark [ 11. Stasheff and Halperin [9] indicated how one should be able to use them to obtain collapse results for the Eilenberg Moore spectral sequence for homogeneous spaces. Gugenheim and the present author 131, [6] made a relatively thorough study of the properties of shm maps. The results enabled the present author to carry out, in If;], the program that Stasheff and Halperin had anticipated. Let DA be the category of differential graded algebras with multiplicative differential maps and let i : DA -*DASHh bt;, the obvious functor. In this paper we prove that i is universal among all functors k : DA-+ Ce which satisfy (i) If f E DA(A, B) is a homology isomorphism then k(f) is an isomorphism. (ii) If f = g E DA(A, B) then k(f) = k(g). As usual such a universal characterization makes life a lot easier in many cases. As an example it follows immediately that a’;ry A, B E DA give functors A @ . @I3 such that the diagrams

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.