Algebraic K-theory with continuous control at infinity

Identifying assembly maps in K - and L -theory

“…Given excision, homotopy invariance is equivalent to showing that C * r B G (CX + × [0, 1); C) has trivial K-theory. The proof of this follows the arguments of [ACFP94], see also [HPR97].…”

“…From our point of view, the primary assembly map (see § 7) is constructed from functors on the continuously controlled categories introduced in [ACFP94]. In § 2 we give some basic material on spectra and introduce the main examples of geometric interest.…”

Coarse homology theories

“…In particular, using coarse structures arising from 'continuous control at infinity' (see [1]) we obtain a result on the Novikov conjecture that appears to be new.…”

Cech homology and the Novikov Conjectures for K- and L-Theory

“…We shall denote this category by cX Y bY R. The subcategory of isomorphisms is also symmetric monoidal and has an associated spectrum cX Y bY R will be denoted by KX Y bY R. Note that if b is the bornology associated to a metric, then cX Y bY R coincides with the bounded category cX Y R of [15] applied to the metric space X . If bX Y dX is the continuously controlled bornology associated to a compacti¢cation X of X then cX Y bX Y dX Y R is the continuously controlled category bX Y dX Y R of [2] and [5]. For brevity we shall continue to use the notation bX Y dX Y R for cX Y bX Y dX Y R.…”

“…We shall use m to denote the category whose objects are quadruples X Y dX Y Y S, where X Y dX is a compact Hausdorff pair, where P RCdX , and where S is a finite subset of dX . A morphism from 2 in m so that 1 refines RCf jdX 1 2 , and so that f S 1 S 2 . Any functor F X m 3 spectra determines a functor F X cE 3 spectra on objects via the formula…”