ISSN:
1573-0514
Keywords:
Geometric modules
;
algebraicK-homology
;
boundedly controlledK-theory
;
controlledK-theory
;
boundedK-theory
;
boundedly controlled topology
;
controlled topology
;
bounded topology
;
Lipschitz spaces
;
open cone
;
spectral sequence
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Letp: X→Z be a continuous map into a (proper) metric space. Using a variation on the geometric modules of Quinn, we associate top (and any reasonable ringR) an additive category ℊℳ(p, R). Mapsp, as above, are the objects of a category on which ℊℳ(-,R) becomes functorial. By composing with an open cone construction, we get a functor which associates to any topological space over a compact Lipschitz space an additive category. Finally, by using the algebraicK-theory spectrum for an additive category, we arrive at a functor which is our main object of study. We show that it is a homology theory in a suitable sense and we derive an Atiyah-Hirzebruch type spectral sequence for its calculation in many cases, including all triangulated objects. On our way, we show that the ‘bounded’K-theory of Pedersen and Weibel is essentially a special case of the ‘boundedly controlledK-theory’ defined earlier by the authors and we establish a close connection, at least philosophically, between the latter theory and the ‘K-theory with ε-control’ developed by Chapman, Ferry and Quinn.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01054451
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