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Malkiewich, Cary; Merling, Mona

Coassembly is a homotopy limit map. (English) Zbl 1452.55011

Ann. \(K\)-Theory 5, No. 3, 373-394 (2020).
Let \(G\) be a group that is the realization of a simplicial group and \(X\) a \(G\)-spectrum. The homotopy limit map is defined as the map \(X^{G}\to X^{hG}\) from fixed points to homotopy fixed points. There are several instances where this appears and it is a basic question when this is an equivalence. On the other hand, given a reduced contravariant homotopy functor \(F\) on the comma category over \(BG\) to spectra, the coassembly map is a natural transformation \(F\to F_{\%}\) where \(F_{\%}\) is a suitable final object and has the universal property of approximating \(F\) by a functor that sends homotopy pushouts to homotopy pullbacks. The main theorem of this work is the following:
Theorem A. Let \(G\) be a group as before. Then the coassembly map on the terminal object \(F(BG)\to F_{\%}(BG)\) is equivalent to the homotopy limit map of the \(G\) spectrum \(F(BG)\to F(BG)^{hG}\).
As an application, the authors prove that the homotopy limit map for \(A_{G}^{coarse}(X)\) is isomorphic to the coassembly map for bivariant \(A\)-theory.
Reviewer: Daniel Juan Pineda (Michoacan)

Cited in 1 Document

MSC:

55P42 Stable homotopy theory, spectra
19D10 Algebraic \(K\)-theory of spaces
55P91 Equivariant homotopy theory in algebraic topology

Keywords:

coassembly; \(A\)-theory; equivariant \(A\)-theory; homotopy limit; bivariant \(A\)-theory
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\textit{C. Malkiewich} and \textit{M. Merling}, Ann. \(K\)-Theory 5, No. 3, 373--394 (2020; Zbl 1452.55011)
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