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

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.

MSC:

 55P42 Stable homotopy theory, spectra 19D10 Algebraic $$K$$-theory of spaces 55P91 Equivariant homotopy theory in algebraic topology
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References:

 [1] 10.2140/gt.2018.22.3761 · Zbl 1409.55013 [2] 10.2140/tunis.2020.2.97 · Zbl 1461.18009 [3] 10.1007/978-3-540-38117-4 [4] 10.4310/HHA.2009.v11.n1.a2 · Zbl 1160.55005 [5] 10.1023/A:1007784106877 · Zbl 0921.19003 [6] 10.1016/S0040-9383(03)00029-6 · Zbl 1047.55004 [7] 10.1007/BF02393236 · Zbl 1077.19002 [8] 10.2307/2152801 · Zbl 0798.57018 [9] 10.1007/978-3-0346-0189-4 · Zbl 1195.55001 [10] 10.2140/gt.1999.3.103 · Zbl 0927.57028 [11] 10.1007/BFb0080003 [12] 10.1007/s00208-003-0454-5 · Zbl 1051.19002 [13] 10.1016/j.aim.2011.05.019 · Zbl 1247.14020 [14] 10.1007/BF01228231 · Zbl 0274.55008 [15] 10.1112/jtopol/jtv003 · Zbl 1373.55012 [16] 10.1016/j.aim.2016.11.017 · Zbl 1360.19001 [17] 10.25537/dm.2019v24.815-855 · Zbl 1423.19003 [18] 10.1112/S0024611501012692 · Zbl 1017.55004 [19] 10.1090/surv/132 [20] 10.1016/0022-4049(86)90094-0 · Zbl 0625.57024 [21] 10.2140/agt.2014.14.299 · Zbl 1299.19001 [22] 10.1017/CBO9781107261457 · Zbl 1317.18001 [23] 10.25537/dm.2018v23.1405-1424 · Zbl 1423.14157 [24] 10.1090/conm/019 · Zbl 0508.00006 [25] 10.1007/BFb0074449 [26] 10.2140/gt.1999.3.67 · Zbl 0927.57027 [27] 10.1017/CBO9780511629365.014 [28] 10.1090/conm/258/1778119
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