×

Excision in equivariant fibred \(G\)-theory. (English) Zbl 1483.19002

The work established in this article is motivated by proving the Borel Conjecture for a group \(\Gamma\), which states that the \(K\)-theory assembly map \(B\Gamma_+ \wedge K(Z) \to K(Z[\Gamma])\) is an equivalence of spectra. This assembly map indeed is just one of a whole family of assembly maps which are conjectured to be equivalences of spectra. In quite a few of the cases the so-called technique of bounded controlled algebra was successfully used to prove the corresponding conjecture. Assembly maps in general arise as the universal approximation of a homotopy invariant functor by a homological functor. In the case at hand the homotopy invariant functor is \(K(Z[\pi_1(X)])\) for a pointed space \(X\), and its homological approximation is \(X_+\wedge K(Z)\). The validity of the Borel conjecture would suggest that objects which are used to describe the assembly map should satisfy certain excision properties. This is precisely the approach to the Borel conjecture which is persued by the authors in the present article.
As it turns out, the desired excision properties in the case at hand more easily can be derived if one switches from algebraic \(K\)-theory to algebraic \(G\)-theory which is built from the category of all finitely presented modules of a ring rather than from the category of all finitely presented projective modules of the given ring. The main result of the present article essentially states that a particular bounded controlled version of equivariant \(G\)-theory satisfies the natural excision properties one would hope for, provided the group \(\Gamma\) is finitely generated and satisfies some natural assumptions. These excision properties then can be used to establish various important steps leading to a proof of the Borel conjecture for the given group, if the \(K\)- and the \(G\)-theory of the group ring \(Z[\Gamma]\) agree. The authors claim that the latter holds in quite a number of interesting cases and refer for the discussion of this property to a future paper.
As in the case of bounded controlled equivariant \(K\)-theory also the bounded controlled equivariant \(G\)-theory that is analyzed in the article is built by applying the non-connective algebraic \(K\)-theory construction to specific categories. In technical terms the categories which are used in the article arise as the relative homotopy fixed points of certain bounded controlled \(G\)-theory categories that can be associated to metric spaces on which the given group \(\Gamma\) acts by bounded coarse equivalences. The corresponding bounded controlled \(G\)-theory categories have been defined and investigated in [G. Carlsson and B. Goldfarb, J. Pure Appl. Algebra 223, No. 12, 5360–5395 (2019; Zbl 1498.19001)], while the basic underlying concept for the formal construction of taking the relative homotopy fixed points of a category is taken from M. Merling [Math. Z. 285, No. 3–4, 1205–1248 (2017; Zbl 1365.19007)].

MSC:

19D50 Computations of higher \(K\)-theory of rings
19L47 Equivariant \(K\)-theory
55P91 Equivariant homotopy theory in algebraic topology
55R91 Equivariant fiber spaces and bundles in algebraic topology
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 10.1023/A:1007726201728 · Zbl 0903.18005
[2] 10.1017/CBO9780511629365.004
[3] 10.1007/978-3-540-27855-9_1
[4] 10.1007/s00222-003-0356-x · Zbl 1071.19003
[5] 10.1016/j.jalgebra.2004.02.006 · Zbl 1057.22013
[6] ; Carlsson, J. Homotopy Relat. Struct., 6, 119 (2011) · Zbl 1278.19002
[7] 10.1142/S0218196716500181 · Zbl 1354.19001
[8] 10.1016/j.jpaa.2019.04.003 · Zbl 1498.19001
[9] 10.1016/0040-9383(94)00033-H · Zbl 0838.55004
[10] 10.7146/math.scand.a-13823 · Zbl 0936.19003
[11] 10.2140/agt.2017.17.2565 · Zbl 1383.55013
[12] ; Kelly, Basic concepts of enriched category theory. London Math. Soc. Lecture Note Ser., 64 (1982) · Zbl 0478.18005
[13] ; Malkiewich, Doc. Math., 24, 815 (2019) · Zbl 1423.19003
[14] 10.1090/surv/132
[15] 10.1007/s00209-016-1745-3 · Zbl 1365.19007
[16] 10.1016/0021-8693(84)90184-4 · Zbl 0545.18003
[17] 10.1007/BFb0074443
[18] 10.1515/crelle-2012-0112 · Zbl 1306.18005
[19] 10.1112/plms/s3-40.2.193 · Zbl 0471.57011
[20] 10.1016/j.top.2004.01.005 · Zbl 1059.18007
[21] ; Thomason, Proceedings of the Northwestern Homotopy Theory Conference. Contemp. Math., 19, 407 (1983) · Zbl 0528.55008
[22] 10.1007/978-0-8176-4576-2_10
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.