Assembly maps for topological cyclic homology of group algebras. (English) Zbl 1428.19002
Let \(\mathbb{A}\) be a connective ring spectrum and \(G\) a discrete group. The authors study topological cyclic homology of the group algebra
\(\mathbb{A}[G]\) at a prime \(p\): \(\mathbf{TC}(\mathbb{A}[G];p)\). The main tool is the analysis of assembly maps
\[\mathbf{A}_{\mathcal{F}}: EG(\mathcal{F})_{+}\wedge _{OrG}\mathbf{TC}(\mathbb{A}[G\smallint{-}];p)\to \mathbf{TC}(\mathbb{A}[G];p),\] where \(\mathcal{F}\) denotes a family of subgroups of \(G\),
\(EG(\mathcal{F})_{+}\) is a model for the universal space for \(G\)-actions with isotropy in \(\mathcal{F}\) and the left hand side may be
interpreted as the homotopy colimit of \(\mathbf{TC}(\mathbb{A}[H];p)\) with \(H\in\mathcal{F}\).
The main results are as follows: the assembly map \(\mathbf{A}_{\mathcal{F}}\) induces an isomorphism in homotopy groups if \(G\) is a finite group and \(\mathcal{F}\) is the family of cyclic subgroups, it also induces an isomorphism of homotopy groups of pro-spectra for general \(G\) and the family of cyclic subgroups. They also prove that the assembly map is injective in the following cases: for the family of finite groups and the existence of a universal space \(EG(Fin)\) of finite type or the family of virtually cyclic subgroups and \(G\) hyperbolic or virtually finitely generated abelian. They also prove that \(\mathbf{A}_{\mathcal{F}}\) fails to induce surjectivity on homotopy groups for the family of virtually cyclic subgroups for some groups.
The main results are as follows: the assembly map \(\mathbf{A}_{\mathcal{F}}\) induces an isomorphism in homotopy groups if \(G\) is a finite group and \(\mathcal{F}\) is the family of cyclic subgroups, it also induces an isomorphism of homotopy groups of pro-spectra for general \(G\) and the family of cyclic subgroups. They also prove that the assembly map is injective in the following cases: for the family of finite groups and the existence of a universal space \(EG(Fin)\) of finite type or the family of virtually cyclic subgroups and \(G\) hyperbolic or virtually finitely generated abelian. They also prove that \(\mathbf{A}_{\mathcal{F}}\) fails to induce surjectivity on homotopy groups for the family of virtually cyclic subgroups for some groups.
Reviewer: Daniel Juan Pineda (Michoacan)
MSC:
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 19D50 | Computations of higher \(K\)-theory of rings |
| 55N15 | Topological \(K\)-theory |
| 55P42 | Stable homotopy theory, spectra |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 20J05 | Homological methods in group theory |
References:
| [1] | V. Angeltveit, On the K-theory of truncated polynomial rings in non-commuting variables, Bull. Lond. Math. Soc. 47 (2015), no. 5, 731-742.; Angeltveit, V., On the K-theory of truncated polynomial rings in non-commuting variables, Bull. Lond. Math. Soc., 47, 5, 731-742 (2015) · Zbl 1329.19003 |
| [2] | V. Angeltveit, T. Gerhardt, M. A. Hill and A. Lindenstrauss, On the algebraic K-theory of truncated polynomial algebras in several variables, J. K-Theory 13 (2014), no. 1, 57-81.; Angeltveit, V.; Gerhardt, T.; Hill, M. A.; Lindenstrauss, A., On the algebraic K-theory of truncated polynomial algebras in several variables, J. K-Theory, 13, 1, 57-81 (2014) · Zbl 1317.19006 |
| [3] | C. Ausoni and J. Rognes, Algebraic K-theory of topological K-theory, Acta Math. 188 (2002), no. 1, 1-39.; Ausoni, C.; Rognes, J., Algebraic K-theory of topological K-theory, Acta Math., 188, 1, 1-39 (2002) · Zbl 1019.18008 |
| [4] | C. Ausoni and J. Rognes, Algebraic K-theory of the first Morava K-theory, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1041-1079.; Ausoni, C.; Rognes, J., Algebraic K-theory of the first Morava K-theory, J. Eur. Math. Soc. (JEMS), 14, 4, 1041-1079 (2012) · Zbl 1253.19001 |
| [5] | B. Bhatt, M. Morrow and P. Scholze, Integral p-adic Hodge theory, preprint (2016), .; <element-citation publication-type=”other“> Bhatt, B.Morrow, M.Scholze, P.Integral p-adic Hodge theoryPreprint2016 <ext-link ext-link-type=”uri“ xlink.href=”>https://arxiv.org/abs/1602.03148 · Zbl 1349.14070 |
| [6] | A. J. Blumberg, D. Gepner and G. Tabuada, Uniqueness of the multiplicative cyclotomic trace, Adv. Math. 260 (2014), 191-232.; Blumberg, A. J.; Gepner, D.; Tabuada, G., Uniqueness of the multiplicative cyclotomic trace, Adv. Math., 260, 191-232 (2014) · Zbl 1297.19002 |
| [7] | A. J. Blumberg and M. A. Mandell, Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol. 16 (2012), no. 2, 1053-1120.; Blumberg, A. J.; Mandell, M. A., Localization theorems in topological Hochschild homology and topological cyclic homology, Geom. Topol., 16, 2, 1053-1120 (2012) · Zbl 1282.19004 |
| [8] | A. J. Blumberg and M. A. Mandell, The homotopy theory of cyclotomic spectra, Geom. Topol. 19 (2015), no. 6, 3105-3147.; Blumberg, A. J.; Mandell, M. A., The homotopy theory of cyclotomic spectra, Geom. Topol., 19, 6, 3105-3147 (2015) · Zbl 1332.19003 |
| [9] | M. Bökstedt, W. C. Hsiang and I. Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), no. 3, 465-539.; Bökstedt, M.; Hsiang, W. C.; Madsen, I., The cyclotomic trace and algebraic K-theory of spaces, Invent. Math., 111, 3, 465-539 (1993) · Zbl 0804.55004 |
| [10] | A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer, Berlin 1972.; Bousfield, A. K.; Kan, D. M., Homotopy limits, completions and localizations (1972) · Zbl 0259.55004 |
| [11] | J. F. Davis and W. Lück, Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory, K-Theory 15 (1998), no. 3, 201-252.; Davis, J. F.; Lück, W., Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory, K-Theory, 15, 3, 201-252 (1998) · Zbl 0921.19003 |
| [12] | B. I. Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 223-242.; Dundas, B. I., Relative K-theory and topological cyclic homology, Acta Math., 179, 2, 223-242 (1997) · Zbl 0913.19002 |
| [13] | B. I. Dundas, T. G. Goodwillie and R. McCarthy, The local structure of algebraic K-theory, Algebr. Appl. 18, Springer, London 2013.; Dundas, B. I.; Goodwillie, T. G.; McCarthy, R., The local structure of algebraic K-theory (2013) · Zbl 1272.55002 |
| [14] | F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993), no. 2, 249-297.; Farrell, F. T.; Jones, L. E., Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc., 6, 2, 249-297 (1993) · Zbl 0798.57018 |
| [15] | T. Geisser and L. Hesselholt, Bi-relative algebraic K-theory and topological cyclic homology, Invent. Math. 166 (2006), no. 2, 359-395.; Geisser, T.; Hesselholt, L., Bi-relative algebraic K-theory and topological cyclic homology, Invent. Math., 166, 2, 359-395 (2006) · Zbl 1107.19002 |
| [16] | R. Geoghegan and M. Varisco, On Thompson’s group T and algebraic K-theory, preprint (2015), ; to appear in Geometric and cohomological group theory, London Math. Soc. Lecture Note Ser., Cambridge University Press.; <element-citation publication-type=”other“> Geoghegan, R.Varisco, M.On Thompson’s group T and algebraic K-theoryPreprint2015 <ext-link ext-link-type=”uri“ xlink.href=”>https://arxiv.org/abs/1401.0357; to appear in Geometric and cohomological group theory, London Math. Soc. Lecture Note Ser., Cambridge University Press |
| [17] | M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York (1987), 75-263.; Gromov, M., Hyperbolic groups, Essays in group theory, 75-263 (1987) · Zbl 0634.20015 |
| [18] | L. Hesselholt, On the p-typical curves in Quillen’s K-theory, Acta Math. 177 (1996), no. 1, 1-53.; Hesselholt, L., On the p-typical curves in Quillen’s K-theory, Acta Math., 177, 1, 1-53 (1996) · Zbl 0892.19003 |
| [19] | L. Hesselholt, Witt vectors of non-commutative rings and topological cyclic homology, Acta Math. 178 (1997), no. 1, 109-141.; Hesselholt, L., Witt vectors of non-commutative rings and topological cyclic homology, Acta Math., 178, 1, 109-141 (1997) · Zbl 0894.19001 |
| [20] | L. Hesselholt, K-theory of truncated polynomial algebras, Handbook of K-theory. Vol. 1, Springer, Berlin (2005), 71-110.; Hesselholt, L., K-theory of truncated polynomial algebras, Handbook of K-theory. Vol. 1, 71-110 (2005) · Zbl 1102.19002 |
| [21] | L. Hesselholt, On the Whitehead spectrum of the circle, Algebraic topology, Abel Symp. 4, Springer, Berlin (2009), 131-184.; Hesselholt, L., On the Whitehead spectrum of the circle, Algebraic topology, 131-184 (2009) · Zbl 1196.19009 |
| [22] | L. Hesselholt and I. Madsen, On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), no. 1, 29-101.; Hesselholt, L.; Madsen, I., On the K-theory of finite algebras over Witt vectors of perfect fields, Topology, 36, 1, 29-101 (1997) · Zbl 0866.55002 |
| [23] | L. Hesselholt and I. Madsen, On the K-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1-113.; Hesselholt, L.; Madsen, I., On the K-theory of local fields, Ann. of Math. (2), 158, 1, 1-113 (2003) · Zbl 1033.19002 |
| [24] | L. Hesselholt and I. Madsen, On the de Rham-Witt complex in mixed characteristic, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), no. 1, 1-43.; Hesselholt, L.; Madsen, I., On the de Rham-Witt complex in mixed characteristic, Ann. Sci. Éc. Norm. Supér. (4), 37, 1, 1-43 (2004) · Zbl 1062.19003 |
| [25] | D. Juan-Pineda and I. J. Leary, On classifying spaces for the family of virtually cyclic subgroups, Recent developments in algebraic topology, Contemp. Math. 407, American Mathematical Society, Providence (2006), 135-145.; Juan-Pineda, D.; Leary, I. J., On classifying spaces for the family of virtually cyclic subgroups, Recent developments in algebraic topology, 135-145 (2006) · Zbl 1107.19001 |
| [26] | W. Lück, The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra 149 (2000), no. 2, 177-203.; Lück, W., The type of the classifying space for a family of subgroups, J. Pure Appl. Algebra, 149, 2, 177-203 (2000) · Zbl 0955.55009 |
| [27] | W. Lück, Survey on classifying spaces for families of subgroups, Infinite groups: Geometric, combinatorial and dynamical aspects, Progr. Math. 248, Birkhäuser, Basel (2005), 269-322.; Lück, W., Survey on classifying spaces for families of subgroups, Infinite groups: Geometric, combinatorial and dynamical aspects, 269-322 (2005) · Zbl 1117.55013 |
| [28] | W. Lück, K- and L-theory of group rings, Proceedings of the International Congress of Mathematicians. Vol. II, Hindustan Book, New Delhi (2010), 1071-1098.; Lück, W., K- and L-theory of group rings, Proceedings of the International Congress of Mathematicians. Vol. II, 1071-1098 (2010) · Zbl 1231.18008 |
| [29] | W. Lück and H. Reich, The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory, Handbook of K-theory. Vol. 2, Springer, Berlin (2005), 703-842.; Lück, W.; Reich, H., The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory, Handbook of K-theory. Vol. 2, 703-842 (2005) · Zbl 1120.19001 |
| [30] | W. Lück, H. Reich, J. Rognes and M. Varisco, Algebraic K-theory of group rings and the cyclotomic trace map, Adv. Math. 304 (2017), 930-1020.; Lück, W.; Reich, H.; Rognes, J.; Varisco, M., Algebraic K-theory of group rings and the cyclotomic trace map, Adv. Math., 304, 930-1020 (2017) · Zbl 1357.19002 |
| [31] | W. Lück, H. Reich and M. Varisco, Commuting homotopy limits and smash products, K-Theory 30 (2003), no. 2, 137-165.; Lück, W.; Reich, H.; Varisco, M., Commuting homotopy limits and smash products, K-Theory, 30, 2, 137-165 (2003) · Zbl 1053.55004 |
| [32] | W. Lück and D. Rosenthal, On the K- and L-theory of hyperbolic and virtually finitely generated abelian groups, Forum Math. 26 (2014), no. 5, 1565-1609.; Lück, W.; Rosenthal, D., On the K- and L-theory of hyperbolic and virtually finitely generated abelian groups, Forum Math., 26, 5, 1565-1609 (2014) · Zbl 1329.19006 |
| [33] | W. Lück and M. Weiermann, On the classifying space of the family of virtually cyclic subgroups, Pure Appl. Math. Q. 8 (2012), no. 2, 497-555.; Lück, W.; Weiermann, M., On the classifying space of the family of virtually cyclic subgroups, Pure Appl. Math. Q., 8, 2, 497-555 (2012) · Zbl 1258.55011 |
| [34] | R. McCarthy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), no. 2, 197-222.; McCarthy, R., Relative algebraic K-theory and topological cyclic homology, Acta Math., 179, 2, 197-222 (1997) · Zbl 0913.19001 |
| [35] | D. Meintrup and T. Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math. 8 (2002), 1-7.; Meintrup, D.; Schick, T., A model for the universal space for proper actions of a hyperbolic group, New York J. Math., 8, 1-7 (2002) · Zbl 0990.20027 |
| [36] | G. Mislin, Classifying spaces for proper actions of mapping class groups, Münster J. Math. 3 (2010), 263-272.; Mislin, G., Classifying spaces for proper actions of mapping class groups, Münster J. Math., 3, 263-272 (2010) · Zbl 1223.57017 |
| [37] | T. von Puttkamer and X. Wu, Linear groups, conjugacy growth, and classifying spaces for families of subgroups, preprint (2017), .; <element-citation publication-type=”other“> von Puttkamer, T.Wu, X.Linear groups, conjugacy growth, and classifying spaces for families of subgroupsPreprint2017 <ext-link ext-link-type=”uri“ xlink.href=”>https://arxiv.org/abs/1704.05304 · Zbl 1515.20237 |
| [38] | F. Quinn, Algebraic K-theory over virtually abelian groups, J. Pure Appl. Algebra 216 (2012), no. 1, 170-183.; Quinn, F., Algebraic K-theory over virtually abelian groups, J. Pure Appl. Algebra, 216, 1, 170-183 (2012) · Zbl 1235.19001 |
| [39] | D. A. Ramras, A note on orbit categories, classifying spaces, and generalized homotopy fixed points, preprint (2017), ; to appear in J. Homotopy Relat. Struct.; <element-citation publication-type=”other“> Ramras, D. A.A note on orbit categories, classifying spaces, and generalized homotopy fixed pointsPreprint2017 <ext-link ext-link-type=”uri“ xlink.href=”>https://arxiv.org/abs/1507.06112; to appear in J. Homotopy Relat. Struct. |
| [40] | H. Reich and M. Varisco, On the Adams isomorphism for equivariant orthogonal spectra, Algebr. Geom. Topol. 16 (2016), no. 3, 1493-1566.; Reich, H.; Varisco, M., On the Adams isomorphism for equivariant orthogonal spectra, Algebr. Geom. Topol., 16, 3, 1493-1566 (2016) · Zbl 1350.55015 |
| [41] | H. Reich and M. Varisco, Algebraic K-theory, assembly maps, controlled algebra, and trace methods, preprint (2017), ; to appear in Space – time – matter. Analytic and geometric structures, De Gruyter, Berlin.; <element-citation publication-type=”other“> Reich, H.Varisco, M.Algebraic K-theory, assembly maps, controlled algebra, and trace methodsPreprint2017 <ext-link ext-link-type=”uri“ xlink.href=”>https://arxiv.org/abs/1702.02218; to appear in Space – time – matter. Analytic and geometric structures, De Gruyter, Berlin · Zbl 1409.19001 |
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.
