Topological manifold bundles and the 𝐴-theory assembly map

Raptis, George; Steimle, Wolfgang

doi:10.1090/proc/15014

Topological manifold bundles and the $A$-theory assembly map
HTML articles powered by AMS MathViewer

by George Raptis and Wolfgang Steimle PDF

Proc. Amer. Math. Soc. 148 (2020), 3787-3799 Request permission

Abstract:

We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized $A$-theory characteristic of such a fiber bundle factors canonically through the assembly map of $A$-theory. Furthermore our main result shows a refinement of this statement by providing such a factorization for an extended $A$-theory characteristic, defined on the parametrized topological cobordism category. The proof uses a convenient framework for bivariant theories and recent results of Gomez-Lopez and Kupers on the homotopy type of the topological cobordism category. We conjecture that this lift of the extended $A$-theory characteristic becomes highly connected as the manifold dimension increases.

References

Jean-Michel Bismut and John Lott, Flat vector bundles, direct images and higher real analytic torsion, J. Amer. Math. Soc. 8 (1995), no. 2, 291–363 (English, with English and French summaries). MR 1303026, DOI 10.1090/S0894-0347-1995-1303026-5
Marcel Bökstedt and Ib Madsen, The cobordism category and Waldhausen’s $K$-theory, An alpine expedition through algebraic topology, Contemp. Math., vol. 617, Amer. Math. Soc., Providence, RI, 2014, pp. 39–80. MR 3243393, DOI 10.1090/conm/617/12282
Michael Cole, Mixing model structures, Topology Appl. 153 (2006), no. 7, 1016–1032. MR 2203016, DOI 10.1016/j.topol.2005.02.004
W. Dwyer, M. Weiss, and B. Williams, A parametrized index theorem for the algebraic $K$-theory Euler class, Acta Math. 190 (2003), no. 1, 1–104. MR 1982793, DOI 10.1007/BF02393236
William Fulton and Robert MacPherson, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 31 (1981), no. 243, vi+165. MR 609831, DOI 10.1090/memo/0243
Mauricio Gomez-Lopez and Alexander Kupers, The homotopy type of the topological cobordism category, Preprint (2018), arXiv: https://arxiv.org/abs/1810.05277v2
Hoang Kim Nguyen, On the infinite loop space structure of the cobordism category, Algebr. Geom. Topol. 17 (2017), no. 2, 1021–1040. MR 3623680, DOI 10.2140/agt.2017.17.1021
George Raptis and Wolfgang Steimle, On the map of Bökstedt-Madsen from the cobordism category to $A$-theory, Algebr. Geom. Topol. 14 (2014), no. 1, 299–347. MR 3158761, DOI 10.2140/agt.2014.14.299
George Raptis and Wolfgang Steimle, Parametrized cobordism categories and the Dwyer-Weiss-Williams index theorem, J. Topol. 10 (2017), no. 3, 700–719. MR 3665408, DOI 10.1112/topo.12019
George Raptis and Wolfgang Steimle, A cobordism model for Waldhausen $K$-theory, J. Lond. Math. Soc. (2) 99 (2019), no. 2, 516–534. MR 3939266, DOI 10.1112/jlms.12182
George Raptis and Wolfgang Steimle, On the $h$-cobordism category. I, To appear in Int. Math. Res. Not. IMRN, https://doi.org/10.1093/imrn/rnz329.
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, DOI 10.1007/BFb0074449
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes, Spaces of PL manifolds and categories of simple maps, Annals of Mathematics Studies, vol. 186, Princeton University Press, Princeton, NJ, 2013. MR 3202834, DOI 10.1515/9781400846528
Michael Weiss and Bruce Williams, Assembly, Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, Cambridge, 1995, pp. 332–352. MR 1388318, DOI 10.1017/CBO9780511629365.014
Bruce Williams, Bivariant Riemann Roch theorems, Geometry and topology: Aarhus (1998), Contemp. Math., vol. 258, Amer. Math. Soc., Providence, RI, 2000, pp. 377–393. MR 1778119, DOI 10.1090/conm/258/1778119