Topological manifold bundles and the $A$-theory assembly map
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- by George Raptis and Wolfgang Steimle
- Proc. Amer. Math. Soc. 148 (2020), 3787-3799
- DOI: https://doi.org/10.1090/proc/15014
- Published electronically: April 22, 2020
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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
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Bibliographic Information
- George Raptis
- Affiliation: Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
- MR Author ID: 875889
- Email: georgios.raptis@ur.de
- Wolfgang Steimle
- Affiliation: Institut für Mathematik, Universität Augsburg, 86135 Augsburg, Germany
- MR Author ID: 908085
- Email: wolfgang.steimle@math.uni-augsburg.de
- Received by editor(s): August 2, 2019
- Received by editor(s) in revised form: January 17, 2020
- Published electronically: April 22, 2020
- Additional Notes: The first author was supported by the SFB 1085 – Higher Invariants (University of Regensburg) funded by the DFG.
The second author was partially supported by the SPP 2026 – Geometry at infinity funded by the DFG - Communicated by: Mark Behrens
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3787-3799
- MSC (2010): Primary 19D10, 57R90
- DOI: https://doi.org/10.1090/proc/15014
- MathSciNet review: 4127825