Topological manifold bundles and the $A$-theory assembly map

George Raptis; Wolfgang Steimle

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arXiv · 1905.01868

Topological manifold bundles and the $A$-theory assembly map

George Raptis·Wolfgang Steimle

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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.

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George Raptis, Wolfgang Steimle. 2019-05-06. Topological manifold bundles and the $A$-theory assembly map. https://arxiv.org/abs/1905.01868

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