### ${L}^{2}$-index theorems, KK-theory, and connections.

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In this paper we give a geometric cobordism description of differential integral cohomology. The main motivation to consider this model (for other models see [, , , ]) is that it allows for simple descriptions of both the cup product and the integration. In particular it is very easy to verify the compatibilty of these structures. We proceed in a similar way in the case of differential cobordism as constructed in []. There the starting point was Quillen’s cobordism description of singular cobordism...

Using methods from coarse topology we show that fundamental classes of closed enlargeable manifolds map non-trivially both to the rational homology of their fundamental groups and to the $K$-theory of the corresponding reduced ${C}^{*}$-algebras. Our proofs do not depend on the Baum–Connes conjecture and provide independent confirmation for specific predictions derived from this conjecture.

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