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arXiv · 2608.26560

Geometric $K$-homology and operator $K$-theory for Hilbert manifolds

Doman Takata
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Abstract

Poincar\'e duality is a classical theorem relating the homology and cohomology of closed oriented manifolds. This theorem has been extended to more general settings and to generalized (co)homology theories, including $K$-theory. In this paper, we construct an infinite-dimensional analogue of the $K$-theoretic Poincar\'e duality homomorphism. More precisely, for an infinite-dimensional Hilbert manifold $\mathcal{M}$, we construct a homomorphism $$K^{geo}_*(\mathcal{M})\to K_*(\mathcal{A}(\mathcal{M})),$$ where $K^{geo}_*(\mathcal{M})$ denotes the geometric $K$-homology of Baum and Douglas, and $\mathcal{A}(\mathcal{M})$ is a $C^*$-algebra associated to $\mathcal{M}$, based on a construction of Gong, Wu, and Yu. We also prove that the constructed homomorphism is non-trivial in certain cases.

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Doman Takata. 2026-08-27. Geometric $K$-homology and operator $K$-theory for Hilbert manifolds. https://arxiv.org/abs/2608.26560

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