AI 中文总结
本文针对希尔伯特流形，构造了Baum-Douglas几何K-同调到Gong-Wu-Yu型$C^*$-代数的K-理论庞加莱对偶同态，并证明其在特定情形下非平凡，拓展了K-理论庞加莱对偶到无穷维情形。
AI 中文摘要
庞加莱对偶是联系闭定向流形同调与上同调的经典定理，该定理已被推广到更一般的情形及广义（上）同调理论，包括K-理论。本文中，我们构造了K-理论庞加莱对偶同态的无穷维类比：对无穷维希尔伯特流形$\boldsymbol{\textit{M}}$，构造同态$\boldsymbol{K^{geo}_*(\textit{M})\to K_*(\textit{A}(\textit{M}))}$，其中$K^{geo}_*(\textit{M})$指Baum与Douglas提出的几何K-同调，$\textit{A}(\textit{M})$是基于Gong、Wu和Yu构造的、与$\textit{M}$关联的$C^*$-代数。我们还证明，该同态在某些情形下是非平凡的。
英文摘要
Poincaré 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 Poincaré 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.
Comments34 pages
