论文标题
无限维lt-manifolds的拓扑方面
Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds
论文作者
论文摘要
令$ t $为圆形组，让$ lt $为其循环组。我们为适当的$ lt $空间制定并调查了$ lt $ equivariant索引理论的几个拓扑方面，其中适当的$ lt $ - 空格是无限二维流形，配备了“适当的cocoCompact” $ lt $ actions。具体而言，我们介绍了“ $ \ nathcal {r} kk $ - 无限二维流形的理论”，通过使用它，我们制定了$ kk $ kk $ theoritypheoticalpoincaréDualityduality atmormorphism的无限二维版本，以及$ \ nthere $ \ nthile country y Mathcal countral countral countral countral country thementhe undery country的ckk kk kk k。适当的$ lt $ - 空格。 庞加莱偶性同态的左侧是由瓜莉安格·尤（Guoliang Yu）引入的“ $ c^*$ - 希尔伯特流形的代数”提出的。因此，本文的结果表明，这种构建带有希尔伯特歧管的一些拓扑信息。为了以经典的方式制定装配图，我们需要交叉产品，这需要一个不变的量度。但是，有一个替代公式可以使用广义的定点代数来定义它们。我们将采用它作为“ $ LT $交叉产品”的定义。
Let $T$ be the circle group and let $LT$ be its loop group. We formulate and investigate several topological aspects of the $LT$-equivariant index theory for proper $LT$-spaces, where proper $LT$-spaces are infinite-dimensional manifolds equipped with "proper cocompact" $LT$-actions. Concretely, we introduce "$\mathcal{R}KK$-theory for infinite-dimensional manifolds", and by using it, we formulate an infinite-dimensional version of the $KK$-theoretical Poincaré duality homomorphism, and an infinite-dimensional version of the $\mathcal{R}KK$-theory counterpart of the assembly map, for proper $LT$-spaces. The left hand side of the Poincaré duality homomorphism is formulated by the "$C^*$-algebra of a Hilbert manifold" introduced by Guoliang Yu. Thus, the result of this paper suggests that this construction carries some topological information of Hilbert manifolds. In order to formulate the assembly map in a classical way, we need crossed products, which require an invariant measure of a group. However, there is an alternative formula to define them using generalized fixed-point algebras. We will adopt it as the definition of "crossed products by $LT$".
