Topological Aspects of the Equivariant Index Theory of Infinite-Dimensional LT-Manifolds

Abstract

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 "RKK-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 RKK-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 ".