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A Thom Isomorphism for Infinite Rank Euclidean Bundles

Jody Trout
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Abstract

An equivariant Thom isomorphism theorem in operator K-theory is formulated and proven for infinite rank Euclidean vector bundles over finite dimensional Riemannian manifolds. The main ingredient in the argument is the construction of a non-commutative C*-algebra associated to a bundle E -> M, equipped with a compatible connection, which plays the role of the algebra of functions on the infinite dimensional total space E. If the base M is a point, we obtain the Bott periodicity isomorphism theorem of Higson-Kasparov-Trout for infinite dimensional Euclidean spaces. The construction applied to an even (finite rank) spin-c-bundle over an even-dimensional proper spin-c-manifold reduces to the classical Thom isomorphism in topological K-theory. The techniques involve non-commutative geometric functional analysis.

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Jody Trout. 2003-06-02. A Thom Isomorphism for Infinite Rank Euclidean Bundles. https://arxiv.org/abs/math/0306048

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