A Thom Isomorphism for Infinite Rank Euclidean Bundles

Jody Trout

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 $\cs$-algebra associated to a bundle $\EE \to M$, equipped with a compatible connection $\nabla$, which plays the role of the algebra of functions on the infinite dimensional total space $\EE$. If the base $M$ is a point, we obtain the Bott periodicity isomorphism theorem of Higson-Kasparov-Trout \cite{HKT98} for infinite dimensional Euclidean spaces. The construction applied to an even {\it finite rank} \spinc-bundle over an even-dimensional proper \spinc-manifold reduces to the classical Thom isomorphism in topological $K$-theory. The techniques involve non-commutative geometric functional analysis.

Homology, Homotopy and Applications, Vol. 5(2003), No. 1, pp. 121-159

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