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# Homology, Homotopy and Applications

## Volume 5 (2003)

### Number 1

### A Thom isomorphism for infinite rank Euclidean bundles

Pages: 121 – 159

DOI: http://dx.doi.org/10.4310/HHA.2003.v5.n1.a7

#### Author

#### 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 ${C^{\star}}$-algebra associated to a bundle ${\mathfrak E} \to M$, equipped with a compatible connection $\nabla$, which plays the role of the algebra of functions on the infinite dimensional total space ${\mathfrak E}$. If the base $M$ is a point, we obtain the Bott periodicity isomorphism theorem of Higson-Kasparov-Trout [19] 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.

#### Keywords

Thom isomorphism, operator algebras, $K$-theory, vector bundles

#### 2010 Mathematics Subject Classification

19-xx, 46-xx, 47-xx, 55-xx, 57-xx, 58-xx