Homology, Homotopy and Applications

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 $\mathfrak{E} \to M$, 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 $\rm{spin}^c$-bundle over an even-dimensional proper $\rm{spin}^c$-manifold reduces to the classical Thom isomorphism in topological $K$-theory. The techniques involve non-commutative geometric functional analysis.

Article information

Source
Homology Homotopy Appl., Volume 5, Number 1 (2003), 121-159.

Dates
First available in Project Euclid: 13 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.hha/1139839930

Mathematical Reviews number (MathSciNet)
MR1989619

Zentralblatt MATH identifier
1024.19004

Subjects
Primary: 58J42: Noncommutative global analysis, noncommutative residues
Secondary: 19K99: None of the above, but in this section 46L99: None of the above, but in this section 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX}

Citation

Trout, Jody. A Thom isomorphism for infinite rank Euclidean bundles. Homology Homotopy Appl. 5 (2003), no. 1, 121--159. https://projecteuclid.org/euclid.hha/1139839930


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