## Homology, Homotopy and Applications

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

### A Thom isomorphism for infinite rank Euclidean bundles

#### 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