Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 8 / 7 / a

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Solution

Fix a separable infinite-dimensional complex Hilbert space

H

and let

U_{K} = {u \in B (H) : u unitary, u - I compact}

with its operator-norm topology. The odd topological K-theory group of a compact metric space is defined by continuous-map homotopy classes

K^{1} (X) = [X, U_{K}] .

Homotopies use the same norm topology. Its finite-dimensional stable realization and group operation are established in the following parts, so this definition does not yet assume that homotopy classes form a group.

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