Difference-element-in-K-theory

An element of the group $ K ( X, A) $ (where $ ( X, A) $ is a pair of spaces and $ X $ is usually supposed to be a finite cellular space, while $ A $ is a cellular subspace of it), constructed from a triple $ ( \xi , \eta , \zeta ) $, where $ \xi $ and $ \eta $ are vector bundles of the same dimension over $ X $ and $ \zeta : \xi | _ {A} \rightarrow \eta | _ {A} $ is an isomorphism of vector bundles (here $ \sigma \mid _ {A} $ is the part of the vector bundle $ \sigma $ over $ X $ located above the subspace $ A $). The construction of a difference element can be carried out in the following way. First one supposes that $ \eta $ is the trivial bundle and that some trivialization of $ \eta $ over $ X $ is fixed. Then $ \zeta $ gives a trivialization of $ \xi \mid _ {A} $ and hence gives an element of the group $ \widetilde{K} ( X/A) = K ( X, A) $. This element is independent of the choice of the trivialization of $ \eta $ above all of $ X $. In the general case one chooses a bundle $ \sigma $ over $ X $ such that the bundle $ \eta \oplus \sigma $ is trivial, and the triple $ ( \xi , \eta , \zeta ) $ is assigned the same element as the triple $ ( \xi \oplus \sigma , \eta \oplus \sigma , \zeta \oplus \mathop{\rm id} \sigma ) $.

References

[a1]	M.F. Atiyah, F. Hirzebruch, "Analytic cycles on complex manifolds" Topology , 1 (1961) pp. 28–45
[a2]	M.F. Atiyah, R. Bott, A. Shapiro, "Clifford modules" Topology , 3. Suppl. 1 (1964) pp. 3–38