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