# 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 ) $.

#### Comments

#### 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 |

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Difference-element-in-K-theory.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Difference-element-in-K-theory&oldid=52381