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An element of the group (where is a pair of spaces and is usually supposed to be a finite cellular space, while is a cellular subspace of it), constructed from a triple , where and are vector bundles of the same dimension over and is an isomorphism of vector bundles (here is the part of the vector bundle over located above the subspace ). The construction of a difference element can be carried out in the following way. First one supposes that is the trivial bundle and that some trivialization of over is fixed. Then gives a trivialization of and hence gives an element of the group . This element is independent of the choice of the trivialization of above all of . In the general case one chooses a bundle over such that the bundle is trivial, and the triple is assigned the same element as the triple .



[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
How to Cite This Entry:
Difference-element-in-K-theory. Yu.B. Rudyak (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098