Difference-element-in-K-theory
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 
.
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 |
Difference-element-in-K-theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Difference-element-in-K-theory&oldid=46651