Friday, 13 April 2007

dg.differential geometry - virtual bundle with compact support

A virtual bundle with compact support is a triple E={E1,E2,a},
where E1 and E2 are bundles over M of the same dimension, M is a closed manifold, a is a bundle map from E1 to E2, which is an isomorphism of bundles over MsmallsetminusX, where X is a compact set in M.



Let nabla1 be a connection on E1, nabla2 a connection on E2. If X is finite number of points in M, and Ch(E) is defined by texttr(exp(nabla1)2)texttr(exp(nabla2)2), we will find Ch(E) integrated on M is an integer multiple of a power of 2pii.



I want to know, if X is a compact submanifold, then how to get the value of the integration? Maybe we can't get the value but how can we analyze the information about X revealed by the integration?

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