Thom isomorphism and push-forward map in twisted K-theory

Abstract

We establish the Thom isomorphism in twisted K-theory for any real vector bundle and develop the push-forward map in twisted K-theory for any differentiable map�f�:�X�?�Y�(not necessarily K-oriented). We also obtain the wrong way functoriality property for the push-forward map in twisted K-theory. For�D-branes satisfying Freed-Witten's anomaly cancellation condition in a manifold with a non-trivial�B-field, we associate a canonical element in the twisted K-group to get the so-called D-brane charges.