Cobordism Invariance of the Index of a Transversally Elliptic Operator

1. The SpinC-Dirac operator and the SpinC-quantization In this section we reformulate the “SpinC-quantization commutes with cobordism” principal in a more analytic language. In Subsection 1.1, we briefly recall the notion of SpinC-Dirac operator. We refer the reader to [2, 6] for details. We also express the SpinC-quantization of an orbifold M = X/G in terms of the index of the lift of the SpinC-Dirac operator to X. In Subsection 1.2, we explain the relationship between such lifts associated with cobordant SpinCstructures. This leads us to a notion of a cobordism between transversally elliptic operators. To show that SpinC-quantization of orbifolds commutes with cobordism it is then enough to prove that the index of transversally elliptic operators commutes with cobordisms, which will be shown in the subsequent sections.