On the Cobordism Invariance of the Index of Dirac Operators

We describe a “tunneling” proof of the cobordism invariance of the index of Dirac operators. The goal of this note is to present a very short proof of the cobordism invariance of the index. More precisely, if D̂ is a Dirac operator on an odd dimensional manifold M̂ with boundary ∂M̂ = M then we show that the index of its restriction D to M is zero. The novelty of this proof consists in the fact that we provide an explicit isomorphism between the kernel and the cokernel of D. This map can be viewed as a sort of “propagator” (see Sect. 4). 1. The setting Consider the following collection of data. (a) A compact, oriented, (2n + 1)-dimensional Riemann manifold (M̂, ĝ) with boundary ∂M̂ = M such that ĝ is a product metric near the boundary. We denote by s the longitudinal coordinate on a collar neighborhood of M . The various orientations are defined as in Figure 1. (b) A bundle of complex self-adjoint Clifford modules Ê → M̂ (in the sense of [BGV]). The Clifford multiplication is denoted by ĉ : T ∗M̂ → End (Ê).