Pseudo-differential projections and the topology of certain spaces of elliptic boundary value problems

We calculate the homotopy groups of the space of elliptic boundary value problems for an elliptic differential operatorA of a first order and of the space of elliptic self-adjoint boundary value problems whenA is a formally self-adjoint. In particular we show that the spectral flow of anS ¹ family of self-adjoint elliptic boundary value problems is well defined. This provides some information on spectral properties along the lines of the Vafa-Witten approach to spectral inequalities.