Question

My understanding is Atiyah-Patodi-Singer solved the index theorem for manifold with boundary only for certain types of Dirac operators, correct? There is still no (or no hope to get) uniform theorem for the Dirac operator associated with any Dirac bundle (in the sense of Gromov-Lawson)?

Answer 1

1. The APS theorem works for any Dirac-type operator; see e.g. the excellent monograph by Booss-Wojchiecowski on this topic.

2. More than four decades ago, Boutet de Monvel has described a general set-up for dealing with boundary value problems that mimicks the K-theoretic approach to the index theorem on closed manifolds. For a modern presentation of this point of view I recommend this paper by Melo-Shrohe-Schick arXiv: 1203.5649 and the references therein. It involves some noncommutative geometry because the symbols in the Boutet-de-Monvel calculus of elliptic boundary value problems define elements in the $K$-theory of a noncommutative $C^*$-algebra. In the case of closed manifolds symbols of elliptic operators lead to elements in the $K$-theory of a commutative $C^*$-algebra.

Answer 2

I don't know such a general theorem. The problem is the appropriate formulation of a generalized pseudodifferential elliptic boundary value problem. They have a product metric and a natural notion of boundary of Dirac operator. So the A.P.S condition can be set up. I suggest you to take a look to the corresponding $L^2$ problem on the manifold with a cylinder attached. In that case the A.P.S. formula has been extended by Richard Melrose to the so called $b$--metrics. These are metrics that are product type only at the infinity. In general the $K$ theory of the boundary contains obstructions to set up LOCAL elliptic boundary value problems.

Answer 3

A general setup for an index theorem on manifolds with boundary, has been developed by the French school in non-commutative geometry, see e.g. http://www.math.univ-toulouse.fr/~monthube/articles/CMcrascorrection.pdf and the references therein.