# Relative index theorem for Clifford linear Dirac operators

Dear community,

there is relative index theorem due to Gromov and Lawson (Thm. 4.18 in POSITIVE SCALAR CURVATURE AND THE DIRAC OPERATOR ON COMPLETE RIEMANNIAN MANIFOLDS) which states that $$ind_a(D^+_1,D^+_0)=ind_t(D^+_1,D^+_0),$$ where $D_0$ and $D_1$ are Dirac operators on complete Riemannian manifolds $X_0$ and $X_1$ which are strictly positive at infinity and coincide outside a compact set.

My question: Is there a $Cl_n$-version of this statement, i.e. does the theorem hold without further modifications for $Cl_n$-linear Dirac operators (then the equality above takes place in $KO_*$)? Is there a nice reference for the $Cl_n$-version, and/or should one simply study the proof in the paper above because it is more or less obvious how the generalization works.

I don't know the answer to your question, but there is another proof of Gromov and Lawson's relative index theorem based on the assembly map from K-homology to coarse K-theory (I think it is outlined in Roe's "Index Theory, Coarse Geometry, and Topology of Manifolds"), and I'm guessing one could generalize that argument to the $Cl_n$-Dirac operator if one carefully understands how to build its real K-homology class. This might also easily follow from Gromov and Lawson's techniques; I don't understand them as well. –  Paul Siegel Jul 10 '12 at 15:23