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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 \begin{equation} ind_a(D^+_1,D^+_0)=ind_t(D^+_1,D^+_0), \end{equation} 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.

Many thanks for any comments in advance.

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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
    
Ulrich Bunke has written a paper "A K-theoretic relative index theorem...", available on his webpage mathematik.uni-regensburg.de/Bunke. He treats the case of an arbitrary (complex or real) C^*-algebra as coeffient algebra. –  Johannes Ebert Apr 25 '13 at 21:03
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