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.