A \(K\)-theoretic relative index theorem and Callias-type Dirac operators
Motivated by the positive scalar curvature problem a relative index theorem for Dirac operators was first proved by Gromov/Lawson (1983). It expresses in topological terms the difference of the \(L^2\)-index of two Dirac operators which live on possibly different complete Riemannian manifolds. The essential assumptions are \(-\) that both operators are positive at infinity and \(-\) that the operators coincide outside of compact sets. In the paper under review the author extends a version of this theorem to operators \(D\) acting on bundles of modules over some \(C^*\)-algebra \(A\). The index \(\text{ind }D\) of such an operator is an element of the \(K\)- theory of \(A\). Let be given two Dirac operators \(D_1\), \(D_2\) on two complete Riemannian manifolds \(M_1\), \(M_2\) which are positive at infinity. Assume that there are compact hypersurfaces \(N_i \subset M_i\), \(i = 1,2\) cutting the \(M_i\) into two pieces and an isomorphism of all structures over a neighbourhood of \(N_1\) to those over a neighbourhood of \(N_2\). Then one can cut at those hypersurfaces and glue the pieces together interchanging the boundary components. One obtains new Dirac operators \(D_3\), \(D_4\). The relative index theorem proved in the paper states: \[ \text{ind } D_1 + \text{ind } D_2 = \text{ind } D_3 + \text{ind } D_4 \in K(A). \] This theorem is also valid for real operators. The above result is applied to show that a homomorphism \[ R_n (\pi) \to \text{KO}_n (C^*_r (\pi)) \] is well defined, where \(\pi\) is a finitely generated group, \(R_n (\pi)\) is the group of \(B \text{Spin} \times B \pi\)-bordisms with distinguished positive scalar curvature metric introduced by Stolz and Hajduk and \(\text{KO}_n (C^*_r (\pi))\) is the \(n\)-th real \(K\)-group of the reduced (real) group-\(C^*\)- algebra of \(\pi\). The above homomorphism is realized by the index of Dirac operators on manifolds with cylindrical ends twisted with flat bundles with fibre \(C^*_r (\pi)\). Another application is the invariance of the real index of a Dirac operator associated with the spin structure twisted with a flat bundle of projective modules over some real \(C^*\)-algebra under cutting and pasting at a compact hypersurface admitting a positive scalar curvature metric. The second part of the paper is concerned with the pairing of the \(K\)- homology class defined by a Dirac operator on a complete Riemannian manifold \(M^n\) with \(K\)-classes coming from the Higson corona space (the boundary of a certain compactification of \(M\)). If the \(K\)-class is given by a matrix function \(F\) on \(M\) which tends to a projection or is unitary at infinity and extends to the Higson corona, an explicit description of a Fredholm operator is given. Its index is the result of the pairing above. It is shown directly that this Fredholm operator represents the Kasparov intersection product between the \(K\)-homology class with compact support of the Dirac operator and the \(K\)-class with compact support which is the image under the boundary map of the class on the Higson corona represented by \(F\). The author also gives another interpretation of the pairing in terms of the relative index pairing (\(n\) even) or the index of a certain Toeplitz type operator (\(n\) odd). The computation of the pairing is reduced to an index (\(n\) odd) or spectral flow problem (\(n\) even) on a suitable compact hypersurface. In the odd-dimensional case this result has been known previously due to Anghel. It is essentially a multiple application of the relative index theorem combined with explicit computations on cylinders. The results just discussed are in turn used to recover the results of Baum/Douglas/Taylor on the boundary map in \(K\)-homology at least rationally. Since a relative index theorem for real Dirac operators is now available the preceding theory is developed partially for real operators too. Here the index is the class of the kernel of the Dirac operator as a module over some Clifford algebra and the index can have values in the torsion group \(\mathbb{Z}_2\). The result is applied to extend previously known (Roe) obstructions against the existence of positive scalar curvature metrics in a given quasiisometry class to cases where the index is an element of a torsion group.
scientific article; zbMATH DE number 840184