Atiyah-Patodi-Singer type index theorems for manifolds with splitting of \(\eta\)-invariants (Q5956638)

The author develops a new approach to index theory on manifolds with corners. The approach is based on a geometric construction that extends the original manifold with corners to a manifold with smooth boundary and wedge singularities (arising from the corners). Under natural general conditions the author shows how to construct self-adjoint extensions of Dirac operators on manifolds with smooth boundary and wedge singularities. These self-adjoint Dirac operators determine self-adjoint extensions of the Dirac operators on manifolds with corners. When the Dirac operator is degree-one with respect to a grading (for example, the full Dirac operator on an even-dimensional manifold), then the self-adjoint operator consists of a Dirac operator \(D_{+}\) and its adjoint in the off-diagonal blocks. The author uses heat equation techniques to prove an index formula for \(D_{+}\). One motivation for the techniques introduced in this paper is the techniques' applicability to corners of codimension higher than two, which will be addressed in a later paper. The paper under review also discusses: another approach to forming self-adjoint extensions on manifolds with corners; analysis of terms in the index formula for some important geometric operators; and a splitting formula for eta invariants. Other recent results in index theory on manifolds with corners appear in \textit{A. Hassell}, \textit{R. Mazzeo}, and \textit{R. Melrose} [Topology 36, No. 5, 1055-1075 (1997; Zbl 0883.58038)] and \textit{W. Müller} [J. Differ. Geom. 44, No. 1, 97-177 (1996; Zbl 0881.58071)].