ABSTRACT
Chapter 5 contains a number of i ndex formulas on m anifolds with singularities. They are mainly obtained by the surgery technique. However, to compute the index we require the operators to have some symmetries. Unfortunately, many operators naturally arising in geometry do not possess such symmetries. In this chapter, we discuss an approach to index formulas for such geometric operators. To simplify the analysis, we consider operators on manifolds with infinite cylindrical ends. To obtain index formulas, we first reduce the operator to a boundary value problem with the same index on the finite part of the manifold. This is a so-called spectral Atiyah-Patodi-Singer boundary value problem. Atiyah, Patodi and Singer (1975) computed its index using the heat equation. We obtain stratified index formulas by analyzing the Atiyah-Patodi-Singer index formula.
