An absolute version of the Gromov-Lawson relative index theorem

Abstract

A Dirac operator on a complete manifold is Fredholm if it is invertible outside a compact set. Assuming a compact group to act on all relevant structure, we express the equivariant index of such a Dirac operator as an Atiyah-Segal-Singer type contribution from inside this compact set, and a contribution from outside this set. Consequences include equivariant versions of the relative index theorem of Gromov and Lawson and the Atiyah-Patodi-Singer index theorem.

Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps

“…We first state an index theorem from [27] for Dirac operators that are invertible outside a compact set, see Theorem 3.1. Then we deduce Theorem 2.16 from this result.…”

“…We review the geometric setting and notation needed to formulate Theorem 2.2 in [27], this is Theorem 3.1 below. The text leading up to Theorem 3.1 is a slight reformulation of corresponding material from [27].…”

“…We review the geometric setting and notation needed to formulate Theorem 2.2 in [27], this is Theorem 3.1 below. The text leading up to Theorem 3.1 is a slight reformulation of corresponding material from [27]. Compared to Theorem 2.16, some assumptions in the main result from [27], Theorem 3.1, are weaker, but there are additional assumptions on D, such as invertibility at infinity.…”

“…In this paper and in [27], we work towards a common framework for studying the three types of index problems mentioned. In [27], we considered a complete Riemannian manifold M , a Clifford module S → M and a Dirac operator D on Γ ∞ (S) that is "invertible at infinity" in the following sense. We assumed that there are a compact subset Z ⊂ M and a b > 0 such that for all s ∈ Γ ∞ c (S) supported outside Z, ∥Ds∥ L 2 ≥ b∥s∥ L 2 .…”

“…We assumed that M has a warped product structure outside Z (the φ-cusps as below, without assumptions on the function φ). Furthermore, we considered an action by a compact group G on M and S, commuting with D, and a group element g ∈ G. The main result in [27] is an expression for the value at g of the equivariant index of such an operator, as an Atiyah-Segal-Singer-type contribution from Z and a contribution from outside Z. This implies an equivariant version of the second index theorem mentioned at the start, and an equivariant version of the first for manifolds with the appropriate warped product form at infinity.…”