Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The G-signature Theorem on Witt spaces

Markus Banagl, Eric Leichtnam, Paolo Piazza

TL;DR

The paper extends the Atiyah–Segal–Singer G-signature formula to Witt G-pseudomanifolds by defining the equivariant signature via intersection cohomology and analyzing the signature operator in the wedge metric setting. It develops an analytic KK-theory framework, proving that the fixed-point data of X^g contributes via a localization formula to Sign(g,X) under a strong normal-non-singularity hypothesis for the fixed-set inclusion, with the L-classes of the fixed components and the characteristic data of the normal bundle driving the formula. Central to the approach are the equivariant K-homology class [D^sign_g], Gysin maps in the equivariant setting, and the localized Chern character machinery of Puschnigg, which together yield a fixed-point expression that specializes to a product of complex characteristic factors when the normal bundle splits into line bundles. The results provide concrete transversality criteria yielding wide classes of Witt G-pseudomanifolds to which the formula applies and fortify the bridge between singular spaces, KK-theory, and index theory. Overall, this work broadens the reach of equivariant index theory to singular spaces with group actions, with implications for geometry and representation theory in contexts where intersection homology and stratified spaces are natural.

Abstract

Let G be a compact Lie group and let X be an oriented Witt G-pseudomanifold. Using intersection cohomology it is possible to define Sign(G,X) in R(G), the G-signature of X. Let g be an element in G. Assuming that the inclusion of the fixed point set associated to g is normally non-singular, we prove a formula for Sign(g,X), the G-signature of X computed at g, thus extending to Witt G-pseudomanifolds the fundamental result proved by Atiyah, Segal and Singer on smooth compact G-manifolds. Along the way, we give a detailed study of the fixed point set of a Thom-Mather G-space X and our main result in this direction is a sufficient condition ensuring that the fixed point set associated to G is included in X in a normally non-singular manner. This latter result provides many examples where our formula applies.

The G-signature Theorem on Witt spaces

TL;DR

The paper extends the Atiyah–Segal–Singer G-signature formula to Witt G-pseudomanifolds by defining the equivariant signature via intersection cohomology and analyzing the signature operator in the wedge metric setting. It develops an analytic KK-theory framework, proving that the fixed-point data of X^g contributes via a localization formula to Sign(g,X) under a strong normal-non-singularity hypothesis for the fixed-set inclusion, with the L-classes of the fixed components and the characteristic data of the normal bundle driving the formula. Central to the approach are the equivariant K-homology class [D^sign_g], Gysin maps in the equivariant setting, and the localized Chern character machinery of Puschnigg, which together yield a fixed-point expression that specializes to a product of complex characteristic factors when the normal bundle splits into line bundles. The results provide concrete transversality criteria yielding wide classes of Witt G-pseudomanifolds to which the formula applies and fortify the bridge between singular spaces, KK-theory, and index theory. Overall, this work broadens the reach of equivariant index theory to singular spaces with group actions, with implications for geometry and representation theory in contexts where intersection homology and stratified spaces are natural.

Abstract

Let G be a compact Lie group and let X be an oriented Witt G-pseudomanifold. Using intersection cohomology it is possible to define Sign(G,X) in R(G), the G-signature of X. Let g be an element in G. Assuming that the inclusion of the fixed point set associated to g is normally non-singular, we prove a formula for Sign(g,X), the G-signature of X computed at g, thus extending to Witt G-pseudomanifolds the fundamental result proved by Atiyah, Segal and Singer on smooth compact G-manifolds. Along the way, we give a detailed study of the fixed point set of a Thom-Mather G-space X and our main result in this direction is a sufficient condition ensuring that the fixed point set associated to G is included in X in a normally non-singular manner. This latter result provides many examples where our formula applies.

Paper Structure

This paper contains 11 sections, 28 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. G-actions and fixed point sets
  3. Transversality and fixed point sets
  4. The equivariant K-homology class defined by $D^{{\rm sign}}$ and its properties
  5. The equivariant K-homology class $[D^{{\rm sign}}_{{\bf g}}]\in K^G_{j} (X)$
  6. G-Stratified Diffeomorphism invariance.
  7. Gysin homomorphisms in the $G$-equivariant setting
  8. The equivariant signature
  9. The G-signature formula on $G$-Witt pseudomanifolds
  10. Localization
  11. The G-signature formula

Key Result

Theorem 1.1

Let $G=\langle g \rangle$ be topologically cyclic and compact. Let $X$ be a compact oriented Witt $G$-pseudomanifold which has the $G$-homotopy type of a finite $G$-CW complex. Assume that the inclusion $X^g\equiv X^G \subset X$ is a strong $G-$equivariant normally non-singular inclusion and that $E with $\mathcal{L}_* (F)$ denoting the (renormalized) Goresky-MacPherson-Siegel homology $L$-class o

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • ...and 65 more