An analytic approach to Lefschetz and Morse theory on stratified pseudomanifolds

Abstract

We develop an analytic framework for Lefschetz fixed point theory and Morse theory for Hilbert complexes on stratified pseudomanifolds. We develop formulas for both global and local Lefschetz numbers and Morse, Poincar\'e polynomials as (polynomial) supertraces over cohomology groups of Hilbert complexes, developing techniques for relating local and global quantities using heat kernel and Witten deformation based methods. We focus on the case where the metric is wedge and the Hilbert complex is associated to a Dirac-type operator and satisfies the Witt condition, constructing Lefschetz versions of Bismut-Cheeger $\mathcal{J}$ forms for local Lefschetz numbers of Dirac operators, with specialized formulas for twisted de Rham, Dolbeault and spin$^{\mathbb{C}}$ Dirac complexes as supertraces of geometric endomorphisms on cohomology groups of local Hilbert complexes. We construct geometric endomorphisms to define de Rham Lefschetz numbers for some self-maps for which the pullback does not induce a bounded operator on $L^2$ forms. A de Rham Witten instanton complex is constructed for Witt spaces with stratified Morse functions, proving Morse inequalities related to other results in the literature including Goresky and MacPherson's in intersection cohomology. We also prove a Lefschetz-Morse inequality for geometric endomorphisms on the instanton complex that is new even on smooth manifolds. We derive $L^2$ Lefschetz-Riemann-Roch formulas, which we compare and contrast with algebraic versions of Baum-Fulton-Quart. In the complex setting, we derive Lefschetz formulas for spin Dirac complexes and Hirzebruch $\chi_y$ genera which we relate to signature, self-dual and anti-self-dual Lefschetz numbers, studying their properties and applications including instanton counting. We compute these invariants in various examples with different features, comparing with versions in other cohomology theories.