Families torsion and Morse functions for covering spaces

Morse functions and cohomology of homogeneous spaces

Morse functions are useful tool in revealing the geometric formation of its domain manifolds M. They define the handle decompositions of M from which the additive homologies H * (M) may be constructed. In these lectures two further questions were emphasized. (1) How to find a Morse function on a given manifold? (2) From Morse functions can one derive the multiplicative coho-mology rather than t...

Analytic Torsion for Families

Torsion Invariants for Families

We give an overview over the higher torsion invariants of Bismut-Lott, Igusa-Klein and Dwyer-Weiss-Williams, including some more or less recent developments. The classical Franz-Reidemeister torsion τFR is an invariant of manifolds with acyclic unitarily flat vector bundles [62], [33]. In contrast to most other algebraic-topological invariants known at that time, it is invariant under homeomorp...

Circle-Valued Morse Theory and Reidemeister Torsion

We compute an invariant counting gradient flow lines (including closed orbits) in S-valued Morse theory, and relate it to Reidemeister torsion for manifolds with χ = 0, b1 > 0. Here we extend the results in [6] following a different approach. However, this paper is written in a self-contained manner and may be read independently of [6]. The motivation of this work is twofold: on the one hand, i...

Holomorphic Morse Inequalities on Covering Manifolds

The goal of this paper is to generalize Demailly’s asymptotic holomorphic Morse inequalities to the case of a covering manifold of a compact manifold. We shall obtain estimates which involve Atiyah’s “normalized dimension” of the square integrable harmonic spaces. The techniques used are those of Shubin who gave a proof for the usual Morse inequalities in the presence of a group action relying ...