Discrete Morse Theory and Extended L 2 Homology

Abstract

A brief overview of Forman's discrete Morse theory is presented, from which analogues of the main results of classical Morse theory can be derived for discrete Morse functions, these being functions mapping the set of cells of a CW complex to the real numbers satisfying some combinatorial relations. The discrete analogue of the strong Morse inequality was proved by Forman for finite CW complexes using a Witten deformation technique. This deformation argument is adapted to provide strong Morse inequalities for infinite CW complexes which have a finite cellular domain under the free cellular action of a discrete group. The inequalities derived are analogous to the L 2 Morse inequalities of Novikov and Shubin and the asymptotic L 2 Morse inequalities of an inexact Morse 1-form as derived by Mathai and Shubin. We also obtain quantitative lower bounds for the Morse numbers whenever the spectrum of the Laplacian contains zero, using the extended category of Farber. Academic Press

L2-Invariants of Regular Coverings of Compact Manifolds and CW-Complexes

“…If Γ has finite order, this follows from 8.). If Γ is infinite, this follows from the fact that More information about Morse inequalities and L 2 -invariants can be found in [89], [90], [157], [172], [173], [196], [197], [234]. Example 1.8 A good source of well understood examples is given by the special case where Γ is the free abelian group Z r of rank r. On one hand everything becomes simple, on the other hand one can already see some of the important phenomenons in this special case.…”

The main theorem of discrete Morse theory for Morse matchings with finitely many rays

“…The present paper deals with the same problem, with a slightly different approach. (It is perhaps worth noting that the discrete Morse theory was also applied to infinite complexes in [16], though in a vastly different spirit than in the present paper; we shall not discuss these investigations. )…”

Ascending and descending regions of a discrete Morse function

“…A model of a geographical terrain Forman [3], [4] to solve this problem. Discrete Morse theory is a PL analogue of classical smooth Morse theory which has gained wide popularity and is used in topological data analysis [5], [6], [7], [8] as well as in addressing purely theoretical topological and combinatorial problems, as for example in [9], [10], [11].…”