Discrete Morse theory and extendedL2 homology

A brief overviewof 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 of the L2 Morse inequalities of Novikov and Shubin ([14]), and the asymptoticL2 Morse inequalities of an inexact Morse 1-form as derived in [13]. We also obtain quantitative lower bounds for the Morse numbers whenever the spectrum of the Laplacian contains zero, using the extended category of Farber [5] (see also [12]).