Morse inequalities for noncompact manifolds
Tsuyoshi Kato, Daisuke Kishimoto, Mitsunobu Tsutaya
TL;DR
The article develops Morse inequalities for noncompact manifolds $M$ with a cocompact discrete group action by $G$, without requiring Morse functions to be $G$-invariant. It introduces a piecewise trace framework and Gaussian-propagation calculus to disassemble Roe-type traces along a fundamental domain, linking $c_k:G o eal$ to the $L^2$-Betti numbers $b_k^{(2)}$ via Witten deformation and an abstract differential framework. The main result yields mean-value Morse inequalities for amenable $G$, reveals that nonzero $b_k^{(2)}$ forces infinitely many critical points of index $k$, and recovers Novikov–Shubin-type relations in the amenable setting. The approach combines fundamental-domain geometry, operator-algebra traces, and deformation arguments to extend Morse theory to large-scale noncompact settings with controlled group actions, offering new tools for understanding critical-point configurations on covers and their $L^2$-invariants.
Abstract
We establish Morse inequalities for a noncompact manifold with a cocompact and properly discontinuous action of a discrete group, where Morse functions are not necessarily invariant under the group action. The inequalities are given in terms of the $L^2$-Betti numbers and functions on the acting group which describe rough configurations of critical points of a Morse function.