Linking and the Morse complex

TL;DR

This paper characterizes the critical points of a Morse function on a compact manifold using linking numbers of links in the manifold, refining Morse inequalities with algebraic and topological insights.

Contribution

It introduces a new link-based criterion for determining the number of critical points of Morse functions, connecting linking numbers with Morse complex algebraic operations.

Findings

Critical points exceed Morse inequalities if certain linking conditions are met.

The number of critical points is characterized by Betti numbers and link behavior.

A refinement of the Rabinowitz Saddle Point Theorem for compact manifolds.

Abstract

For a Morse function f on a compact oriented manifold M, we show that f has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in M whose components have nontrivial linking number, such that the minimal value of f on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of f in terms of the Betti numbers of M and the behavior of f with respect to links. This can be viewed as a refinement, in the case of compact manifolds, of the Rabinowitz Saddle Point Theorem. Our approach, inspired in part by techniques of chain-level symplectic Floer theory, involves associating to collections of chains in M algebraic operations on the Morse complex of f, which yields relationships between the linking numbers of homologically trivial (pseudo)cycles in M and an algebraic linking pairing on the Morse complex.