A Morse deformation lemma at infinity

TL;DR

This paper extends Morse Theory's deformation lemma to functions that tend to negative infinity at a compact set, broadening its applicability under certain growth conditions.

Contribution

It introduces a generalized deformation lemma accommodating functions with unbounded below behavior at infinity, expanding Morse Theory tools.

Findings

Proves a deformation lemma for functions going to -infinity at a compact set.

Allows the lower deformation value to be -infinity.

Applicable to functions satisfying specific growth conditions.

Abstract

We prove a generalized version of the classic deformation lemma from Morse Theory that considers functions going to $-\infty$ at a compact set, and allowing the lower value of the deformation to be $-\infty$. The result is valid for a class of functions satisfying a suitable growth condition.