An Introduction to Parametric Morse Theory and Pseudo-Isotopies

Abstract

Parametric Morse theory is a refinement of classical Morse theory that studies families of smooth functions on a manifold parametrized by an auxiliary space. This approach is particularly useful in understanding the topology of function spaces and has applications in areas such as pseudo-isotopies and h-cobordism theory.

Pseudo-isotopies, a central concept in differential topology, generalize isotopies by allowing deformations that are not necessarily level-preserving. Understanding pseudo-isotopies is crucial in high-dimensional topology, particularly in the study of diffeomorphism groups and automorphisms of manifolds.

This thesis provides an introduction to the basic ideas of parametric Morse theory and its connection to pseudo-isotopies. More precisely, this thesis provides the foundational knowledge required to understand the work of Allen E. Hatcher and John B. Wagoner in their 1973 paper, ”Pseudo-isotopies of Compact Manifolds” ([Hat73a]). We discuss almost all the background material necessary to understand their work. Specifically, we have reorganized a way to understand the stratification of function spaces, the proof of the h-cobordism theorem, and a framework for understanding the s-cobordism theorem.