Stabilizations of $s$-cobordisms of dimension $5$

TL;DR

This paper investigates the stabilization process required to convert 5-dimensional s-cobordisms into product cobordisms, utilizing advanced 4D topology tools like Gabai's light bulb theorem and Freedman-Quinn invariants.

Contribution

It applies 4D light bulb theorems and Freedman-Quinn invariants to analyze stabilization in 5D s-cobordisms, providing new insights into their structure.

Findings

Determines the number of stabilizations needed for 5D s-cobordisms.

Connects stabilization process to 4D light bulb theorem.

Uses Freedman-Quinn invariants to refine analysis.

Abstract

It has long been known that the $s$-cobordism theorem fails for $5$-dimensional $s$-cobordisms. In this article we study how many times of "stabilizations" are needed to turn a $5$-dimensional $s$-cobordism to a product cobordism. The question is analogous to asking how many times of stabilizations are needed to turn an exotic pair of four manifolds into diffeomorphic ones. The main tools in this article are Gabai's $4$D light bulb theorem and its applications, and we also use a refinement of $4$D light bulb theorem by Freedman Quinn invariant.