Homotopy Equivalences of 3-Manifolds

TL;DR

This paper proves that for any oriented closed 3-manifold, there is a uniform bound on the number of iterations needed for a self-homotopy equivalence to become homotopic to a homeomorphism, depending only on the manifold.

Contribution

It establishes a uniform bound on the iteration of self-homotopy equivalences that become homotopic to homeomorphisms for all closed 3-manifolds.

Findings

Existence of a manifold-dependent constant A_M

Bound on the number of iterations to reach a homeomorphism

Applicable to all oriented closed 3-manifolds

Abstract

Let $M$ be an oriented closed $3$-manifold. We prove that there exists a constant $A_M$, depending only on the manifold $M$, such that for every self-homotopy equivalence $f$ of $M$ there is an integer $k$ such that $1 \leq k \leq A_M$ and $f^k$ is homotopic to a homeomorphism.