Pseudo-isotopies of 3-manifolds with infinite fundamental groups

TL;DR

This paper demonstrates that the groups of pseudo-isotopies and concordance automorphisms of certain 3-manifolds with infinite fundamental groups are infinitely generated abelian groups, revealing complex algebraic structures in their topology.

Contribution

It establishes that the pseudo-isotopy groups of 3-manifolds with infinite fundamental groups have infinite rank and are abelian, extending understanding of their algebraic and topological properties.

Findings

The canonical map between pseudo-isotopy groups of diffeomorphisms and homeomorphisms has infinite rank.

The groups of pseudo-isotopies and concordance automorphisms are infinite rank abelian groups.

The study uses actions of barbell diffeomorphisms on embedded arcs and configuration spaces.

Abstract

Suppose $Y$ is a compact, connected, oriented 3-manifold possibly with boundary, such that $\pi_1(Y)$ is infinite. Let $\operatorname{Diff}_\partial(I\times Y)$ denote the group of self-diffeomorphisms of $I\times Y$ that are equal to the identity near the boundary. Let $\operatorname{Diff}_{PI}(I\times Y)$ denote the subgroup of $\operatorname{Diff}_\partial(I\times Y)$ consisting of elements pseudo-isotopic to the identity. Define $\operatorname{Homeo}_\partial(I\times Y)$, $\operatorname{Homeo}_{PI}(I\times Y)$ similarly for homeomorphisms. We show that the canonical map $\pi_0\operatorname{Diff}_{PI}(I\times Y) \to \pi_0\operatorname{Homeo}_{PI}(I\times Y)$ is of infinite rank. As a consequence, $\pi_0\operatorname{Diff}_{PI}(I\times Y)$, $\pi_0\operatorname{Diff}_{\partial}(I\times Y)$, $\pi_0\operatorname{Homeo}_{PI}(I\times Y)$, $\pi_0\operatorname{Homeo}_{\partial}(I\times Y)$ are all abelian groups of infinite rank. We also prove that $\pi_0\,C(Y)$ contains an abelian subgroup of infinite rank, and $\pi_0\,C(I\times Y)$ admits a surjection to an abelian group of infinite rank, where $C(X)$ denotes the concordance automorphism group $\operatorname{Diff}(I\times X, \{0\}\times X\cup I\times \partial X)$ or $\operatorname{Homeo}(I\times X, \{0\}\times X\cup I\times \partial X)$. These results are proved by studying the actions of barbell diffeomorphisms on the spaces of embedded arcs and configuration spaces.