Pseudo-isotopies of 3-manifolds with infinite fundamental groups

TL;DR

This work analyzes the smooth and topological mapping class groups of cylinders $I\times Y$ with $Y$ a connected, compact, oriented $3$-manifold of infinite fundamental group. By constructing Budney–Gabai barbell diffeomorphisms and examining their actions on spaces of embedded arcs, the authors obstruct isotopies via configuration-space invariants derived from the Dax isomorphism and the Bousfield–Kan spectral sequence, producing infinite-rank subgroups in $\pi_0\mathrm{Diff}_{PI}(I\times Y)$, $\pi_0\mathrm{Diff}_{\partial}(I\times Y)$, $\pi_0\mathrm{Homeo}_{PI}(I\times Y)$, and $\pi_0\mathrm{Homeo}_{\partial}(I\times Y)$. They also establish that $\pi_0\,C(Y)$ contains an abelian subgroup of infinite rank and that $\pi_0\,C(I\times Y)$ surjects onto an abelian infinite-rank group, connecting concordance automorphisms to configuration-space obstructions and algebraic K-theory in the stable regime. The paper treats irreducible and reducible cases after filling $D^3$'s, uses finite covers to transfer obstructions to the closed setting, and handles special manifolds such as $S^1\times S^2$, $\mathbb{RP}^3\#\mathbb{RP}^3$, and connected sums, obtaining a comprehensive infinite-rank picture for the mapping class and concordance groups in dimension three and their concordance structure in higher dimensions.

Abstract

Suppose $Y$ is a compact, connected, oriented 3-manifold possibly with boundary, such that $π_1(Y)$ 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 $π_0\operatorname{Diff}_{PI}(I\times Y) \to π_0\operatorname{Homeo}_{PI}(I\times Y)$ is of infinite rank. As a consequence, $π_0\operatorname{Diff}_{PI}(I\times Y)$, $π_0\operatorname{Diff}_{\partial}(I\times Y)$, $π_0\operatorname{Homeo}_{PI}(I\times Y)$, $π_0\operatorname{Homeo}_{\partial}(I\times Y)$ are all abelian groups of infinite rank. We also prove that $π_0\,C(Y)$ contains an abelian subgroup of infinite rank, and $π_0\,C(I\times Y)$ 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.

Theorems & Definitions (96)

• Theorem A
• Corollary 1.1
• Remark 1.2
• Corollary 1.3
• Theorem B
• Lemma 2.1
• proof
• Proposition 3.1
• proof
• Remark 3.2