$\pi_1$-injective bounding and application to 3- and 4-manifolds

TL;DR

The paper investigates $$-injective bounds for manifolds, showing how fundamental group properties influence bounding manifolds and applying these results to 3- and 4-manifold topology, including group actions and cobordism.

Contribution

It proves residual finiteness and finiteness of $$-injective bounding manifolds based on the fundamental group, and applies these to study group actions, cobordism, and bounding indices in 3- and 4-manifolds.

Findings

Closed 3-manifolds can bound 4-manifolds with specific fundamental group properties.

Two lens spaces are $$-isomorphic cobordant iff there is a degree one map between them.

Prime 3-manifolds can be realized as minimal bounding indices.

Abstract

Suppose a closed oriented $n$-manifold $M$ bounds an oriented $(n+1)$-manifold. It is known that $M$ $\pi_1$-injectively bounds an oriented $(n+1)$-manifold $W$. We prove that $\pi_1(W)$ can be residually finite if $\pi_1(M)$ is, and $\pi_1(W)$ can be finite if $\pi_1(M)$ is. In particular, each closed 3-manifold $M$ $\pi_1$-injectively bounds a 4-manifold with residually finite $\pi_1$, and bounds a 4-manifold with finite $\pi_1$ if $\pi_1(M)$ is finite. Applications to 3- and 4-manifolds are given: (1) We study finite group actions on closed 4-manifolds and $\pi_1$-isomorphic cobordism of 3-dimensional lens spaces. Results including: (a) Two lens spaces are $\pi_1$-isomorphic cobordant if and only if there is a degree one map between them. (b) Each spherical 3-manifold $M\ne S^3$ can be realized as the unique non-free orbit type for a finite group action on a closed 4-manifold. (2) The minimal bounding index $O_b(M)$ for closed 3-manifolds $M$ are defined, %and bounding Euler charicteristic $\chi_b(M)$. the relations between finiteness of $O_b(M)$ and virtual achirality of aspherical (hyperbolic) $M$ are addressed. We calculate $O_b(M)$ for some lens spaces $M$. Each prime is realized as a minimal bounding index. (3) We also discuss some concrete examples:Surface bundle often bound surface bundles, and prime 3-manifolds often virtually bound surface bundles, $W$ bounded by some lens spaces realizing $O_b$ is constructed.