${\pi }_1$-injective bounding and application to 3- and 4-manifolds

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.