Maps from 3-manifolds to 4-manifolds that induce isomorphisms on $\pi_1$

TL;DR

This paper proves that most closed orientable 3-manifolds cannot be mapped to 4-manifolds in a way that induces isomorphisms on their fundamental groups, highlighting fundamental topological obstructions.

Contribution

It establishes new constraints on maps from 3-manifolds to 4-manifolds that preserve fundamental groups, extending understanding of 3- and 4-manifold relationships.

Findings

Most 3-manifolds do not admit fundamental group isomorphisms via maps to 4-manifolds.

Certain 3-manifolds cannot be bounded by 4-manifolds with the same fundamental group.

Results extend to higher-dimensional manifolds.

Abstract

In this paper, we prove that any closed orientable 3-manifold $M$ other than $\#^k S^1\times S^2$ and $S^3$ satisfies the following properties: (1) For any compact orientable 4-manifold $N$ bounded by $M$, the inclusion does not induce an isomorphism on their fundamental groups $\pi_1$. (2) For any map $f:M\to N$ from $M$ to a closed orientable 4-manifold $N$, $f$ does not induce an isomorphism on $\pi_1$. Relevant results on higher dimensional manifolds are also obtained.