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

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.