On $\pi_1$-injectivity of self-maps in low dimensions

Abstract

We show that all self-maps of non-zero degree of $3$-manifolds not covered by $S^3$ and of Thurston geometric $4$-manifolds and their connected sums not covered by $N\#(\#_{p\geq0}S^2\times S^2)\#(\#_{q\geq0}\mathbb C P^2)$, where $N$ is an $S^2\times\mathbb X^2$ or $S^3\times\mathbb R$ manifold, are $\pi_1$-injective. We thus determine when these maps induce $\pi_1$-isomorphisms. The results in dimension three were previously established by Shicheng Wang. We give a uniform group theoretic proof in all cases based only on the residual finiteness of the fundamental groups for the $\pi_1$-injectivity and then only on numerical invariants for the $\pi_1$-isomorphisms.