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 $N\#({\#}_{p\ge 0}{S}^2\times S^2)\#({\#}_{q\ge 0}\mathbb{C}{P}^2)$, 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.

arXiv comment13 pages