Prescribed Virtual Homological Torsion of 3-Manifolds

Abstract

Let M be an irreducible $3$ -manifold M with empty or toroidal boundary which has at least one hyperbolic piece in its geometric decomposition, and let A be a finite abelian group. Generalizing work of Sun [20] and of Friedl–Herrmann [7], we prove that there exists a finite cover $M' \to M$ so that A is a direct factor in $H_1(M',{\mathbb Z})$ .

Virtual domination of 3‐manifolds II

“…The proof of the $$\pi _1$$‐injectivity result in [9] does not need such an interpolation, since the topology of the convex core is simple for surface groups. In [4], Chu and Groves used a local‐global argument to prove the $$\pi _1$$‐injectivity of mapped‐in 2‐complexes in cusped hyperbolic 3‐manifolds, but they do not need to figure out the topology of the convex core, so they did not use an interpolation either.…”

“…In [4], Chu and Groves proved that, for any compact irreducible 3‐manifold $$M$$ with positive simplicial volume and empty or tori boundary, and any finite abelian group $$A$$, $$M$$ has a finite cover $$M^{\prime }$$ such that $$A$$ is a direct summand of $$H_1(M^{\prime };\mathbb {Z})$$. Theorem 1.2 does not fully recover Chu and Groves' result, but it can be used to give an alternative proof for the case of closed 3‐manifolds.…”