The complexity of prime 3-manifolds and the first $\mathbb{Z}_{/2\mathbb{Z}}$-cohomology of small rank

TL;DR

This paper establishes lower bounds for the complexity of prime 3-manifolds based on the $bZ/2bZ$-cohomology Thurston norm, removing the previous atoroidal assumption.

Contribution

It extends known inequalities relating manifold complexity and cohomology norms to prime manifolds without the atoroidal condition.

Findings

Lower bounds for complexity in rank-1 subgroups

Lower bounds for complexity in rank-2 subgroups

Results hold without atoroidal assumption

Abstract

For a closed orientable connected 3-manifold $M$, its complexity $\boldsymbol{T}(M)$ is defined to be the minimal number of tetrahedra in its triangulations. Under the assumption that $M$ is prime (but not necessarily atoroidal), we establish a lower bound for the complexity $\boldsymbol{T}(M)$ in terms of the $\mathbb{Z}_{/2\mathbb{Z}}$-coefficient Thurston norm for $H^1(M;\mathbb{Z}_{/2\mathbb{Z}})$: (1) for any rank-1 subgroup $\{0,\varphi\} \leqslant H^1(M;\mathbb{Z}_{/2\mathbb{Z}})$, we have $\boldsymbol{T}(M) \geqslant 2+2||\varphi||$ unless $M$ is a lens space with $\boldsymbol{T}(M)=1+2||\varphi||$; (2) for any rank-2 subgroup $\{0,\varphi_1,\varphi_2,\varphi_3\} \leqslant H^1(M;\mathbb{Z}_{/2\mathbb{Z}})$, we have $\boldsymbol{T}(M) \geqslant 2+||\varphi_1||+||\varphi_2||+||\varphi_3||$. Under the extra assumption that $M$ is atoroidal, these inequalities had already been shown by Jaco, Rubinstein, and Tillmann. Our work here shows that we do not need to require $M$ to be atoroidal.