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

Abstract

For a closed orientable connected 3-manifold $M$, its complexity $\mathbf{T}(M)$ 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 $\mathbf{T}(M)$ 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 $\{0,\phi \}⩽H^1(M;{\mathbb{Z}}_{/2\mathbb{Z}})$, we have $\mathbf{T}(M)⩾2+2||\phi ||$ unless $M$ is a lens space with $\mathbf{T}(M)=1+2||\phi ||$; (2) for any rank-2 subgroup $\{0,{\phi }_1,{\phi }_2,{\phi }_3\}⩽H^1(M;{\mathbb{Z}}_{/2\mathbb{Z}})$, we have $\mathbf{T}(M)⩾2+||{\phi }_1||+||{\phi }_2||+||{\phi }_3||$. 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.

30 pages, 7 figures