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

For a closed orientable connected 3-manifold $M$, its complexity $\mathit{T}(M)$ 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 $\mathit{T}(M)$ in terms of the ${\mathbb{Z}}_{\mathrm{/}2\mathbb{Z}}$-coefficient Thurston norm for $H^1(M;{\mathbb{Z}}_{\mathrm{/}2\mathbb{Z}})$: (1) for any rank-1 subgroup ${0,\varphi} \leqslant H^1(M;$\mathbb{Z}${/2$\mathbb{Z}$})$,wehave$\boldsymbol{T}(M) \geqslant 2+2||\varphi||$unless$M$isalensspacewith$\boldsymbol{T}(M)=1+2||\varphi||$;(2)foranyrank-2subgroup${0,\varphi_1,\varphi_2,\varphi_3} \leqslant H^1(M;$\mathbb{Z}${/2$\mathbb{Z}$})$,wehave$\boldsymbol{T}(M) \geqslant 2+||\varphi_1||+||\varphi_2||+||\varphi_3||$\mathrm{.}Undertheextraassumptionthat$M$ 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.