English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a closed orientable connected 3-manifold $M$, its complexity$\boldsymbol{T}(M)$ is defined to be the minimal number of tetrahedra in itstriangulations. Under the assumption that $M$ is prime (but not necessarilyatoroidal), 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$\{0,\varphi_1,\varphi_2,\varphi_3\} \leqslantH^1(M;\mathbb{Z}_{/2\mathbb{Z}})$, we have $\boldsymbol{T}(M) \geqslant2+||\varphi_1||+||\varphi_2||+||\varphi_3||$. 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.

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