2017/12/07 by Kei Nakamura, Nakamura, Kei
Mathematics · #57N10 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1712.02607
openalex publication_date 2017/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a closed orientable connected 3-manifold M, its complexity \boldsymbolT(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 \boldsymbolT(M) in terms of the ℤ/2ℤ-coefficient Thurston norm for H1(M;ℤ/2ℤ): (1) for any rank-1 subgroup \0,φ\ \leqslant H1(M;ℤ/2ℤ), we have \boldsymbolT(M) \geqslant 2+2||φ|| unless M is a lens space with \boldsymbolT(M)=1+2||φ||; (2) for any rank-2 subgroup \0,φ1,φ2,φ3\ \leqslant H1(M;ℤ/2ℤ), we have \boldsymbolT(M) \geqslant 2+||φ1||+||φ2||+||φ3||. 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.