A complexity of compact 3-manifold via immersed surfaces
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866909446940131328 |
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| author | Amendola, Gennaro |
| author_facet | Amendola, Gennaro |
| contents | We define an invariant, which we call surface-complexity, of compact 3-manifolds by means of Dehn surfaces. The surface-complexity is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on $\mathbb{P}^2$-irreducible and boundary-irreducible manifolds without essential annuli and Möbius strips. Moreover, for these manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere, the ball, the projective space and the lens space $\mathbb{L}_{4,1}$, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of ideal triangulations and Matveev complexity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_05899 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A complexity of compact 3-manifold via immersed surfaces Amendola, Gennaro Geometric Topology 57M27 (primary), 57M20 (secondary) We define an invariant, which we call surface-complexity, of compact 3-manifolds by means of Dehn surfaces. The surface-complexity is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on $\mathbb{P}^2$-irreducible and boundary-irreducible manifolds without essential annuli and Möbius strips. Moreover, for these manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere, the ball, the projective space and the lens space $\mathbb{L}_{4,1}$, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of ideal triangulations and Matveev complexity. |
| title | A complexity of compact 3-manifold via immersed surfaces |
| topic | Geometric Topology 57M27 (primary), 57M20 (secondary) |
| url | https://arxiv.org/abs/2102.05899 |