A 3-manifold complexity via immersed surfaces

Details A 3-manifold complexity via immersed surfaces A 3-manifold complexity via immersed surfaces

2008-04-04

arxiv.org/abs/0804.0695v1

J. Knot Theory Ramifications 19 (2010), no. 12, 1549-1569

Mathematics

Geometric Topology

20 pages, 18 figures, 1 table

Scientific paper

10.1142/S0218216510008558

We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold 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 P2-irreducible manifolds. Moreover, for P2-irreducible manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere S3, the projective space RP3 and the lens space L41, which have surface-complexity zero. We will also give estimations of the surface-complexity by means of triangulations, Heegaard splittings, surgery presentations and Matveev complexity.

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