Stable complexity and simplicial volume of manifolds

Details Stable complexity and simplicial volume of manifolds Stable complexity and simplicial volume of manifolds

2012-01-03

arxiv.org/abs/1201.0660v2

Mathematics

Geometric Topology

33 pages, 5 figures, minor corrections in the second version

Scientific paper

Let the Delta-complexity of a closed manifold M be the minimum number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the appropriate infimum on all finite coverings of M we can promote it to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which we call the "stable Delta-complexity" of M. We study here the relation between the stable Delta-complexity of M and Gromov's simplicial volume ||M||. We prove that the two characteristic numbers coincide on any irreducile 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. We do not know if they coincide on any irreducible 3-manifolds with infinite fundamental group: they do if a particular three-dimensional version of the Ehrenpreis conjecture holds. On the other hand, we show that ||M|| is strictly smaller than C_n times the stable Delta-complexity on any hyperbolic manifold M of dimension greater or equals than 4, for some constant C_n < 1 which depends only on the dimension n. We also discuss the relationship between our results and a conjecture of Gromov that states that the Euler characteristic of an aspherical manifold vanishes whenever its simplicial volume does.

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