Stable complexity and simplicial volume of manifolds

TL;DR

This paper explores the relationship between stable complexity and Gromov's simplicial volume of manifolds, revealing differences in higher dimensions and specific cases where they coincide, with implications for understanding manifold invariants.

Contribution

It establishes that stable complexity and simplicial volume do not always coincide, providing bounds and conditions under which they are equal, especially in 3-manifolds.

Findings

For hyperbolic manifolds of dimension ≥4, simplicial volume is less than a constant times stable complexity.

Stable complexity equals simplicial volume for certain 3-manifolds with specific JSJ decompositions.

The equality of the two invariants in all hyperbolic 3-manifolds depends on a version of the Ehrenpreis conjecture.

Abstract

Let the complexity of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree we can promote it to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which call the "stable complexity" of M. We study here the relation between the stable complexity of M and Gromov's simplicial volume ||M||. It is immediate to show that ||M|| is smaller or equal than the stable complexity of M and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental group. We show that this is not always the case: there is a constant C_n<1 such that ||M|| is smaller than C_n times the stable complexity for any hyperbolic manifold M of dimension at least 4. The question in dimension 3 is still open in general. We prove that the stable complexity equals ||M|| for any aspherical irreducible 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. The equality holds for all closed hyperbolic 3-manifolds if a particular three-dimensional version of the Ehrenpreis conjecture is true.