Simplicial spheres, maps between them, and the simplicial volume of Davis’ manifolds
Poster for the workshop “Combinatorial Algebraic Topology and Applications”, Centro di Ricerca Matematica Ennio De Giorgi, Pisa, 27–28 November 2023.
Note: What I present in this poster is part of my PhD project, and is not to be intended as a finished work, as I am still working on it.
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov, related to other invariants coming, for example, from differential geometry, like the minimal volume. Gromov predicted a relation between the simplicial volume and the Euler characteristic of aspherical manifolds; this has become a central question in the mathematical community studying this invariant. The poster describes an approach for the study of this conjecture for a class of manifolds arising from a construction by M. Davis. This approach leads to the study of certain triangulated spheres and simplicial maps between them, and the poster focuses mostly on this “combinatorial component” of the approach.
Digital copy of the poster:
Flag spheres and Davis' construction. For a description of Davis' construction (which is more general than what appears in the poster), see Davis' book [1]. The theorem by Davis and Okun about the positivity of γ for flag 3-spheres is proved in [2]. The flag 3-sphere with 12 vertices that happens to be a minimal element of the poset considered in the poster is described in [3].
- M. W. Davis, “The Geometry and Topology of Coxeter Groups”, London Mathematical Society Monographs, vol. 32, Princeton University Press, Princeton, NJ, 2007.
- M. W. Davis and B. Okun, Vanishing theorems and conjectures for the ℓ²-homology of right-angled Coxeter groups, Algebr. Geom. Topol. 5 (2001), 7–74.
- L. Venturello, On flag spheres with few equators, arXiv preprint (2022).
Simplicial volume. For an introduction to simplicial volume, see [4]. In [6] there is a proof of the fact that the vanishing of the simplicial volume of a triangulated manifold cannot be decided algorithmically. For a survey about Gromov's question, see [5].
- R. Frigerio, “Bounded Cohomology of Discrete Groups”, Mathematical Surveys and Monographs, vol. 227, American Mathematical Society, Providence, RI, 2017.
- C. Löh, M. Moraschini and G. Raptis, On the simplicial volume and the Euler characteristic of (aspherical) manifolds, Res. Math. Sci. 9 (2022).
- S. Weinberger, “Computers, Rigidity, and Moduli”, Porter Lectures, Princeton University Press, Princeton, NJ, 2005.
Graph minors. The graph minor theorem is proved in [7].