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2020 How many simplices are needed to triangulate a Grassmannian?
Dejan Govc, Wacław Marzantowicz, Petar Pavešić
Topol. Methods Nonlinear Anal. 56(2): 501-518 (2020). DOI: 10.12775/TMNA.2020.027

Abstract

In this paper we use recently developed methods to compute a lower bound for the number of simplices that are needed to triangulate the Grassmann manifold $G_k(\mathbb{R}^n)$. We first estimate the number of vertices that are needed for such a triangulation by giving a general lower bound and some more precise bounds for $k=2,3,4$. By applying the Lower Bound Theorem (LBT) of Barnette for triangulated manifolds, we then obtain estimates for the number of simplices in all dimensions. For higher-dimensional simplices these estimates can be considerably improved by using the recent progress on the Generalized Lower Bound Theorem for triangulated manifolds, which states that the $h''$-numbers of triangulated manifolds are unimodal, together with the computation of the Poincaré polynomial. For example, we are able to prove that the number of top-dimensional simplices in a triangulation of $G_k(\mathbb{R}^n)$ grows exponentially with $n$. Our method can be used to estimate the minimal size of triangulations for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc.

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Dejan Govc. Wacław Marzantowicz. Petar Pavešić. "How many simplices are needed to triangulate a Grassmannian?." Topol. Methods Nonlinear Anal. 56 (2) 501 - 518, 2020. https://doi.org/10.12775/TMNA.2020.027

Information

Published: 2020
First available in Project Euclid: 17 December 2020

Digital Object Identifier: 10.12775/TMNA.2020.027

Rights: Copyright © 2020 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.56 • No. 2 • 2020
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