On the number of simplices required to triangulate a Lie group

Abstract

We estimate the number of simplices required for triangulations of compact Lie groups. As in the previous work [11], our approach combines the estimation of the number of vertices by means of the covering type via a cohomological argument from [10], and application of the recent versions of the Lower Bound Theorem of combinatorial topology. For the exceptional Lie groups, we present a complete calculation using the description of their cohomology rings given by the first and third authors. For the infinite series of classical Lie groups of growing dimension d, we estimate the growth rate of number of simplices of the highest dimension, which extends to the case of simplices of (fixed) codimension $d-i$.

Introduction

Every smooth manifold X admits an (essentially unique) compatible piecewise linear structure, i.e. a triangulation. Obviously it is finite if X is compact. But the existence theorem says nothing about the number of simplices, e.g. vertices, we need for a triangulation of X. The problem of finding minimal triangulation, i.e. a triangulation which has minimal number of vertices, was a subject of many studies of combinatorial topology (see [5] and [16] for references). Consequently, it is important to give any estimate of the number of vertices, or more generally the number of simplices of given dimension $0\le i\le d=\dim X$.

Notation 1.1

For a triangulation of a d-dimensional closed manifold by a simplicial complex K, we denote by $f_i$, $i=0,\dots ,d$ the number of i-dimensional simplices in K.

In this work we present estimates of coordinates of the vector $(f_0,f_1,\dots ,f_d)$ in the case where X is a compact Lie group. We restrict our study to classical Lie groups for which the cohomology rings have complete description. The case of five exceptional Lie groups is equipped with complete calculation of the formula based on the complete description of their cohomology rings given by the first and third author in [7]. For the remaining infinite series of classical Lie groups we describe the asymptotic growth of the number $f_i$ of vertices of dimension i, according to the dimension d. According to our knowledge there is no result in literature which gives estimate of the number of simplices of triangulation of Lie groups in general. Let us recall that every compact Lie group has finite-sheeted covering which is a product of a torus and some simple and simply-connected Lie groups, see [27, App. 1.2]. Simple and simply-connected Lie groups have four infinite series $A_n$, $B_n$, $C_n$, $D_n$, and finite family of exceptional Lie groups $G_2$, $F_4$, $E_6,E_7,E_8$. The series A, B, C, and D correspond to the groups $SU(n+1)$, $SO(2n+1)$, $Sp(n)$, and $SO(2n)$ respectively.

Our approach has two factors. In [14] M. Karoubi and Ch. Weibel defined a homotopy invariant of a space X called the covering type of X and denoted $\mathrm{ct}(X)$. Directly from the definition it follows that $\mathrm{ct}(X)$ is a lower bound for $f_0(X)$ (cf. [14], also [10]). In [10], a method estimating $\mathrm{ct}(X)$ from below was presented. The main result of this paper provides a formula in terms of multiplicative structure of the cohomology ring $H^{⁎}(X;R)$ in any coefficient ring R. More precisely, it estimates $\mathrm{ct}(X)$ by the maximal weighted length of a non-zero product in ${\tilde{H}}^{⁎}(X;R)$ (Theorem 2.1).

The second component of our approach is based on the recent much more sharper versions of the Lower Bound Theorem, shortly called LBT (see [11] for an exposition). Purely combinatorial in arguments LBT (cf. [13]) states that the number i-dimensional simplices of K grows as $f_0$ times the number of i-dimensional simplices of the standard simplex ${\mathrm{\Delta }}_d$ lowered by a term $i\left(\begin{array}{c}d+1\\ i\end{array}\right)$, which does not depend on $f_0$. Recent versions of LBT, called GLBT (cf. [15], [19], [20], and [21], [22]) or g-conjecture confirmed in [1], increase the formula of LBT by adding terms which depend on the reduced Betti numbers of X (cf. Theorem 3.1). The latter not only involves the topology of X but also essentially improves the estimate.

The paper is organized in the following way. In the second section we derive or estimate the main formula of [10] (Theorem 2.1) for the classical Lie groups estimating the covering type of spaces in problem. Next, in the third section we adapt Theorem 3.1 to the discussed spaces by substituting the estimate of $f_0$ from second section and the values (or estimates) of Betti numbers of studied spaces. At the end we include a Mathematica notebook which derives the value of main formula provided values of $f_0$ and the Betti numbers ${\beta }_i$ are known. We present the result of computation of estimates of $f_i$, $0\le i\le 14$ for the group $G_2$, and $F_4$, leaving the reader a possibility to compute this for the remaining exceptional compact Lie groups $E_6$, $E_7$, and $E_8$.