On the number of simplices required to triangulate a Lie group
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 .
Notation 1.1 For a triangulation of a d-dimensional closed manifold by a simplicial complex K, we denote by , the number of i-dimensional simplices in K.
In this work we present estimates of coordinates of the vector 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 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 , , , , and finite family of exceptional Lie groups , , . The series A, B, C, and D correspond to the groups , , , and 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 . Directly from the definition it follows that is a lower bound for (cf. [14], also [10]). In [10], a method estimating from below was presented. The main result of this paper provides a formula in terms of multiplicative structure of the cohomology ring in any coefficient ring R. More precisely, it estimates by the maximal weighted length of a non-zero product in (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 times the number of i-dimensional simplices of the standard simplex lowered by a term , which does not depend on . 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 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 and the Betti numbers are known. We present the result of computation of estimates of , for the group , and , leaving the reader a possibility to compute this for the remaining exceptional compact Lie groups , , and .
Section snippets
Computations of value of covering type by use of (2.1)
In this section derive the value of formula of Theorem 2.1 spaces which are the object of our investigation. To do it we restate the some facts presented already in [10], derived the value from a description of cohomology rings of exceptional Lie groups given by the first and third author in [6], and [7]. We also adapt the classical results on the cohomology rings of Lie groups (for example see [9]). Finally we include the results of direct computations done for some other spaces.
Theorem 2.1 If there are[10, Theorem 3.5]
Estimates of number of simplices of given dimension
In this section we estimate the number of simplices of a given dimension, e.g. of facets, and of all simplices that are needed to triangulate a Lie group or flag manifold. In our approach we follow our previous work [11]. A classical tool for such an estimation is the Lower Bound Theorem of Kalai [13] and also Gromov [12] (see also [3], [17], [18], and [11] for more information). Note that LBT is purely combinatorial and does not take into account the homology of the manifold. As in [11] we are
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Cited by (3)
- 1
Supported by National Natural Science Foundation of China (No. 11961131004).
- 2
Supported by the Polish Research Grant NCN Sheng 1 UMO-2018/30/Q/ST1/00228.