Chinese Journal of Mathematics

Volume 2018, Article ID 9842324, 4 pages

https://doi.org/10.1155/2018/9842324

## The Configuration Space of -Tuples of Equiangular Unit Vectors for , , and

Department of Mathematics, University of the Ryukyus, Nishihara-cho, Okinawa 903-0213, Japan

Correspondence should be addressed to Yasuhiko Kamiyama; pj.ca.uykuyr-u.ics@amayimak

Received 3 November 2017; Accepted 1 February 2018; Published 1 March 2018

Academic Editor: Marc Coppens

Copyright © 2018 Yasuhiko Kamiyama. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Let be the configuration space of -tuples of unit vectors in such that all interior angles are . The space is an -dimensional space. This paper determines the topological type of for , , and .

#### 1. Introduction

Recently, starting in [1], the topology of the configuration space of spatial polygons of arbitrary edge lengths has been considered by many authors. In the equilateral case, the definition is given as follows. For , we setHere denote the unit vectors in the directions of the edges of a polygon; the group acts diagonally on .

Many topological properties of are already known: First, it is clear that there is a homeomorphism

Second, it is proved in [2] that is homeomorphic to del Pezzo surface of degree 5.

Third, when is odd, the integral cohomology ring was determined in [3]. We refer to [4] for other properties of , which is an excellent survey of linkages.

In another direction, we consider the space of -tuples of equiangular unit vectors in . More precisely, we define the following: We fix and setwhere denotes the standard inner product on . Using (3), we define

It is expected that the space is much more difficult than . For example, the following trivial observation shows that does not admit a similar property to (2): when is odd, we have but .

We claim that is a hypersurface of the torus . In fact, if we forget the condition in (3), the space corresponding to (4) is as observed in [5, 6]. Hence the claim follows.

We recall previous results on . First, [7] considered the case for . The main result is that, realizing as a homotopy colimit of a diagram involving and , we inductively computed . In particular, we obtained a homeomorphism , where denotes a connected closed orientable surface of genus 5.

Second, we set Note that is the configuration space of equilateral and equiangular -gons. Crippen [8] studied the topological type of for , and 5. The result is that is either , one point, or two points depending on . Later, O’Hara [9] studied the topological type of . The result is that is disjoint union of a certain number of ’s and points.

The purpose of this paper is to determine the topological type of for , and 5. In contrast to the fact that at most one-dimensional spaces appear in the results of [8, 9], surfaces appear in our results.

This paper is organized as follows. In Section 2, we state our main results and in Section 3 we prove them.

#### 2. Main Results

Theorem A. *The topological type of is given in Table 1.*