Chinese Journal of Mathematics
Volume 2018 (2018), Article ID 9842324, 4 pages
https://doi.org/10.1155/2018/9842324

Research Article

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

Yasuhiko Kamiyama

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

Correspondence should be addressed to Yasuhiko Kamiyama; kamiyama@sci.u-ryukyu.ac.jp

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 set

Here 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 set

where 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.

Table 1: The topological type of .

Topological type

Theorem B. (i) The topological type of is given in Table 2.
(ii) As approaches , point in Figure 1(a) approaches point .

Table 2: The topological type of .

Topological type

Figure 1(a)

Figure 1(b)

Figure 1(a)

(a)

(b)

Figure 1: (a) for or . (b) .

Theorem C. (i) The topological type of is given in Table 3. Let be a connected closed orientable surface of genus .
(ii) (a) Let satisfy that . We study the situation where approaches . We identify the torus with the Dupin cyclide, which we denote by . (See Figure 2.)
Using this, we identify with , where the connected sum is formed by cutting a small circular hole away from the narrow part of . As approaches , the center of each narrow part pinches to a point. Thus the five singular points appear.
(b) We consider the situation where increases from . Then each pinched point of separates. Thus we obtain .
(c) Let satisfy that . We consider the situation where approaches . In contrast to (a), the center of exactly one narrow part pinches to a point. Thus one singular point appears.

Table 3: The topological type of .

Topological type

Contains five singular points

Contains one singular point

Figure 2: The Dupin cyclide.

Corollary D. As a subspace of , we define the space

Then is a singular point of and has a neighborhood , where denotes the cone.

Remark 1. Cone-type singularities appear in Theorems B and C and Corollary D. We note that singularities of configuration spaces of mechanical linkages have been studied extensively by Blanc and Shvalb [10].

3. Proofs of the Main Results

We fix and set

Normalizing and to be and , respectively, we have the following description:

Hereafter we use (8).

In order to prove our main results, we use the following fact, whose proof is left to the reader.

Fact 2. Let satisfy that

Then, there exists such that

Now we first consider the case . Consider Fact 2 for , and . Then there exists such that

Next, we consider Fact 2 for , and . Then there exists such that

Finally, we consider Fact 2 for in (12),

and . Then there exists such that

Now we define the function by

We can understand as a level set. More precisely, we define the function

by . Then we have

if .

Remark 3. Since for all , and , we have . On the other hand, it is clear that . Hence (17) does not hold for . Apart from this point, there is an identification

where is defined in (6).

Lemma 4. We set

Then is given in Table 4.

Table 4: The set .

Proof. The lemma is proved by direct computations.

Proof of Theorem C. We consider in (16) as a Morse function on . First, Table 4 and (17) show that .
Second, direct computation shows that

Since this is nonzero, the space is smooth at . Actually, we can prove that the point is a nondegenerate critical point of the function . Hence Morse lemma shows that there is a homeomorphism for . But if we use [11, Corollary B], we need not check that is nondegenerate at . For our reference, we draw the figure of in Figure 3.
Third, the other parts of Table 3 follow from Table 4. This completes the proof of Theorem C.

Figure 3: .

Proof of Corollary D. The corollary is an immediate consequence of Theorem C.

Proof of Theorem B. We define as in (11). We also define to be the right-hand side of (12). We define the function by

Similarly to (17), we have . Since is one-dimensional, it is easy to draw its figure. Thus Theorem B follows.

Proof of Theorem A. We define the function by . Since , Theorem A follows.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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