#### 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.*

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

**(a)**

**(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.*

Corollary D. *As a subspace of , we define the spaceThen 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 thatThen, 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 byWe can understand as a level set. More precisely, we define the functionby . Then we haveif .

*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.*

*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.

*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.