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Journal of Mathematics
Volume 2019, Article ID 6958218, 6 pages
https://doi.org/10.1155/2019/6958218
Research Article

The Topological Type of Equilateral and Almost Equiangular Polygon Spaces

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

Received 16 June 2019; Accepted 20 August 2019; Published 9 September 2019

Academic Editor: Andrei V. Kelarev

Copyright © 2019 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 equilateral spatial n-gons. For and , let be the subspace of consisting of elements whose first k bond angles are θ. Recently, the topological type of was determined for small n, special θ, and or . In this paper, we determine the topological type of for general n and θ.

1. Introduction

Starting in , the topology of polygon spaces in the Euclidean space of dimension three has been studied extensively by many authors. For example, Milgram and Trinkle  obtained results by making excellent use of Morse surgery. We refer to  for an excellent exposition with emphasis on Morse theory.

Let be the configuration space of equilateral spatial n-gons. Recently, in order to construct a mathematical model of chemical objects, attempts are made to impose restrictions on the bond angles of an element of . For example, an equilateral and equiangular polygon can be considered as a mathematical model of a cycloalkane. We fix the number n of the vertices and the bond angle θ. Then, the set of all the possible shapes of a model is called conformations in chemistry and configuration spaces in mathematics (for more about chemistry, see Example 1 and the papers therein).

Generalizing the configuration space which was already constructed, we give the following filtration of .

Definition 1. For and , we set(1).(2), for .Here denotes the standard inner product on . Moreover, when , we understand to be .
Let act on diagonally, and we set . Note that is the subspace of consisting of elements whose first k bond angles are θ. In particular, we have .

Example 1. (i) An equilateral and equiangular n-gon can be considered as a mathematical model of a cycloalkane. Such a cycloalkane is called cyclobutanes, cyclopentanes, or cyclohexanes according as or 6. The space is the configuration space of such a model. The topological type of for various θ was determined by Crippen  for and 5 and by O’Hara  for .(ii)Consider an equilateral polygon whose bond angles are the same except for the last two ones. It can be considered as a mathematical model of a ringed hydrocarbon molecule. The space is the configuration space of such a model. Goto et al.  studied the topological type of for the case that n is small and θ is slightly smaller than . Here, note that is the bond angle of the regular n-gon in (see Theorem 1).The space is tough, and the only technical tools to analyze it are the implicit function theorem and Reeb’s theorem. For this reason, the known result is limited to the case that is homeomorphic to a sphere (see Theorem 1).
The purpose of this paper is to study the topology of . Since we can use usual techniques in topology, for example, fiber bundles, we can determine the topological type of for general n and θ.
This paper is organized as follows. In Section 2, we prove preliminary results. In Section 3, we state our main theorem. In Section 4, we prove auxiliary lemmas. In Section 5, we prove our main theorem. Finally, in Section 6, we state our conclusions. (Note: throughout this paper, the notation means that X is homeomorphic to Y.)

2. Preliminary Results

We first recall the results about .

Theorem 1  (). (i)There is a homeomorphism:Note that consists of the regular n-gon in .(ii)Let ε be a sufficiently small positive real number. Then, for , we have .

In the definition of , we may normalize that and . Next, we set

Then, we obtain the following description of :(1) and .(2).(3) and for .

Hereafter, we use description (4).

The following definition and lemma will be used in Section 3.

Definition 2. Let n be an integer greater than or equal to 5.(i)We define the function byThen, we define to be largest θ which satisfies the following equation:(ii)We define the function byThen, we define as follows:(a).(b)When , we define to be largest θ which satisfies the following equation:

Example 2. (i), and .(ii), and .

Lemma 1. We define and to be the elements of , which satisfy the following conditions:(a)The linkages are planar polygons such that .(b)The linkages do not have self-intersection points.(c)We set and . Then,(i)For , we require that and .(ii)For , we require that and .(iii)For , we require that , and .Figures 1(a)1(c) are , , and for , respectively. Then, the following assertions hold:(i)For , the value realizes (ii)For , the value realizes (iii)For , the value realizes