The Configuration Space of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math>-Tuples of Equiangular Unit Vectors for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn fontstyle="italic">3</mml:mn></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mrow><mml:mn fontstyle="italic">4</mml:mn></mml:mrow></mml:math>, and <mml:math xmlns:mml="http://www.w3.org

Kamiyama, Yasuhiko

doi:10.1155/2018/9842324

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“…(i) It is not true that critical points of the same critical value have same index. For example, Γ 9 contains elements (3, 2) and (9,6) such that their critical values are 2π/3. On the other hand, the index of the former is 4 but that of the latter is 5.…”

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The Euler characteristic of the regular spherical polygon spaces

Kamiyama

2020

Homology, Homotopy and Applications

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Let a be a real number satisfying 0 < a < π. We denote by M n (a) the configuration space of regular spherical n-gons with side-lengths a. The purpose of this paper is to determine χ(M n (a)) for all a and odd n. To do so, we construct a manifold X n and a function µ : X n → R such that µ −1 (a) = M n (a). In fact, the function µ is different from the well-known "wall-crossing" function. We determine the index of each critical point of µ. Since a level set is obtained by successive Morse surgeries, we can determine χ(M n (a)).2000 Mathematics Subject Classification. Primary 58E05; Secondary 58D29.

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“…On the other hand, for our case, it will suffice to consider just one manifold X n and one map µ in order to obtain information on all regular spherical polygon spaces. In [6], we reproved the results in [11, §6] along our lines.…”

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The Euler characteristic of the regular spherical polygon spaces

Kamiyama

2020

Homology, Homotopy and Applications

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The Configuration Space of n-Tuples of Equiangular Unit Vectors for n=3, 4, and
Yasuhiko Kamiyama
¹

Abstract: Let Mn(θ) be the configuration space of n-tuples of unit vectors in R3 such that all interior angles are θ. The space Mn(θ) is an (n-3)-dimensional space. This paper determines the topological type of Mn(θ) for n=3, 4, and 5.

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References 7 publications

The Euler characteristic of the regular spherical polygon spaces

The Euler characteristic of the regular spherical polygon spaces

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The Configuration Space of n-Tuples of Equiangular Unit Vectors for n=3, 4, and Yasuhiko Kamiyama1

Abstract: Let Mn(θ) be the configuration space of n-tuples of unit vectors in R3 such that all interior angles are θ. The space Mn(θ) is an (n-3)-dimensional space. This paper determines the topological type of Mn(θ) for n=3, 4, and 5.

Cited by 1 publication

References 7 publications

The Euler characteristic of the regular spherical polygon spaces

The Euler characteristic of the regular spherical polygon spaces

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The Configuration Space of n-Tuples of Equiangular Unit Vectors for n=3, 4, and
Yasuhiko Kamiyama
¹