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Yasuhiko Kamiyama

Publications and source records attributed to Yasuhiko Kamiyama.

6 recordsLinked to original sources
arXiv2018

The Euler characteristic of the regular spherical polygon spaces

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 $μ: X_n \to \mathbf{R}$ 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 $χ(M_n(a))$.

math.GT↗
arXiv2015

On the middle dimensional homology classes of equilateral polygon spaces

Let $M_n$ be the configuration space of equilateral polygonal linkages with $n$ vertices in the Euclidean plane ${\mathbb R}^2$. We consider the case that $n$ is odd and set $n=2m+1$. In spite of the long history of research, the homology classes in $H_{m-1}(M_n;{\mathbb Z})$ are mysterious and not well-understood. Let $τ\colon M_n \to M_n$ be the involution induced by complex conjugation. In this paper, we determine the representation matrix of the homomorphism $τ_\ast\colon H_{m-1}(M_n;{\mathbb Z}) \to H_{m-1}(M_n;{\mathbb Z})$ with respect to a basis of $H_{m-1}(M_n;{\mathbb Z})$.

math.AT↗
arXiv2009

Configurations and parallelograms associated to centers of mass

The purpose of this article is to 1. define M(t,k) the t-fold center of mass arrangement for k points in the plane, 2. give elementary properties of M(t,k) and 3. give consequences concerning the space M(2,k) of k distinct points in the plane, no four of which are the vertices of a parallelogram. The main result proven in this article is that the classical unordered configuration of k points in the plane is not a retract up to homotopy of the space of k unordered distinct points in the plane, no four of which are the vertices of a parallelogram. The proof below is homotopy theoretic without an explicit computation of the homology of these spaces. In addition, a second, speculative part of this article arises from the failure of these methods in the case of odd primes p. This failure gives rise to a candidate for the localization at odd primes p of the double loop space of an odd sphere obtained from the p-fold center of mass arrangement. Potential consequences are listed.

math.AT↗
arXiv2009

Spaces of real polynomials with common roots

Let RX_{k,n}^l be the space consisting of all (n+1)-tuples (p_0(z),...,p_n(z)) of monic polynomials over R of degree k and such that there are at most l roots common to all p_i(z). In this paper, we prove a stable splitting of RX_{k,n}^l.

math.AT↗