Mitsunobu Tsutaya
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View article: Vector fields on noncompact manifolds
Vector fields on noncompact manifolds Open
View article: Morse inequalities for noncompact manifolds
Morse inequalities for noncompact manifolds Open
We establish Morse inequalities for a noncompact manifold with a cocompact and properly discontinuous action of a discrete group, where Morse functions are not necessarily invariant under the group action. The inequalities are given in ter…
View article: Homotopy commutativity in quasitoric manifolds
Homotopy commutativity in quasitoric manifolds Open
We prove that the loop space of a quasitoric manifold is homotopy commutative if and only if the underlying polytope is $(Δ^3)^n$ and the characteristic matrix is equivalent to a matrix of certain type. We also construct for each $n\ge 2$ …
View article: Higher homotopy normalities in topological groups
Higher homotopy normalities in topological groups Open
The purpose of this paper is to introduce ‐maps (), which describe higher homotopy normalities, and to study their basic properties and examples. An ‐map is defined with higher homotopical conditions. It is shown that a homomorphism is an …
View article: The space of commuting elements in a Lie group and maps between classifying spaces
The space of commuting elements in a Lie group and maps between classifying spaces Open
Let $π$ be a discrete group, and let $G$ be a compact connected Lie group. Then there is a map $Θ\colon\mathrm{Hom}(π,G)_0\to\mathrm{map}_*(Bπ,BG)_0$ between the null-components of the spaces of homomorphism and based maps, which sends a h…
View article: Homotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$
Homotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$ Open
The index theory for the space of finite propagation unitary operators was\ndeveloped by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks\nin mathematical physics. In particular, they proved that $\\pi_0$ of the space\nis…
View article: Vector fields on non-compact manifolds
Vector fields on non-compact manifolds Open
Let $M$ be a non-compact connected manifold with a cocompact and properly discontinuous action of a discrete group $G$. We establish a Poincaré-Hopf theorem for a bounded vector field on $M$ satisfying a mild condition on zeros. As an appl…
View article: Hilbert bundles with ends
Hilbert bundles with ends Open
Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed ba…
View article: Higher homotopy normalities in topological groups
Higher homotopy normalities in topological groups Open
The purpose of this paper is to introduce $N_k(\ell)$-maps ($1\le k,\ell\le\infty$), which describe higher homotopy normalities, and to study their basic properties and examples. An $N_k(\ell)$-map is defined with higher homotopical condit…
View article: Homotopy type of the unitary group of the uniform Roe algebra on ℤn
Homotopy type of the unitary group of the uniform Roe algebra on ℤn Open
We study the homotopy type of the space of the unitary group [Formula: see text] of the uniform Roe algebra [Formula: see text] of [Formula: see text]. We show that the stabilizing map [Formula: see text] is a homotopy equivalence. Moreove…
View article: De Rham cohomology of the weak stable foliation of the geodesic flow of a hyperbolic surface
De Rham cohomology of the weak stable foliation of the geodesic flow of a hyperbolic surface Open
We compute the de Rham cohomology of the weak stable foliation of the geodesic flow of a connected orientable closed hyperbolic surface with various coefficients. For most of the coefficients, we also give certain "Hodge decompositions" of…
View article: Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^n$
Homotopy type of the unitary group of the uniform Roe algebra on $\mathbb{Z}^n$ Open
We study the homotopy type of the space of the unitary group $\U_1(C^\ast_u(|\mathbb{Z}^n|))$ of the uniform Roe algebra $C^\ast_u(|\mathbb{Z}^n|)$ of $\mathbb{Z}^n$. We show that the stabilizing map $\U_1(C^\ast_u(|\mathbb{Z}^n|))\to\U_\i…
View article: Tverberg's theorem for cell complexes
Tverberg's theorem for cell complexes Open
The topological Tverberg theorem states that any continuous map of a $(d+1)(r-1)$-simplex into the Euclidean $d$-space maps some points from $r$ pairwise disjoint faces of the simplex to the same point whenever $r$ is a prime power. We sub…
View article: $G$-index, topological dynamics and marker property
$G$-index, topological dynamics and marker property Open
Given an action of a finite group $G$, we can define its index. The $G$-index roughly measures a size of the given $G$-space. We explore connections between the $G$-index theory and topological dynamics. For a fixed-point free dynamical sy…
View article: UPPER BOUND FOR MONOIDAL TOPOLOGICAL COMPLEXITY
UPPER BOUND FOR MONOIDAL TOPOLOGICAL COMPLEXITY Open
We show that tcM(M) ≤ 2 cat(M) for a finite simplicial complex M. For example,we have tcM(Sn ∨ Sm) = 2 for any positive integers n and m.
View article: A short proof for tc(K)=4
A short proof for tc(K)=4 Open
View article: SAMELSON PRODUCTS IN <i>p</i>-REGULAR SO(2<i>n</i>) AND ITS HOMOTOPY NORMALITY
SAMELSON PRODUCTS IN <i>p</i>-REGULAR SO(2<i>n</i>) AND ITS HOMOTOPY NORMALITY Open
A Lie group is called p -regular if it has the p -local homotopy type of a product of spheres. (Non)triviality of the Samelson products of the inclusions of the factor spheres into p -regular SO(2 n ( p ) is determined, which completes the…
View article: Coincidence Reidemeister trace and its generalization
Coincidence Reidemeister trace and its generalization Open
We give a homotopy invariant construction of the Reidemeister trace for the coincidence of two maps between closed manifolds of not necessarily the same dimensions. It is realized as a homology class of the homotopy equalizer, which coinci…