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Sho Hasui

Publications and source records attributed to Sho Hasui.

13 recordsLinked to original sources
arXiv2024

Homotopy commutativity in quasitoric manifolds

We prove that the loop space of a quasitoric manifold is homotopy commutative if and only if the underlying polytope is a product of $3$-simplices $(\Delta^3)^n$ and the characteristic matrix is equivalent to a matrix of certain type. Quasitoric manifolds over $(\Delta^3)^n$ include generalized Bott manifolds, and we also construct an infinite family of homotopy nonequivalent generalized Bott manifolds over $(\Delta^3)^n$, only half of them have homotopy commutative loop spaces. In particular, for each $n\ge 2$, there are infinitely many homotopy types in $6n$-dimensional quasitoric manifolds having homotopy (non)commutative loop spaces.

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arXiv2021

Tverberg's theorem for cell complexes

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 substantially generalize this theorem to continuous maps of certain CW complexes, including simplicial $((d+1)(r-1)-1)$-spheres, into the Euclidean $d$-space. We also discuss the atomicity of the Tverberg property.

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arXiv2017

Classification of toric manifolds over an $n$-cube with one vertex cut

We say that a complete nonsingular toric variety (called a toric manifold in this paper) is over $P$ if its quotient by the compact torus is homeomorphic to $P$ as a manifold with corners. Bott manifolds (or Bott towers) are toric manifolds over an $n$-cube $I^n$ and blowing them up at a fixed point produces toric manifolds over $\mathrm{vc}(I^n)$ an $n$-cube with one vertex cut. They are all projective. On the other hand, Oda's $3$-fold, the simplest non-projective toric manifold, is over $\mathrm{vc}(I^n)$. In this paper, we classify toric manifolds over $\mathrm{vc}(I^n)$ $(n\ge 3)$ as varieties and also as smooth manifolds. As a consequence, it turns out that (1) there are many non-projective toric manifolds over $\mathrm{vc}(I^n)$ but they are all diffeomorphic, and (2) toric manifolds over $\mathrm{vc}(I^n)$ in some class are determined by their cohomology rings as varieties among toric manifolds.

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arXiv2017

Samelson products in quasi-$p$-regular exceptional Lie groups

There is a product decomposition of a compact connected Lie group $G$ at the prime $p$, called the mod $p$ decomposition, when $G$ has no $p$-torsion in homology. Then in studying the multiplicative structure of the $p$-localization of $G$, the Samelson products of the factor space inclusions of the mod $p$ decomposition are fundamental. This paper determines (non-)triviality of these fundamental Samelson products in the $p$-localized exceptional Lie groups when the factor spaces are of rank $\le 2$, that is, $G$ is quasi-$p$-regular.

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arXiv2016

Higher homotopy commutativity in localized Lie groups and gauge groups

The first aim of this paper is to study the $p$-local higher homotopy commutativity of Lie groups in the sense of Sugawara. The second aim is to apply this result to the $p$-local higher homotopy commutativity of gauge groups. Although the higher homotopy commutativity of Lie groups in the sense of Williams is already known, the higher homotopy commutativity in the sense of Sugawara is necessary for this application. The third aim is to resolve the $5$-local higher homotopy non-commutativity problem of the exceptional Lie group $\mathrm{G}_2$, which has been open for a long time.

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arXiv2014

$p$-local stable splitting of quasitoric manifolds

We show a homotopy decomposition of $p$-localized suspension $\Sigma M_{(p)}$ of a quasitoric manifold $M$ by constructing power maps. As an application we investigate the $p$-localized suspension of the projection $\pi$ from the moment-angle complex onto $M$, from which we deduce its triviality for $p>\dim M/2$. We also discuss non-triviality of $\pi_{(p)}$ and $\Sigma^\infty\pi$.

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arXiv2014

On the cohomology equivalences between bundle-type quasitoric manifolds over a cube

The aim of this article is to establish the notion of bundle-type quasitoric manifolds and provide two classification results on them: (1) ($\mathbb{C}P^2\sharp\mathbb{C}P^2$)-bundle type quasitoric manifolds are weakly equivariantly homeomorphic if their cohomology rings are isomorphic, and (2) quasitoric manifolds over $I^3$ are homeomorphic if their cohomology rings are isomorphic. In the latter case, there are only four quasitoric manifolds up to weakly equivariant homeomorphism which are not bundle-type.

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arXiv2014

Samelson products in p-regular exceptional Lie groups

The (non)triviality of Samelson products of the inclusions of the spheres into p-regular exceptional Lie groups is completely determined, where a connected Lie group is called p-regular if it has the p-local homotopy type of a product of spheres.

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arXiv2013

Strong cohomological rigidity of quasitoric manifolds over the 3-cube

A quasitoric manifold is a smooth manifold with a locally standard torus action for which the orbit space is identified with a simple polytope. For a class of topological spaces, the class is called strongly cohomologically rigid if any isomorphism of cohomology rings can be realized as a homeomorphism. This paper shows the strong cohomological rigidity of the class of quasitoric manifolds over $I^3$.

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arXiv2013

On the classification of quasitoric manifolds over the dual cyclic polytopes

For a simple $n$-polytope $P$, a quasitoric manifold over $P$ is a $2n$-dimensional smooth manifold with a locally standard action of the $n$-dimensional torus for which the orbit space is identified with $P$. This paper shows the topological classification of quasitoric manifolds over the dual cyclic polytope $C^n(m)^*$, when $n>3$ or $m-n=3$. Besides, we classify small covers, the "real version" of quasitoric manifolds, over all dual cyclic polytopes.

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