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Daisuke Kishimoto

Publications and source records attributed to Daisuke Kishimoto.

At least 37 records · Page 2Linked to original sources
arXiv2023

Homotopy type of the space of finite propagation unitary operators on $\mathbb{Z}$

The index theory for the space of finite propagation unitary operators was developed by Gross, Nesme, Vogts and Werner from the viewpoint of quantum walks in mathematical physics. In particular, they proved that $π_0$ of the space is determined by the index. However, nothing is known about the higher homotopy groups. In this article, we describe the homotopy type of the space of finite propagation unitary operators on the Hilbert space of square summable $\mathbb{C}$-valued $\mathbb{Z}$-sequences, so we can determine its homotopy groups. We also study the space of (end-)periodic finite propagation unitary operators.

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arXiv2023

Upper bounds for virtual dimensions of Seiberg-Witten moduli spaces

Given a closed four-manifold with $b_1=0$ and a prime number $p$, we prove that for any mod $p^r$ basic class, the virtual dimension of the Seiberg-Witten moduli space is bounded above by $2r(p-1)-2$ under some conditions on $r$ and $b_2^+$. As an application, we obtain adjunction inequalities for embedded surfaces with negative self-intersection number.

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arXiv2022

Torsion in the space of commuting elements in a Lie group

Let $G$ be a compact connected Lie group, and let $\mathrm{Hom}(\mathbb{Z}^m,G)$ be the space of pairwise commuting $m$-tuples in $G$. We study the problem of which primes $p$ $\mathrm{Hom}(\mathbb{Z}^m,G)_1$, the connected component of $\mathrm{Hom}(\mathbb{Z}^m,G)$ containing the element $(1,\ldots,1)$, has $p$-torsion in homology. We will prove that $\mathrm{Hom}(\mathbb{Z}^m,G)_1$ for $m\ge 2$ has $p$-torsion in homology if and only if $p$ divides the order of the Weyl group of $G$ for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\mathrm{Hom}(\mathbb{Z}^m,G)_1$ and show that $\mathrm{Hom}(\mathbb{Z}^m,G)_1$ always has 2-torsion in homology whenever $G$ is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\mathrm{Hom}(\mathbb{Z}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

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arXiv2022

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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arXiv2022

Cellular cosheaf homology are cosheaf homology

A cosheaf is the dual notion of a sheaf, but we cannot define its homology as the formal dual of sheaf cohomology, in general, because of the lack of the cosheafification. A cellular cosheaf is a contravariant functor from the face poset of a CW complex to the category of abelian groups. We show that given a cellular cosheaf $F$, there is a natural way to associate a cosheaf $\widehat{F}$, for which we can define homology as the formal dual of sheaf cohomology, such that the Borel-Moore homology of $F$ is isomorphic to the homology of $\widehat{F}$ whenever the underlying CW complex of $F$ is a simplicial complex.

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arXiv2022

Homotopy commutativity in Hermitian symmetric spaces

Ganea proved that the loop space of $\mathbb{C}P^n$ is homotopy commutative if and only if $n=3$. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C}P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency of the loop spaces of flag manifolds.

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arXiv2021

Jacobi identity in polyhedral products

We show that a relation among minimal non-faces of a fillable complex $K$ yields an identity of iterated (higher) Whitehead products in a polyhedral product over $K$. In particular, for the $(n-1)$-skeleton of a simplicial $n$-sphere, we always have such an identity, and for the $(n-1)$-skeleton of a $(n+1)$-simplex, the identity is the Jacobi identity of Whitehead products ($n=1$) and Hardie's identity of higher Whitehead products ($n\ge 2$).

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arXiv2021

Hilbert bundles with ends

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 basis also defines unitary operators of finite propagation, and these operators preserve an end of a Hilbert space. Then, we can define a Hilbert bundle with end, which lightens up new structures of Hilbert bundles. In a special case, we can define characteristic classes of Hilbert bundles with ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with ends appear in natural contexts. First, we generalize the pushforward of a vector bundle along a finite covering to an infinite covering, which is a Hilbert bundle with end under a mild condition. Then we compute characteristic classes of some pushforwards along infinite coverings. Next, we will show the spectral decompositions of nice differential operators give rise to Hilbert bundles with ends, which elucidate new features of spectral decompositions. The spectral decompositions we will consider are the Fourier transform and the harmonic oscillators.

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arXiv2021

Homotopy types of gauge groups over Riemann surfaces

Let $G$ be a compact connected Lie group with $π_1(G)\cong\mathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of the moduli spaces of stable vector bundles over Riemann surfaces.

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arXiv2021

Golod and tight 3-manifolds

The notions Golodness and tightness for simplicial complexes come from algebra and geometry, respectively. We prove these two notions are equivalent for 3-manifold triangulations, through a topological characterization of a polyhedral product for a tight-neighborly manifold triangulation of dimension $\ge 3$.

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arXiv2021

The mod-$p$ homology of the classifying spaces of certain gauge groups

Let $G$ be a simply-connected, simple compact Lie group of type $\{n_{1},\ldots,n_{\ell}\}$, where $n_{1}\le\cdots \le n_{\ell}$. Let $\mathcal{G}_k$ be the gauge group of the principal $G$-bundle (namedright{P}{}{S^{4}}) whose isomorphism class is determined by the the second Chern class having value $k\in\mathbb{Z}$. We calculate the mod-$p$ homology of the classifying space $B\mathcal{G}_k$ provided that $n_{\ell}<p-1$ and $p$ does not divide $k$.

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arXiv2020

Relative phantom maps and rational homotopy

We generalize some results of Gray and McGibbon-Roitberg on relations between phantom maps and rational homotopy to relative phantom maps. Since the $\lim^1$ and the profinite completion techniques do not apply to relative phantom maps, we develop new techniques.

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arXiv2020

Two-dimensional Golod complexes

We characterize two-dimensional Golod complexes combinatorially by vertex-breakability and topologically by the fat-wedge filtration of a polyhedral product. Applying the characterization, we consider a difference between Golodness over fields and rings, which enables us to give a two-dimensional simple Golod complex over any field such that the corresponding moment-angle complex is not a suspension.

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arXiv2020

Monoids of self-maps of topological spherical space forms

A topological spherical space form is the quotient of a sphere by a free action of a finite group. In general, their homotopy types depend on specific actions of a group. We show that the monoid of homotopy classes of self-maps of a topological spherical space form is determined by the acting group and the dimension of the sphere, not depending on a specific action.

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arXiv2020

On the growth of topological complexity

Let $\mathrm{TC}_r(X)$ denote the $r$-th topological complexity of a space $X$. In many cases, the generating function $\sum_{r\ge 1}\mathrm{TC}_{r+1}(X)x^r$ is a rational function $\frac{P(x)}{(1-x)^2}$ where $P(x)$ is a polynomial with $P(1)=\mathrm{cat}(X)$, that is, the asymptotic growth of $\mathrm{TC}_r(X)$ with respect to $r$ is $\mathrm{cat}(X)$. In this paper, we introduce a lower bound $\mathrm{MTC}_r(X)$ of $\mathrm{TC}_r(X)$ for a rational space $X$, and estimate the growth of $\mathrm{MTC}_r(X)$.

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arXiv2020

Note on Samelson products in exceptional Lie groups

We determine (non-)triviality of Samelson products of inclusions of factors of the mod $p$ decomposition of $G_{(p)}$ for $(G,p)=(E_7,5),(E_7,7),(E_8,7)$. This completes the determination of (non-)triviality of those Samelson products in $p$-localized exceptional Lie groups when $G$ has $p$-torsion free homology.

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arXiv2020

Spaces of commuting elements in the classical groups

Let $G$ be the classical group, and let Hom$(\mathbb{Z}^m,G)$ denote the space of commuting $m$-tuples in $G$. First, we refine the formula for the Poincaré series of Hom$(\mathbb{Z}^m,G)$ due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincaré series, and apply it to prove the dependence of the topology of Hom$(\mathbb{Z}^m,G)$ on the parity of $m$ and the rational hyperbolicity of Hom$(\mathbb{Z}^m,G)$ for $m\ge 2$. Next, we give a minimal generating set of the cohomology of Hom$(\mathbb{Z}^m,G)$ and determine the cohomology in low dimensions. We apply these results to prove homological stability for Hom$(\mathbb{Z}^m,G)$ with the best possible stable range. Baird proved that the cohomology of Hom$(\mathbb{Z}^m,G)$ is identified with a certain ring of invariants of the Weyl group of $G$, and our approach is a direct calculation of this ring of invariants.

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