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Ryuma Orita

Publications and source records attributed to Ryuma Orita.

12 recordsLinked to original sources
arXiv2026

Maximal topological complexity of monotone symplectic 4-manifolds

We continue the study of Farber's topological complexity for monotone symplectic manifolds initiated in \cite{Or25}. First, we show that a closed spherically monotone symplectic manifold whose fundamental group contains no subgroup isomorphic to $\ZZ\oplus\ZZ$ is automatically toroidally monotone, with the same monotonicity constant. As a consequence, every closed $4$-dimensional spherically monotone symplectic manifold whose Kodaira dimension is not $-\infty$ and whose fundamental group contains no $\ZZ\oplus\ZZ$ (for instance, is Gromov hyperbolic) has maximal topological complexity $\TC(M)=9$. This settles, under strictly weaker hypotheses, the dichotomy $\TC(M)\in\{8,9\}$ left open there. Second, we compute the topological complexity and the Lusternik--Schnirelmann category of all blowups of $S^2$-bundles over closed orientable surfaces of genus $g\geq 2$: they satisfy $\cat(M)=4$ and $\TC(M)=7$. In particular, the hypothesis on the Kodaira dimension in the first result cannot be removed, and closed symplectic $4$-manifolds realize the pairs $(\cat(M),\TC(M))=(3,5)$, $(4,7)$, $(5,9)$ in the three regimes considered in this paper. Throughout, $\TC$ and $\cat$ are taken in the unreduced convention.

math.AT↗
arXiv2025

Morse-Bott-Smale chain complex

Banyaga and Hurtubise defined the Morse-Bott-Smale chain complex as a quotient of a large chain complex by introducing five degeneracy relations. However, their five degeneracy relations are in fact redundant. In the present paper, we unify these five conditions into a single degeneracy condition and resolve the issue of the well-definedness of the Morse-Bott-Smale chain complex. This provides an appropriate definition of the Morse-Bott homology and more computable examples. Moreover, we show that our chain complex for a Morse-Smale function is quasi-isomorphic to the usual Morse-Smale-Witten chain complex. As a consequence, we obtain an alternative proof of the Morse Homology Theorem.

math.AT↗
arXiv2024

Relative simplicity of the universal coverings of transformation groups and Tsuboi's metric

Many transformation groups on manifolds are simple, but their universal coverings are not. In the present paper, we study the concept of relatively simple group, that is, a group with the maximum proper normal subgroup. We show that many examples of universal coverings of transformation groups are relatively simple, including the universal covering $\widetilde{\mathrm{Ham}}(M,\omega)$ of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,\omega)$. Tsuboi constructed a metric space $\mathcal{M}(G)$ for a simple group $G$. We generalize his construction to relatively simple groups, and study their large scale geometric structure. In particular, Tsuboi's metric space of $\widetilde{\mathrm{Ham}}(M, \omega)$ is not quasi-isometric to the half line for every closed symplectic manifold $(M,\omega)$.

math.GR↗
arXiv2023

Floer-type bipersistence modules and rectangle barcodes

We study the bipersistence modules obtained from the action-window homology of Floer-type complexes over $\Lambda^{\mathbb{F},\{0\}}=\mathbb{F}$. We prove that these modules are rectangle-decomposable and establish an explicit dictionary between their graded rectangle barcodes and the classical one-parameter barcodes. This dictionary is an isometry for the bottleneck distance. Consequently, the interleaving and bottleneck distances coincide on this class of bipersistence modules: the optimal constant in algebraic stability improves from $3$ for general rectangle-decomposable $\mathbb{R}^2$-indexed modules to $1$, even though the rectangles that occur are not blocks. The resulting rectangle barcodes depend $1$-Lipschitz-continuously on the Morse function in the uniform norm and on the Hamiltonian diffeomorphism in Hofer's metric. Their corner multiplicities record critical points and one-periodic Hamiltonian orbits degree by degree and filtration level by filtration level, while their geometry displays spectral invariants, boundary depth, and spectral spreads. Thus the rectangle barcode gives a geometric presentation, rather than a refinement, of the classical barcode.

math.SG↗
arXiv2023

Topological complexity of monotone symplectic manifolds

We study Farber's topological complexity for monotone symplectic manifolds. More precisely, we estimate the topological complexity of 4-dimensional spherically monotone manifolds whose Kodaira dimension is not $-\infty$.

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arXiv2020

Existence of pseudo-heavy fibers of moment maps

In the present paper, we introduce the notion of pseudo-heaviness of closed subsets of closed symplectic manifolds and prove the existence of pseudo-heavy fibers of moment maps. In particular, we generalize Entov and Polterovich's theorem, which ensures the existence of non-displaceable fibers, and provide a partial answer to a problem posed by them, which asks the existence of heavy fibers. Moreover, we apply our results to prove that some generalized coupled angular momenta have more than two non-displaceable fibers.

math.SG↗
arXiv2020

Rigid fibers of spinning tops

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show that a singular level set of a convex Hamiltonian is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.

math.SG↗
arXiv2019

Disjoint superheavy subsets and fragmentation norms

We present a lower bound for a fragmentation norm and construct a bi-Lipschitz embedding $I\colon \mathbb{R}^n\to\mathrm{Ham}(M)$ with respect to the fragmentation norm on the group $\mathrm{Ham}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $(M,ω)$. As an application, we provide an answer to Brandenbursky's question on fragmentation norms on $\mathrm{Ham}(Σ_g)$, where $Σ_g$ is a closed Riemannian surface of genus $g\geq 2$

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arXiv2017

Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories

The first author introduced a relative symplectic capacity $C$ for a symplectic manifold $(N,ω_N)$ and its subset $X$ which measures the existence of non-contractible periodic trajectories of Hamiltonian isotopies on the product of $N$ with the annulus $A_R=(R,R)\times\mathbb{R}/\mathbb{Z}$. In the present paper, we give an exact computation of the capacity $C$ of the $2n$-torus $\mathbb{T}^{2n}$ relative to a Lagrangian submanifold $\mathbb{T}^n$ which implies the existence of non-contractible Hamiltonian periodic trajectories on $A_R\times\mathbb{T}^{2n}$. Moreover, we give a lower bound on the number of such trajectories.

math.SG↗
arXiv2017

On the existence of infinitely many non-contractible periodic orbits of Hamiltonian diffeomorphisms of closed symplectic manifolds

We show that the presence of a non-contractible one-periodic orbit of a Hamiltonian diffeomorphism of a connected closed symplectic manifold $(M,\omega)$ implies the existence of infinitely many non-contractible simple periodic orbits, provided that the symplectic form $\omega$ is aspherical and the fundamental group $\pi_1(M)$ is either a virtually abelian group or an $\mathrm{R}$-group. We also show that a similar statement holds for Hamiltonian diffeomorphisms of closed monotone or negative monotone symplectic manifolds under the same conditions on their fundamental groups. These results generalize some works by Ginzburg and G\"urel. The proof uses the filtered Floer--Novikov homology for non-contractible periodic orbits.

math.SG↗