arXiv subjects
Publications and source records attributed to Takahiro Matsushita.
For each integer $k \ge 2$, we prove that an abelian group $Γ$ satisfies $\exp(Γ) \in \{1,\dots,k\} \cup \{2,4,\dots,2k\}$ if and only if it admits no non-bipartite Cayley graph whose circular chromatic number is strictly less than $2 + \frac{2}{k-1}$. This refines the algebraic characterization by Krebs and Sankar of abelian groups that admit $3$-chromatic Cayley graphs.
In 1996, Youngs proved that any quadrangulation of the real projective plane is not 3-chromatic. This result has been extended in various directions over the years, including to other non-orientable closed surfaces, higher-dimensional analogues of quadrangulations and circular colorings. In this paper, we provide a generalization which yields some of these extensions of Youngs' theorem.
Let $G$ be a group and $N$ its normal subgroup. On the mixed commutator subgroup $[G,N]$, the mixed stable commutator length $\mathrm{scl}_{G,N}$ and the restriction of the ordinary stable commutator length $\mathrm{scl}_{G}$ are defined. We characterize when they are bi-Lipschitz equivalent by the vanishing of a certain $\mathbb{R}$-linear space $\mathrm{W}(G,N)$ related to invariant quasimorphisms. For the proof, we obtain a refined version of the generalized mixed Bavard duality theorem, and perform functional analysis on the completion of a certain space of $1$-chains.
The Hom complex $\mathrm{Hom}(G, H)$ of graphs is a simplicial complex associated to a pair of graphs $G$ and $H$, and its homotopy type is of interest in the graph coloring problem and the homomorphism reconfiguration problem. In this paper, we show that if $G$ is a connected graph and $H$ is a square-free connected graph, then every connected component of $\mathrm{Hom}(G, H)$ is homotopy equivalent to a point, a circle, $H$ or a connected double cover over $H$. We also obtain a certain relation between the fundamental group of $\mathrm{Hom}(G,H)$ and realizable walks studied in the homomorphism reconfiguration problem.
The closed neighborhood complex $\mathcal{N}[G]$ of a simple graph $G$ is the simplicial complex whose simplices are finite sets of vertices contained in a closed neighborhood of a vertex in $G$. We reveal that the closed neighborhood complex has close connections with other concepts, including the independence complex of the canonical double covering and the independence complex of the neighborhood hypergraph. Furthermore, we show that the fundamental group of the closed neighborhood complex is isomorphic to Grigor'yan--Lin--Muranov--Yau's fundamental group of a graph introduced in the study of path homology.
This article provides an expository account of the celebrated duality theorem of Bavard and three its strengthenings. The Bavard duality theorem connects scl (stable commutator length) and quasimorphisms on a group. Calegari extended the framework from a group element to a chain on the group, and established the generalized Bavard duality. Kawasaki, Kimura, Matsushita and Mimura studied the setting of a pair of a group and its normal subgroup, and obtained the mixed Bavard duality. The first half of the present article is devoted to an introduction to these three Bavard dualities. In the latter half, we present a new strengthening, the generalized mixed Bavard duality, and provide a self-contained proof of it. This third strengthening recovers all of the Bavard dualities treated in the first half; thus, we supply complete proofs of these four Bavard dualities in a unified manner. In addition, we state several results on the space $\mathrm{W}(G,N)$ of non-extendable quasimorphisms, which is related to the comparison problem between scl and mixed scl via the mixed Bavard duality.
We show a criterion for a skeleton of a manifold triangulation being embeddable into Euclidean space in terms of the complement of a submanifold. As an application, we obtain embeddability of a $(q-1)$-skeleton of a triangulation of an $S^p$-bundle over $S^q$ into $\mathbb{R}^{p+q}$.
Let $G$ be a group and $N$ a normal subgroup of $G$. We study the large scale behavior, not the exact values themselves, of the stable mixed commutator length $scl_{G,N}$ on the mixed commutator subgroup $[G,N]$; when $N=G$, $scl_{G,N}$ equals the stable commutator length $scl_G$ on the commutator subgroup $[G,G]$. For this purpose, we regard $scl_{G,N}$ not only as a function from $[G,N]$ to $\mathbb{R}_{\geq 0}$, but as a bi-invariant metric function $d^+_{scl_{G,N}}$ from $[G,N]\times [G,N]$ to $\mathbb{R}_{\geq 0}$. Our main focus is coarse group theoretic structures of $([G,N],d^+_{scl_{G,N}})$. Our preliminary result (the absolute version) connects, via the Bavard duality, $([G,N],d^+_{scl_{G,N}})$ and the quotient vector space of the space of $G$-invariant quasimorphisms on $N$ over one of such homomorphisms. In particular, we prove that the dimension of this vector space equals the asymptotic dimension of $([G,N],d^+_{scl_{G,N}})$. Our main result is the comparative version: we connect the coarse kernel, formulated by Leitner and Vigolo, of the coarse homomorphism $ι_{G,N}\colon ([G,N],d^+_{scl_{G,N}})\to ([G,N],d^+_{scl_{G}})$; $y\mapsto y$, and a certain quotient vector space $W(G,N)$ of the space of invariant quasimorphisms. Assume that $N=[G,G]$ and that $W(G,N)$ is finite dimensional with dimension $\ell$. Then we prove that the coarse kernel of $ι_{G,N}$ is isomorphic to $\mathbb{Z}^{\ell}$ as a coarse group. In contrast to the absolute version, the space $W(G,N)$ is finite dimensional in many cases, including all $(G,N)$ with finitely generated $G$ and nilpotent $G/N$. As an application of our result, given a group homomorphism $φ\colon G\to H$ between finitely generated groups, we define an $\mathbb{R}$-linear map `inside' the groups, which is dual to the naturally defined $\mathbb{R}$-linear map from $W(H,[H,H])$ to $W(G,[G,G])$ induced by $φ$.
Given a closed connected symplectic manifold $(M,ω)$, we construct an alternating $\mathbb{R}$-bilinear form $\mathfrak{b}=\mathfrak{b}_{μ_{\mathrm{Sh}}}$ on the real first cohomology of $M$ from Shelukhin's quasimorphism $μ_{\mathrm{Sh}}$. Here $μ_{\mathrm{Sh}}$ is defined on the universal cover of the group of Hamiltonian diffeomorphisms on $(M,ω)$. This bilinear form is invariant under the symplectic mapping class group action, and $\mathfrak{b}$ yields a constraint on the fluxes of commuting two elements in the group of symplectomorphisms on $(M,ω)$. These results might be seen as an analog of Rousseau's result for an open connected symplectic manifold, where he recovered the symplectic pairing from the Calabi homomorphism. Furthermore, $\mathfrak{b}$ controls the extendability of Shelukhin's quasimorphisms, as well as the triviality of a characteristic class of Reznikov. To construct $\mathfrak{b}$, we build general machinery for a group $G$ of producing a real-valued $\mathbb{Z}$-bilinear form $\mathfrak{b}_μ$ from a $G$-invariant quasimorphism $μ$ on the commutator subgroup of $G$.
Shelukhin constructed a quasimorphism on the universal covering of the group of Hamiltonian diffeomorphisms for a general closed symplectic manifold. In the present paper, we prove the non-extendability of that quasimorphism for certain symplectic manifolds, such as a blow-up of torus and the product of a surface of genus at least two and a closed symplectic manifold. As its application, we prove the non-vanishing of Reznikov's characteristic class for the above symplectic manifolds.
In his work on molecular spaces, Ivashchenko introduced the notion of an $\mathfrak{I}$-contractible transformation on a graph $G$, a family of addition/deletion operations on its vertices and edges. Chen, Yau, and Yeh used these operations to define the $\mathfrak{I}$-homotopy type of a graph, and showed that $\mathfrak{I}$-contractible transformations preserve the simple homotopy type of $C(G)$, the clique complex of $G$. In other work, Boulet, Fieux, and Jouve introduced the notion of $s$-homotopy of graphs to characterize the simple homotopy type of a flag simplicial complex. They proved that $s$-homotopy preserves $\mathfrak{I}$-homotopy, and asked whether the converse holds. In this note, we answer their question in the affirmative, concluding that graphs $G$ and $H$ are $\mathfrak{I}$-homotopy equivalent if and only if $C(G)$ and $C(H)$ are simple homotopy equivalent. We also show that a finite graph $G$ is $\mathfrak{I}$-contractible if and only if $C(G)$ is contractible, which answers a question posed by the first author, Espinoza, Frías-Armenta, and Hernández. We use these ideas to give a characterization of simple homotopy for arbitrary simplicial complexes in terms of links of vertices.
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 $\widetilde{\mathrm{Ham}}(M,ω)$ of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M,ω)$. 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 $\widetilde{\mathrm{Ham}}(M, ω)$ is not quasi-isometric to the half line for every closed symplectic manifold $(M,ω)$.
We show that the dominance complex $\mathcal{D}(G)$ of a graph $G$ coincides with the combinatorial Alexander dual of the neighborhood complex $\mathcal{N}(\overline{G})$ of the complement of $G$. Using this, we obtain a relation between the chromatic number $χ(G)$ of $G$ and the homology group of $\mathcal{D}(G)$. We also obtain several known results related to dominance complexes from well-known facts of neighborhood complexes. After that, we suggest a new method for computing the homology groups of the dominance complexes, using independence complexes of simple graphs. We show that several known computations of homology groups of dominance complexes can be reduced to known computations of independence complexes. Finally, we determine the homology group of $\mathcal{D}(P_n \times P_3)$ by determining the homotopy types of the independence complex of $P_n \times P_3 \times P_2$.
A homogeneous quasimorphism $ϕ$ on a normal subgroup $N$ of $G$ is said to be $G$-invariant if $ϕ(gxg^{-1}) = ϕ(x)$ for every $g \in G$ and for every $x \in N$. Invariant quasimorphisms have naturally appeared in symplectic geometry and the extension problem of quasimorphisms. Moreover, it is known that the existence of non-extendable invariant quasimorphisms is closely related to the behavior of the stable mixed commutator length $\mathrm{scl}_{G,N}$, which is a certain generalization of the stable commutator length $\mathrm{scl}_G$. In this survey, we review the history and recent developments of invariant quasimorphisms and stable mixed commutator length. The topics we treat include several examples of invariant quasimorphisms, Bavard's duality theorem for invariant quasimorphisms, Aut-invariant quasimorphisms, and the estimation of the dimension of spaces of non-extendable quasimorphisms. We also mention the extension problem of partial quasimorphisms.
The van Kampen-Flores theorem states that the $d$-skeleton of a $(2d+2)$-simplex does not embed into $\mathbb{R}^{2d}$. We prove the van Kampen-Flores theorem for triangulations of manifolds satisfying a certain condition on their Stiefel-Whitney classes. In particular, we show that the $d$-skeleton of a triangulation of a $(2d+1)$-manifold with non-trivial total Stiefel-Whitney class does not embed into $\mathbb{R}^{2d}$.
For a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the space of quasimorphisms and quasi-cocycles on $N$ non-extendable to $G$. To treat this space, we establish the five-term exact sequence of cohomology relative to the bounded subcomplex. As its application, we study the spaces associated with the kernel of the (volume) flux homomorphism, the IA-automorphism group of a free group, and certain normal subgroups of Gromov-hyperbolic groups. Furthermore, we employ this space to prove that the stable commutator length is equivalent to the stable mixed commutator length for certain pairs of a group and its normal subgroup.
The van Kampen-Flores theorem states that the $n$-skeleton of a $(2n+2)$-simplex does not embed into $\mathbb{R}^{2n}$. We give two proofs for its generalization to a continuous map from a skeleton of a certain regular CW complex (e.g. a simplicial sphere) into a Euclidean space. We will also generalize Frick and Harrison's result on the chirality of embeddings of the $n$-skeleton of a $(2n+2)$-simplex into $\mathbb{R}^{2n+1}$.
Let $(S,ω)$ be a closed connected oriented surface whose genus $l$ is at least two equipped with a symplectic form. Then we show the vanishing of the cup product of the fluxes of commuting symplectomorphisms. This result may be regarded as an obstruction for commuting symplectomorphisms. In particular, the image of an abelian subgroup of $\mathrm{Symp}_0^c(S, ω)$ under the flux homomorphism is isotropic with respect to the natural intersection form on $H^1(S;\mathbb{R})$. The key to the proof is a refinement of the non-extendability result, previously given by the first-named and second-named authors, for Py's Calabi quasimorphism $μ_P$ on $\mathrm{Ham}(S, ω)$.