Mitsuaki Kimura
We generalize the result of Brandenbursky and Marcinkowski for the bounded cohomology of transformation groups to infinite volume case. To state the result, we introduce the notion of norm controlled cohomology as a generalization of bounded cohomology. This cohomology theory also provides a framework for relative quasimorphisms.
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We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In the appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed …
We introduce a norm on the real 1-cohomology of finite 2-complexes determined by the Euler characteristics of graphs on these complexes. We also introduce twisted Alexander-Fox polynomials of groups and show that they give rise to norms on the real 1-cohomology of groups. Our main theorem states that for a finite 2-complex X, the norm on H^1(X; R) determined by graphs on X majorates the Alexander-Fox norms derived from π_1(X).
Symplectic Groups, Mapping Class Groups and the Stability of Bounded Cohomology
本研究探讨映射类群与辛群的有界上同调稳定性，证明映射类群有界上同调不稳定、Mumford-Morita-Miller类稳定多项式无界，辛群2次有界上同调非等距稳定并给出相关范数计算与估计。
The aim of this paper is to discuss some applications of the relation between Seiberg-Witten theory and two natural norms defined on the first cohomology group of a closed 3-manifold N - the Alexander and Thurston norms. We start by giving a "new" proof of McMullen's inequality between these norms, and then use these norms to study two problems related to symplectic 4-manifolds of the form S^1xN. First we prove that - as long as N is irreducible - the unit balls of these norms are related in a way similar to the case of fibered 3-manifolds, supporting the conjecture that N is fibered. Second, …
We establish a new homological lower bound for the Thurston norm on 1-cohomology of 3-manifolds. This generalizes previous results of C. McMullen, S. Harvey, and the author. We also establish an analogous lower bound for 1-cohomology of 2-dimensional CW-complexes.
We study the relationship between two norms on the first cohomology of a hyperbolic 3-manifold: the purely topological Thurston norm and the more geometric harmonic norm. Refining recent results of Bergeron, Şengün, and Venkatesh as well as older work of Kronheimer and Mrowka, we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3-manifold itself. Moreover, we give families of examples showing that some (but not all) qualitative aspects of our estimates are sharp. Finally, we exhibit closed hyperbolic 3-…
Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic $3$-Manifolds
本文用Thurston范数界定双曲3-流形上L²调和1-形式的L²-范数，推广Brock-Dunfield不等式，并研究其锐性，统一结果并提出猜想。
In this paper we define, for each aspherical orientable 3-manifold $M$ endowed with a \emph{torus splitting} $Ţ$, a 2-dimensional fundamental $l_1$-class $[M]^{Ţ}$ whose $l_1$-norm has similar properties as the Gromov simplicial volume of $M$ (additivity under torus splittings and isometry under finite covering maps). Next, we use the Gromov simplicial volume of $M$ and the $l_1$-norm of $[M]^{Ţ}$ to give a complete characterization of those nonzero degree maps $f\co M\to N$ which are homotopic to a ${\rm deg}(f)$-covering map. As an application we characterize those degree one maps $f\co M\to…
Johnson's characterization of amenable groups states that a discrete group $Γ$ is amenable if and only if $H_b^{n \geq 1}(Γ; V) = 0$ for all dual normed $\mathbb{R}[Γ]$-modules V. In this paper, we extend the previous result to homomorphisms by proving the converse of the Mapping Theorem: a surjective group homomorphism $ϕ\colon Γ\to K$ has amenable kernel H if and only if the induced inflation map $H^\bullet_b(K; V^H) \to H^\bullet_b(Γ; V)$ is an isometric isomorphism for every dual normed $\mathbb{R}[Γ]$-module V. In addition, we obtain an analogous characterization for the (smaller) class o…
The bounded cohomology of transformation groups of Euclidean spaces and discs
本文证明了 $\mathbb{R}^n$ 的保向同胚群和微分同胚群在所有正则性下均有界无环，这是对非紧支撑变换群有界上同调的首次完整计算，并据此得出平坦丛的许多示性类是无界的。
In 1986, W. Thurston introduced a (possibly degenerate) norm on the first cohomology group of a 3-manifold. Inspired by this definition, Turaev introduced in 2002 a analogous norm on the first cohomology group of a finite 2-complex. We show that if N is the exterior of a link in a rational homology sphere, then the Thurston norm agrees with a suitable variation of Turaev's norm defined on any 2-skeleton of N.
Multiplicative relations in the cohomology ring of a manifold impose constraints upon its stable systoles. Given a compact Riemannian manifold (X,g), its real homology H_*(X,R) is naturally endowed with the stable norm. Briefly, if h\in H_k(X,R) then the stable norm of h is the infimum of the Riemannian k-volumes of real cycles representing h. The stable k-systole is the minimum of the stable norm over nonzero elements in the lattice of integral classes in H_k(X,R). Relying on results from the geometry of numbers due to W. Banaszczyk, and extending work by M. Gromov and J. Hebda, we prove metr…
The canonical lamination calibrated by a cohomology class
本文研究了由上同调类校准的规范分层，揭示了稳定范数与地震范数之间的几何类比。
This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometri…
For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H^{<=2}(X), to the second nilpotent quotient, G/G_3. We define invariants of G/G_3 by counting normal subgroups of a fixed prime index p, according to their abelianization. We show how to compute this distribution from the resonance varieties of the Orlik-Solomon algebra mod p. As an application, we establish the cohomology classification of 2-arrangements of n<=6 planes in R^4.
Heaps are para-associative ternary operations bijectively exemplified by groups via the operation $(x,y,z) \mapsto x y^{-1} z$. They are also ternary self-distributive, and have a diagrammatic interpretation in terms of framed links. Motivated by these properties, we define para-associative and heap cohomology theories and also a ternary self-distributive cohomology theory with abelian heap coefficients. We show that one of the heap cohomologies is related to group cohomology via a long exact sequence. Moreover we construct maps between second cohomology groups of normalized group cohomology a…
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