A characteristic class of Homeo(X)0\mathrm{Homeo(X)_0}-bundles and an abelian extension of the homeomorphism group

A $\mathrm{Homeo(X)_0}$-bundle is a fiber bundle with fiber $X$ whose structure group reduces to the identity component $\mathrm{Homeo(X)_0}$ of the homeomorphism group of $X$. We construct a characteristic class of $\mathrm{Homeo(X)_0}$-bundles as a third cohomology class with coefficients in $\mathbb{Z}$. We also investigate the relation between the universal characteristic class of flat fiber bundles and the gauge group extension of the homeomorphism group. Furthermore, under some assumptions, we construct and study the central $S^1$-extension and the corresponding group two-cocycle of $\mathrm{Homeo(X)_0}$.Comment: 15 page

The translation number and quasi-morphisms on groups of symplectomorphisms of the disk

On groups of symplectomorphisms of the disk, we construct two homogeneous quasi-morphisms which relate to the Calabi invariant and the flux homomorphism respectively. We also show the relation between the quasi-morphisms and the translation number introduced by Poincar\'{e}.Comment: 9 pages, to appear in Ann. Inst. Fourier (Grenoble

THE FLUX HOMOMORPHISM AND CENTRAL EXTENSIONS OF DIFFEOMORPHISM GROUPS

Let D be a closed unit disk in dimension two and G_＜rel＞ the group of symplectomorphisms on D preserving the origin and the boundary ∂D pointwise. We consider the flux homomorphism on G_＜rel＞ and construct a central R-extension called the flux extension. We determine the Euler class of this extension and investigate the relation among the extension, the group 2-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of D

The space of non-extendable quasimorphisms

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.Comment: 58 pages, 1 figure. Major revision. Theorem 1.12 in v3 has been generalized to Theorem 1.2 in the current version: this new theorem treats hyperbolic mapping tori in general cases, and it serves as a leading application of our main theore

Survey on invariant quasimorphisms and stable mixed commutator length

In this survey, we review the history and recent developments of invariant quasimorphisms and stable mixed commutator length.Comment: 26 pages, 1 figure; minor revisio