By a classical result of Schafer and Kahn ([12], [6]), oriented topological sphere bundles have well defined rational Pontrjagin classes. In the smooth case, we show that the corresponding Pontrjagin character is proportional to the higher Franz-Reidemeister torsion invariant in each degree when the fiber is even dimensional and we discuss the relationship in the odd dimensional case.

In this paper, we study a series of $L^2$-torsion invariants from the viewpoint of the mapping class group of a surface. We establish some vanishing theorems for them. Moreover we explicitly calculate the first two invariants and compare them with hyperbolic volumes.

In this note we introduce parameterized Gromov-Witten invariants for symplectic fiber bundles and study the topology of the symplectomorphism group. We also give sample applications showing the non-triviality of certain homotopy groups of some symplectomorphism groups.

Recent projects on pronilpotent and profinite completions of surface groups are discussed, and common threads are then compared in the context of general group completions. In particular, a profinite version of the Torelli groups and a pronilpotent version of the punctured solenoid are introduced.

The Torelli group and the Johnson kernel are important subgroups of the mapping class group of a surface. Using Johnson's homomorphisms as their free abelian quotients together with the representation theory of the symplectic groups, we will give some descriptions of cup products of rational cohomology classes of degree one obtained from the dual of Johnson's homomorphisms.

We provide affirmative evidences for conjectural mod $p$ Riemann-Roch formulae for mapping class groups by considering Steenrod operations on mod $p$ Morita-Mumford classes.

Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping class group to $\frac{1}{2} \wedge^3 H \rtimes \mathrm{S_p}(H)$. This Johnson-Morita homomorphism is not surjective, but its image is finite index in $\frac{1}{2} \wedge^3 H \rtimes \mathrm{S_p}(H)$ [11]. Here we give a description of the exact image of Morita's homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.

A group is said to be bounded if it has a finite diameter with respect to any bi-invariant metric. In the present paper we discuss boundedness of various groups of diffeomorphisms.

Chakiris [5] constructed examples of holomorphic Lefschetz fibrations of genus 2 with separating singular fibers and proved a classification theorem for such fibrations in the late 1970's. We generalize some parts of his construction topologically to give new examples of hyperelliptic Lefschetz fibrations of arbitrary genus with separating singular fibers which include homeomorphic but non-diffeomorphic 4-manifolds.

Suppose $G$ acts acylindrically by isometries on a $\delta$-hyperbolic graph $\Gamma$. We discuss subgroups generated by two hyperbolic elements in $G$ and give sufficient conditions for them to be free of rank two.

We apply our results to the mapping class group $\mathrm{Mod}(S)$ of a compact orientable surface $S$ and its action on the curve graph such that $S$ is non-sporadic. There exists a constant $Q$, depending only on $S$, with the following property. If $a, b \in \mathrm{Mod}(S)$ are pseudo-Anosovs such that $\langle a, b \rangle$ is not virtually cyclic, then there exists $M \gt 0$, which depends on $a, b$, such that either $\langle a^n, b^m \rangle$ is free of rank two for all $n \ge Q, m \ge M$, or $\langle a^m, b^n \rangle$ is free of rank two for all $n \ge Q, m \ge M$ (Theorem 3.1).

At the end we ask a question in connection to the uniformly exponential growth of subgroups in a mapping class group (Question 3.4).

The groups of differential characters of Cheeger and Simons admit a natural multiplicative structure. The map given by the squares of degree $2k$ differential characters reduces to a homomorphism of ordinary cohomology groups. We prove that the homomorphism factors through the Steenrod squaring operation of degree $2k$. A simple application shows that five-dimensional Chern-Simons theory for pairs of $B$-fields is $SL(2, \mathbb{Z})$-invariant on spin manifolds.

We consider the linear representations of the mapping class group of an $n$–punctured 2–sphere constructed by V. F. R. Jones using Iwahori–Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori–Hecke algebra representation of Artin's braid group of $n-1$ strands. In the case of $n = 6$, we provide a further restriction for the kernel using our previous result, as well as a certain relation to the Burau representation of degree 4.

Let $\Sigma_{g,1}$ be an oriented compact surface of genus $g$ with 1 boundary component, and $\Gamma_{g,1}$ the mapping class group of $\Sigma_{g,1}$. We determine the stable cohomology group of $\Gamma_{g,1}$ with coefficients in $H^1 (\Sigma_{g ,1} ; \mathbb{Z})^{\otimes n}$, $n \ge 1$, explicitly modulo the stable cohomology group with trivial coefficients. As a corollary the rational stable cohomology algebra of the semi-direct product $\Gamma_{g,1} \ltimes H_1 (\Sigma_{g,1} ; \mathbb{Z})$ (which we call the extended mapping class group) is proved to be freely generated by the generalized Morita-Mumford classes $\widetilde{m_{i,j}}$'s $(i \ge 0,\, j \ge 1,\, i+j \ge 2)$ [11] over the rational stable cohomology algebra of the group $\Gamma_{g,1}$.

We show that the stable commutator length vanishes for certain groups defined as infinite unions of smaller groups. The argument uses a group-theoretic analogue of the Mazur swindle, and goes back to the works of Anderson, Fisher, and Mather on homeomorphism groups.

We introduce an easily tractable cohomological criterion for the existence of 2-dimensional foliations with a prescribed compact leaf on a 4-manifold relying on standard methods, Milnor's inequality for the existence of a flat connection on an $\mathbb{R}^2$-bundle over a surface, and Thurston's $h$-principle. This is used to investigate the self-intersection numbers of compact leaves of foliations on the product of two surfaces, in particular the question whether these numbers are bounded on a given 4-manifold.

We describe the group version of the trace maps given in [29]. This gives rise to abelian quotients of symplectic IA-automorphism groups of nilpotent quotients of the fundamental groups of compact surfaces. By making use of them, we construct a representation of the group $\mathcal{H}_{g,1}$ of homology cobordism classes of homology cylinders introduced by Garoufalidis and Levine [6]. We define various cohomology classes of $\mathcal{H}_{g,1}$ and propose a few problems concerning them. In particular, we mention a possible relation to additive invariants for the group $\Theta_{\mathbb{Z}}^{3}$ of homology cobordism classes of homology 3-spheres.

From our previous paper, it is known that the Magnus representation of the Torelli group is not faithful. In this paper, we show a certain kind of elements in the kernel of the representation.

We consider the group $\mathrm{Cont}_c^r (M^{2n+1}, \alpha)$ of $C^r$ contactomorphisms with compact support of a contact manifold $(M^{2n+1}, \alpha)$ of dimension $(2n+1)$ with the $C^r$ topology. We show that the first homology group of the classifying space $B \overline{\mathrm{Cont}}{}_c^r (M^{2n+1}, \alpha)$ for the $C^r$ foliated $M^{2n+1}$ products with compact support with transverse contact structure $\alpha$ is trivial for $1 \le r \lt n+(3/2)$. This implies that the identity component $\mathrm{Cont}_c^r (M^{2n+1}, \alpha)_0$ of the group $\mathrm{Cont}_c^r (M^{2n+1}, \alpha)$ of contactomorphisms with compact support of a connected contact manifold $(M^{2n+1}, \alpha)$ is a simple group for $1 \le r \lt n+(3/2)$.

We show that any element of the identity component of the group of $C^r$ diffeomorphisms $\mathrm{Diff}_c^r (\boldsymbol{R}^n)_0$ of the $n$-dimensional Euclidean space $\boldsymbol{R}^n$ with compact support $(1 \leqq r \leqq \infty,\ r \ne n+1)$ can be written as a product of two commutators. This statement holds for the interior $M^n$ of a compact $n$-dimensional manifold which has a handle decomposition only with handles of indices not greater than $(n-1)/2$. For the group $\mathrm{Diff}^r (M)$ of $C^r$ diffeomorphisms of a compact manifold $M$, we show the following for its identity component $\mathrm{Diff}^r (M)_0$. For an even-dimensional compact manifold $M^{2m}$ with handle decomposition without handles of the middle index $m$, any element of $\mathrm{Diff}^r (M^{2m})_0$ $(1 \leqq r \leqq \infty,\ r \ne 2m+1)$ can be written as a product of four commutators. For an odd-dimensional compact manifold $M^{2m+1}$, any element of $\mathrm{Diff}^r (M^{2m+1})_0$ $(1 \leqq r \leqq \infty,\ r \ne 2m+2)$ can be written as a product of six commutators.