Cohomology of symplectic groups and Meyer's signature theorem

Meyer showed that the signature of a closed oriented surface bundle over a surface is a multiple of $4$, and can be computed using an element of $H^2(\mathsf{Sp}(2g, \mathbb{Z}),\mathbb{Z})$. Denoting by $1 \to \mathbb{Z} \to \widetilde{\mathsf{Sp}(2g,\mathbb{Z})} \to \mathsf{Sp}(2g,\mathbb{Z}) \to 1$ the pullback of the universal cover of $\mathsf{ Sp}(2g,\mathbb{R})$, Deligne proved that every finite index subgroup of $\widetilde{\mathsf {Sp}(2g, \mathbb{Z})}$ contains $2\mathbb{Z}$. As a consequence, a class in the second cohomology of any finite quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ can at most enable us to compute the signature of a surface bundle modulo $8$. We show that this is in fact possible and investigate the smallest quotient of $\mathsf{Sp}(2g, \mathbb{Z})$ that contains this information. This quotient $\mathfrak{H}$ is a non-split extension of $\mathsf {Sp}(2g,2)$ by an elementary abelian group of order $2^{2g+1}$. There is a central extension $1\to \mathbb{Z}/2\to\tilde{{\mathfrak{H}}}\to\mathfrak{H}\to 1$, and $\tilde{\mathfrak{H}}$ appears as a quotient of the metaplectic double cover $\mathsf{Mp}(2g,\mathbb{Z})=\widetilde{\mathsf{Sp}(2g,\mathbb{Z})}/2\mathbb{Z}$. It is an extension of $\mathsf{Sp}(2g,2)$ by an almost extraspecial group of order $2^{2g+2}$, and has a faithful irreducible complex representation of dimension $2^g$. Provided $g\ge 4$, $\widetilde{\mathfrak{H}}$ is the universal central extension of $\mathfrak{H}$. Putting all this together, we provide a recipe for computing the signature modulo $8$, and indicate some consequences.Comment: 18 pages. Minor corrections. The most important one is in the table for $g=1$ on page 16: two columns had been swapped in the previous version. This is the version accepted for publication in Algebraic and Geometric Topolog

On an action of the braid group B_{2g+2} on the free group F_{2g}

We construct an action of the braid group B_{2g+2} on the free group F_{2g} extending an action of B_4 on F_2 introduced earlier by Reutenauer and the author. Our action induces a homomorphism from B_{2g+2} into the symplectic modular group Sp_{2g}(Z). In the special case g=2 we show that the latter homomorphism is surjective and determine its kernel, thus obtaining a braid-like presentation of Sp_4(Z).Comment: 11 pages. Minor changes in v

Projective Normality Of Algebraic Curves And Its Application To Surfaces

Let $L$ be a very ample line bundle on a smooth curve $C$ of genus $g$ with $\frac{3g+3}{2}<\deg L\le 2g-5$. Then $L$ is normally generated if $\deg L>\max\{2g+2-4h^1(C,L), 2g-\frac{g-1}{6}-2h^1(C,L)\}$. Let $C$ be a triple covering of genus $p$ curve $C'$ with $C\stackrel{\phi}\to C'$ and $D$ a divisor on $C'$ with $4p<\deg D< \frac{g-1}{6}-2p$. Then $K_C(-\phi^*D)$ becomes a very ample line bundle which is normally generated. As an application, we characterize some smooth projective surfaces.Comment: 7 pages, 1figur

Splittings of knot groups

Let K be a knot of genus g. If K is fibered, then it is well known that the knot group pi(K) splits only over a free group of rank 2g. We show that if K is not fibered, then pi(K) splits over non-free groups of arbitrarily large rank. Furthermore, if K is not fibered, then pi(K) splits over every free group of rank at least 2g. However, pi(K) cannot split over a group of rank less than 2g. The last statement is proved using the recent results of Agol, Przytycki-Wise and Wise.Comment: 28 pages, 2 figure