417,391 research outputs found

### 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

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