  In this paper we define a $\mathbf{QP}^1$-valued class function on the
mapping class group $\mathcal{M}_{g,2}$ of a surface $\Sigma_{g,2}$ of genus
$g$ with two boundary components. Let $E$ be a $\Sigma_{g,2}$ bundle over a
pair of pants $P$. Gluing to $E$ the product of an annulus and $P$ along the
boundaries of each fiber, we obtain a closed surface bundle over $P$. We have
another closed surface bundle by gluing to $E$ the product of $P$ and two
disks.
  The sign of our class function cobounds the 2-cocycle on $\mathcal{M}_{g,2}$
defined by the difference of the signature of these two surface bundles over
$P$.
