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In this paper we define a $\mathbf{QP}^1$-valued class function on themapping 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 apair of pants $P$. Gluing to $E$ the product of an annulus and $P$ along theboundaries of each fiber, we obtain a closed surface bundle over $P$. We haveanother closed surface bundle by gluing to $E$ the product of $P$ and twodisks.  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$P$.

A class function on the mapping class group of an orientable surface and the Meyer cocycle