The $\mu$-invariant of $3$-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented $2$-manifold

Let $\mathcal{H}(n)$ be the group of orientation-preserving selfhomeomorphisms of a closed oriented surface Bd U of genus n, and let $\mathcal{K}(n)$ be the subgroup of those elements which induce the identity on ${H_1}({\text{Bd}}\;U;{\mathbf{Z}})$. To each element $h \in \mathcal{H}(n)$ we associate a 3-manifold $M(h)$ which is defined by a Heegaard splitting. It is shown that for each $h \in \mathcal{H}(n)$ there is a representation $\rho$ of $\mathcal{K}(n)$ into ${\mathbf{Z}}/2{\mathbf{Z}}$ such that if $k \in \mathcal{K}(n)$, then the $\mu$-invariant $\mu (M(h))$ is equal to the $\mu$-invariant $\mu (M(kh))$ if and only if k $\in$ kernel $\rho$. Thus, properties of the 4-manifolds which a given 3-manifold bounds are related to group-theoretical structure in the group of homeomorphisms of a 2-manifold. The kernels of the homomorphisms from $\mathcal{K}(n)$ onto ${\mathbf{Z}}/2{\mathbf{Z}}$ are studied and are shown to constitute a complete conjugacy class of subgroups of $\mathcal{H}(n)$. The class has nontrivial finite order.