Remarks on the sections of universal hyperelliptic curves

In this paper, we study the obstruction for the sections of the universal hyperelliptic curves of genus $g\ge 3$. The obstruction of our interest comes from the relative completion of the hyperelliptic mapping class groups and the Lie algebra of the unipotent completion of the fundamental group of the configuration space of a compact oriented surface. Using the obstruction, we prove that the Birman exact sequence for the hyperelliptic mapping class groups does not split for $g\ge 3$.