The hyperelliptic mapping class group SMod(S) for a surface S is the subgroup of the mapping class group of S consisting of elements that commute with some fixed hyperelliptic involution. If S is a closed surface of genus g, then a theorem of Birman-Hilden gives that SMod(S) is closely related to the braid group on 2g+1 strands. In this talk we study the hyperelliptic Torelli group, that is, the subgroup SI(S) of SMod(S) consisting of elements that act trivially on the homology of S. This group is the fundamental group of the branch locus of the period mapping. Hain has conjectured that SI(S) is generated by Dehn twists. We show that if SI(S) is generated by reducible elements then it is generated by Dehn twists. We also show that SI(S) is generated by reducible elements if a particular simple-to-define complex is simply connected.