Point pushing, homology, and the hyperelliptic involution (Q376661)

The authors study the hyperelliptic Torelli subgroup of the mapping class group of a closed surface \(S_{g}\). The hyperelliptic Torelli group \(SI(S_g)\) is the group of mapping classes of \(S_g\) which commute with a fixed hyperelliptic involution of \(S_g\). In the authors' notation, \(S_{g,1}\) denotes a once--punctured surface with an involution--invariant puncture, and \(S_{g,2}\) denotes a twice punctured surface with the punctures permuted by the involution.NEWLINENEWLINEThe authors develop a Birman exact sequence for the hyperelliptic Torelli group. Their first result is that the forgetful map \(SI(S_{g,1})\to SI(S_g)\) is an isomorphism. Then, the authors consider the kernel \(SIBK(S_{g,2})\) of the forgetful map \(SI(S_{g,2})\to SI(S_g)\). They show that these groups fit into a split exact sequence and that \(SIBK(S_{g,2})\) is free of infinite rank, so that \(SI(S_{g,2})\) is a semidirect product of \(SI(S_g)\) and an infinite rank free group.NEWLINENEWLINEThe authors then study applications of their results to the study of generators for hyperelliptic Torelli groups, showing among other things that every element of \(SIBK(S_{g,2})\) is a product of Dehn twists about symmetric separating simple closed curves.