Journal of the
Korean Mathematical Society JKMS

The disjoint union of mapping class groups $\Gamma_{g,1}$ gives us a braided monoidal category so that it gives rise to a double loop space structure. We show that there exists a natural twist in this category, so it gives us a ribbon category. We explicitly express this structure by showing how the twist acts on the fundamental group of the surface $S_{g,1}$. We also make an explicit description of this structure in terms of the standard Dehn twists, as well as in terms of Wajnryb's Dehn twists. We show that the inverse of the twist $\tau_g$ for the genus $g$ equals the Dehn twist along the fixed boundary of the surface $S_{g,1}$.

Keywords: mapping class group, Dehn twist, monoidal category, braiding, twist

MSC numbers: 14H10, 18D50, 57N16