maps are precisely those that act trivially on top cohomology H2(Σ) ≅ Z. H1(Σ) has a symplectic structure, coming from the cup product; since these maps are automorphisms, and maps preserve the cup product, the mapping class group acts as symplectic automorphisms, and indeed all symplectic automorphisms are realized, yielding the short exact sequence: One can extend this to The symplectic group is well understood. Hence understanding the algebraic structure of the mapping class group often reduces to questions about the Torelli group. Note that for the torus (genus 1) the map to the symplectic group is an isomorphism, and the Torelli group vanishes. Stable mapping class group One can embed the surface of genus g and 1 boundary component into by attaching an additional hole on the end (i.e., gluing together and ), and thus automorphisms of the small surface fixing the boundary extend to the larger surface. Taking the direct limit of these groups and inclusions yields the stable mapping class group, whose rational cohomology ring was conjectured by David Mumford (one of conjectures called the Mumford conjectures). The integral (not just rational) cohomology ring was computed in 2002 by Ib Madsen and Michael Weiss, proving Mumford's conjecture. See also Braid groups, the mapping class groups of punctured discs Homotopy groups Homeotopy groups Lantern relation References Stable mapping class group External links Madsen-Weiss MCG Seminar; many references Geometric topology Homeomorphisms