The Cohomology Ring of the Real Locus of the Moduli Space of Stable Curves of Genus 0 with Marked Points

Let M0,n be the Deligne-Mumford compactification of the moduli space of algebraic curves of genus 0 with n marked points, labeled by 1, ..., n. This is a smooth projective variety defined over Q ([DM]), equipped with a natural action of the symmetric group Sn (by permuting the labels); its points are stable, possibly singular, curves of genus 0 with n labeled points. The geometry of this variety is very well understood; in particular, its cohomology ring (i.e., the singular cohomology of the manifold of complex points M0,n(C)) was computed by Keel [Keel], who also showed that it coincides with the Chow ring of M0,n. In this paper we will be interested in the topology of the manifold Mn := M0,n(R) of real points of this variety. There are a number of prior results concerning the topology of this manifold. Kapranov and Devadoss ([Kap], [Dev]) found a cell decomposition of Mn, and Devadoss used it to determine its Euler characteristic. Davis-Januszkiewicz-Scott ([DJS]) found a presentation of the fundamental group Γn of Mn (“the pure cactus group”) and proved that Mn is a K(π, 1) space for this group. Nevertheless, the topology of Mn is less well understood than that of M0,n(C). In particular, the cohomology groups (or even the Betti numbers) of Mn have been unknown until present.