| The moduli space Mn0 C of Riemann spheres with n punctures has an extensive history. We study the real points Mn0 R of this space, the set of fixed points of Mn0 C under complex conjugation. It is known that Mn0 R is naturally tessellated by open copies of the Stasheff associahedron Kn-1 . We describe the combinatorial structure of Kn-1 , a convex polytope of dimension n - 3, with the property that each face is a product of lower dimensional associahedra.; Under the Deligne-Mumford-Knudsen compactification of Mn0 R , the tiles glue to form Mn0 R , a connected manifold without boundary. We show the points of Mn0 R to be parameterized by polygons with non-intersecting diagonals (modulo an equivalence relation), enabling us to describe how these domains glue. This leads to an elementary computation of the Euler characteristic of Mn0 R .; Another approach in obtaining Mn0 R is through iterated blow-ups of the braid hyperplane arrangement. Incorporating the description of Mn0 R using polygons, {09}we show a product structure of lower dimensional moduli spaces sitting in Mn0 R , similar to the property of the associahedron. As a consequence, another perspective is given to the truncation of the n - 3 simplex leading to Kn-1 .; We also construct a new cyclic operad of mosaics defined by polygons with marked diagonals whose underlying spaces are Mn0 R , and show that the fundamental groups of Mn0 R form an operad with similarities to the operad of braid groups. |
