Components and Cohomology of Fock's Baby Teichmüller Space

At an Oberwolfach problem session in 2006, V. Fock asked for the cohomology and the number of components of the space B_n of maps f: ℤ/n → ℝP¹ with f(i) ≠ f(i+1) and at least three values, modulo SL(2,ℝ). We answer both questions for all n ≥ 3. The space B_n is a real-analytic (n−3)-manifold with exactly n−1 components B_{n,k}, indexed by the winding number k. For k ≠ n/2 they are Hausdorff and contractible, and for k = 1 and k = n−1 they are the Teichmüller space of the ideal n-gon. For even n the middle component is not Hausdorff: two distinct points have no disjoint neighbourhoods exactly when one is represented by a map constant on the even positions and the other by a map constant on the odd positions, and both loci are spheres S^{n/2−2}. The middle component is weakly homotopy equivalent to S^{n−3}. Hence H⁰(B_n; ℤ) ≅ ℤ^{n−1}, H^{n−3}(B_n; ℤ) ≅ ℤ for even n, and all other singular cohomology vanishes; the cohomology of the constant sheaf agrees, and so does its Čech cohomology on the Hausdorff components and, on the middle component, in degrees at most 2. For n = 4 the de Rham cohomology of smooth forms differs, and B_{4,2} is weakly homotopy equivalent to a circle but not homotopy equivalent to it. Modulo PGL(2,ℝ) there are ⌊n/2⌋ components, and for even n the middle one is weakly homotopy equivalent to ℝP^{n−3}; for odd n the space of real tame frieze patterns of width n−3 has (n−1)/2 components, all contractible. This is an unrefereed note.