Simplicial Cohomology of Smooth Orbifolds in GAP

This short research announcement briefly describes the simplicial method underlying the GAP package SCO for computing the socalled orbifold cohomology of topological resp. smooth orbifolds. SCO can be used to compute the lower dimensional group cohomology of some infinite groups. Instead of giving a complete formal definition of an orbifold, we start with a simple construction, general enough to give rise to any orbifoldM. Let X be a (smooth) manifold and G a Lie group acting (smoothly and) properly onX , i.e., the action graph α : G ×X → X ×X : (g, x) → (x, gx) is a proper map. In particular, G acts with compact stabilizersGx = α−1({(x, x)}). Further assume thatG is either – discrete (acting discontinuously), or – compact acting almost freely (i.e. with discrete stabilizers) on X . In both cases G acts with finite stabilizers. “Enriching” M := X/G with these finite isotropy groups leads to the so-called fine orbit space M := [X/G], also called the global quotient orbifold. We call the orbifold reduced if the action is faithful. The topological space M is then called the coarse space underlying the orbifold M. Roughly speaking, a reduced orbifold M locally looks like the orbit space R/V , where V is a finite subgroup of GLn(R). It is easy to show that every effective orbifold M arises as a global quotient orbifold by a compact Lie group1 G. By considering (countable) discrete groups G acting properly and discontinuously on X we still obtain a subclass of orbifolds which includes many interesting moduli spaces. The most prominent one is the (non-compactified) moduli space Mg,n of curves of genus g with n marked points and 2g + n ≥ 3. It is the global quotient [Tg,n/Γg,n] of the contractible Teichmüller space Tg,n ≈ C3g−3+n by the proper discontinuous action of the mapping class group Γg,n [7]. 1 Define X as the bundle of orthonormal frames on M. It is a manifold on which the orthogonal group G := On(R) acts almost freely with faithful slice representations [8, Thm. 4.1]. K. Fukuda et al. (Eds.): ICMS 2010, LNCS 6327, pp. 46–49, 2010. c © Springer-Verlag Berlin Heidelberg 2010 Simplicial Cohomology of Smooth Orbifolds in GAP 47 A convenient way to define a cohomology theory on M is to consider the category Ab(M) of Abelian sheaves on M, together with the global section functor ΓM : Ab(M) → Ab to the category of Abelian groups. The Abelian category Ab(M) has enough injectives and the orbifold cohomology of M with values in a sheaf2 A is then simply defined as the derived functor cohomology