Homology Computations for Mapping Class Groups, in particular for Γ⁰_3,1

Abstract

In this thesis we compute the homology of mapping class groups of orientable and non-orientable surfaces. The surfaces we consider are of genus <i>g</i>, have one boundary curve and <i>m</i> permutable punctures. The corresponding moduli spaces $M_{g,1}^m$ in the orientable and $N_{g,1}^m$ in the non-orientable case are classifying spaces for the mapping class groups. <br /> We are able to compute the integral homology of the moduli spaces $M_{g,1}^m$ for <i>h=2g+m<6</i> and of $N_{g,1}^m$ for <i>h=g+m+1<5</i> (Note that we give a non-orientable surface the genus <i>g</i> if it is the connected sum of <i>g+1</i> projective planes). For <i>h=6</i> in the orientable case and <i>h=5</i> in the non-orientable case (these are the cases $M_{3,1}^0$, $M_{2,1}^2$ and $M_{1,1}^4$ resp. $N^0_{4,1}$, $N^1_{3,1}$, $N^2_{2,1}$ and $N^3_{1,1}$) we can compute some <i>p</i>-torsion in the homology and the mod-<i>p</i> Betti numbers for several primes. But this is enough evidence to conjecture that we have indeed the entire integral homology in these cases, too. <br /> The computations are based on a cell structure of the moduli spaces. This cell structure is bi-simplicial and the associated chain complex $Q_{••}(h,m)$ resp. $NQ_{••}(h,m)$ can be described by parts of the classifying spaces of symmetric groups <i>S₂</i>, ..., <i>S_2h</i> resp. by parts of the classifying space of a category of pairings. <br /> Motivated by B. Visy's Dissertation, we investigate ways to simplify the homology computation for $M_{g,1}^m$ and $N_{g,1}^m$. On the one hand, we extend the notion of factorable groups to factorable categories and study the homology of the norm complex associated to a factorable category; moreover, similar to the fact that a symmetric group is factorable, we prove that the category of pairings is a factorable category. On the other hand, from the cell structures of $M_{g,1}^m$ and $N_{g,1}^m$ with their orientation systems, we construct the double complexes $\tilde{Q}_{••}(h,m)$ and $\tilde{NQ}_{••}(h,m)$ and study their homology. <br /> For the actual computations, we implemented the new algorithms in a computer program.