Wang, Rui: Homology Computations for Mapping Class Groups, in particular for Γ03,1. - Bonn, 2011. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5N-26108
@phdthesis{handle:20.500.11811/5018,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5N-26108,
author = {{Rui Wang}},
title = {Homology Computations for Mapping Class Groups, in particular for Γ03,1},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2011,
month = sep,

note = {In this thesis we compute the homology of mapping class groups of orientable and non-orientable surfaces. The surfaces we consider are of genus g, have one boundary curve and m 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.
We are able to compute the integral homology of the moduli spaces $M_{g,1}^m$ for h=2g+m<6 and of $N_{g,1}^m$ for h=g+m+1<5 (Note that we give a non-orientable surface the genus g if it is the connected sum of g+1 projective planes). For h=6 in the orientable case and h=5 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 p-torsion in the homology and the mod-p Betti numbers for several primes. But this is enough evidence to conjecture that we have indeed the entire integral homology in these cases, too.
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 S2, ..., S2h resp. by parts of the classifying space of a category of pairings.
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.
For the actual computations, we implemented the new algorithms in a computer program.},

url = {https://hdl.handle.net/20.500.11811/5018}
}

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