The first homology group with twisted coefficients for the mapping class group of a non-orientable surface with boundary (Q6621552)

The authors determined the first homology group with coefficients in \(H_1(N;\mathbf{Z})\) for variuos mapping class groups of non-orientable surface \(N\) with punctures and/or boundary. The organization of work is as follows: First section provides brief introduction about the surface, backgrund about homological computations and statements of main results proved later in article. Section two presents preliminaries required, i.e Non-orientable surface,and computation of first homology group with twisted coefficients. The action of \(\mathcal{M}(N_{g,s}^n)\) on \(H_1(N_{g,s}^n;\mathbb{Z})\)is described in section three. Goal of section four is to obtain simple generating sets for \(\mathcal{PM}^k (N_{g,s}^n)\) and \(\mathcal{M}(N_{g,s}^n)\). In Section five they computed \(\left<\overline{X} \right> \cap ker \overline{\partial}_1\). In sections six to ten the boundings \(H_1(\mathcal{PM}^+(N_{g,s}^n) ;H_1(N_{g,s}^n; \mathbb{Z}))\), \(H_1(\mathcal{PM}^k(N_{g,s}^n) ;H_1(N_{g,s}^n; \mathbb{Z}))\), \(H_1(\mathcal{M}(N_{g,s}^n) ;H_1(N_{g,s}^n; \mathbb{Z}))\), \(H_1(\mathcal{PM^k}(N_{g,s}^n) ;H_1(N_{g,s}^n;\mathbb{Z}))\) and \(H_1(\mathcal{M}(N_{g,s}^n) ;H_1(N_{g,s}^n; \mathbb{Z}))\) are done with completion of proofs of main results.