On the non-vanishing of the powers of the Euler class for mapping class groups. (English) Zbl 1490.57015
Let \(\Gamma_{g}^{1}\) be the mapping class group of a surface \(\Sigma_{g}^{1}\) of genus \(g\geq 1\) and one marked point. Using the monomorphism \(\rho:\Gamma_{g}^{1}\hookrightarrow \mathrm{Homeo}_{+}(\mathbb{S}^{1})\) one pulls back the discrete universal Euler class \(\mathbf{E}\in H^{2} (\mathrm{Homeo}_{+}(\mathbb{S}^{1});\mathbb{Z})\) to \(E\in H^{2}(\Gamma_{g}^{1};\mathbb{Z})\). By naturality \(\mathbf{E}^{n}\) pulls back to \(E^{n}\in H^{2n}(\Gamma_{g}^{1};\mathbb{Z})\). The main theorem asserts that the classes \(E^{n}\) are nonzero for \(g\geq1\) and \(n\geq 1\). Furthermore, for \(n\geq g\) the subgroup of \(H^{2n}(\Gamma_{g}^{1};\mathbb{Z})\) generated by the class \(E^{n}\) is a finite cyclic group of order a multiple of \(4g(2g+1)\). The authors detect torsion classes by pulling back the Euler class to finite subgroups of \(\mathrm{Homeo}_{+}(\mathbb{S}^{1})\). The authors also prove a vanishing result for powers of the Euler class in the cohomology of pure mapping class groups. Namely, let \(1\leq i\leq k\), then the powers \(E^{n}_{i}\in H^{2n}(\Gamma_{g}^{k};\mathbb{Q})\) vanish for all \(n\geq g\).
Reviewer: Daniel Juan Pineda (Michoacán)
MSC:
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |
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