The first integral cohomology of pure mapping class groups
arXiv:1711.03132 · doi:10.1093/imrn/rnaa229
Abstract
It is a classical result of Powell that pure mapping class groups of connected, orientable surfaces of finite type and genus at least three are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface's simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.
v2: Updated to include referees' comments, which includes a significant rewriting of the exposition in the introduction. To appear in Int. Math. Res. Not. IMRN. 23 pages, 5 figures. v1: 21 pages, 4 figures
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- First cohomology of pure mapping class groups of big genus one and zero surfaces
- Coarse Geometry of Pure Mapping Class Groups of Infinite Graphs
- Large-Scale Geometry of Pure Mapping Class Groups of Infinite-Type Surfaces
- Three perfect mapping class groups
- Mapping class groups of surfaces with noncompact boundary components
- Infinite-type loxodromic isometries of the relative arc graph
- On the homology of big mapping class groups