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.

中文翻译：

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纯映射类群的第一整上同调

鲍威尔的经典结果是，由至少三个有限类型和属的连通的、可定向的曲面组成的纯映射类群是完美的。与此形成鲜明对比的是，我们构造了从无限属映射类群到整数的非平凡同态。此外，我们根据表面的单纯同调计算与至少为 2 的属的任何连接的可定向表面的纯映射类组相关联的第一个积分上同调组。为了做到这一点，我们展示了无限属曲面的纯映射类群分裂为半直积。