The First Integral Cohomology of Pure Mapping Class Groups

It is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomo...

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Bibliographic Details
Published in:International mathematics research notices 2020-11
Main Authors: Aramayona, Javier, Patel, Priyam, Vlamis, Nicholas G
Format: Article
Language:English
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Summary:It is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 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.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnaa229