Theoretical analysis demonstrates complete classification of braid group homomorphisms into pure mapping class groups of nonorientable surfaces, highlighting rigid geometric constraints.
We classify homomorphisms from the braid group on n strands to the pure mapping class group of a nonoriantable surface of genus g. For n≥ 14 and g≤ 2n/2+1 every such homomorphism is either cyclic, or it maps standard generators of the braid group to either distinct Dehn twists, or distinct crosscap transpositions, possibly multiplied by the same element of the centralizer of the image.
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Stukow et al. (2024) studied this question.
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