{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T01:20:18Z","timestamp":1767835218410,"version":"3.49.0"},"reference-count":9,"publisher":"World Scientific Pub Co Pte Ltd","issue":"06","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2022,9]]},"abstract":"<jats:p> The mod 4 braid group, [Formula: see text], is defined to be the quotient of the braid group by the subgroup of the pure braid group generated by squares of all elements, [Formula: see text]. Kordek and Margalit proved [Formula: see text] is an extension of the symmetric group by [Formula: see text]. For [Formula: see text], we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying [Formula: see text]. We show this cocycle is the mod\u00a02 reduction of the 2-cocycle corresponding to the extension of the symmetric group by the abelianization of the pure braid group. We also construct the 2-cocycle corresponding to this second extension and show that it represents an order two element in the cohomology of the symmetric group. Furthermore, we give presentations for both extensions and a normal generating set for [Formula: see text]. <\/jats:p>","DOI":"10.1142\/s0218196722500485","type":"journal-article","created":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T03:42:25Z","timestamp":1651722145000},"page":"1125-1159","source":"Crossref","is-referenced-by-count":3,"title":["The cohomology class of the mod 4 braid group"],"prefix":"10.1142","volume":"32","author":[{"given":"Trevor","family":"Nakamura","sequence":"first","affiliation":[{"name":"University of Arkansas, Fayetteville, AR 72701 Arkansas, USA"}]}],"member":"219","published-online":{"date-parts":[[2022,6,10]]},"reference":[{"key":"S0218196722500485BIB002","series-title":"Annals of Mathematica Studies","volume-title":"Braids, Links, and Mapping Class Groups","volume":"82","author":"Birman J. S.","year":"1974"},{"key":"S0218196722500485BIB003","doi-asserted-by":"publisher","DOI":"10.1006\/aima.1998.1761"},{"key":"S0218196722500485BIB004","doi-asserted-by":"publisher","DOI":"10.1515\/crelle-2015-0032"},{"key":"S0218196722500485BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9327-6"},{"key":"S0218196722500485BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/978-88-7642-431-1_9"},{"key":"S0218196722500485BIB007","doi-asserted-by":"publisher","DOI":"10.2307\/1969174"},{"key":"S0218196722500485BIB008","volume-title":"A Primer on Mapping Class Groups","volume":"49","author":"Farb B.","year":"2012"},{"key":"S0218196722500485BIB010","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61896-3"},{"key":"S0218196722500485BIB011","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196718500169"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196722500485","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,26]],"date-time":"2022-08-26T07:08:10Z","timestamp":1661497690000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0218196722500485"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,10]]},"references-count":9,"journal-issue":{"issue":"06","published-print":{"date-parts":[[2022,9]]}},"alternative-id":["10.1142\/S0218196722500485"],"URL":"https:\/\/doi.org\/10.1142\/s0218196722500485","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,10]]}}}