{"id":"https://openalex.org/W7222202387","doi":"https://doi.org/10.48550/arxiv.2610.20478","title":"Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds","display_name":"Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds","publication_year":2026,"publication_date":"2026-10-14","ids":{"openalex":"https://openalex.org/W7222202387","doi":"https://doi.org/10.48550/arxiv.2610.20478"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.20478","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20478","pdf_url":"https://arxiv.org/pdf/2610.20478","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.20478","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5004350357","display_name":"Oscar Ocampo","orcid":"https://orcid.org/0000-0002-4021-7859","observed_orcids":["https://orcid.org/0000-0002-4021-7859"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Ocampo, Oscar","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5004350357"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T13080","display_name":"Advanced Differential Geometry Research","score":0.0003000000142492354,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11680","display_name":"Advanced Algebra and Geometry","score":0.00019999999494757503,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/artin-braid-group","display_name":"Artin braid group","score":0.9490000009536743},{"id":"https://openalex.org/keywords/torus-bundles","display_name":"torus bundles","score":0.9459999799728394},{"id":"https://openalex.org/keywords/fundamental-group","display_name":"fundamental group","score":0.42100000381469727},{"id":"https://openalex.org/keywords/translation-subgroup","display_name":"translation subgroup","score":0.49300000071525574},{"id":"https://openalex.org/keywords/group-extensions","display_name":"group extensions","score":0.6620000004768372},{"id":"https://openalex.org/keywords/affine-equivalence","display_name":"affine equivalence","score":0.6729999780654907},{"id":"https://openalex.org/keywords/crystallographic-groups","display_name":"crystallographic groups","score":0.6949999928474426}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8736000061035156},{"id":"https://openalex.org/C33840335","wikidata":"https://www.wikidata.org/wiki/Q220409","display_name":"Braid group","level":2,"score":0.6952000260353088},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.6376000046730042},{"id":"https://openalex.org/C124194741","wikidata":"https://www.wikidata.org/wiki/Q320809","display_name":"Braid","level":2,"score":0.5663999915122986},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.517300009727478},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.4733000099658966},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4648999869823456},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.4397999942302704},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.43959999084472656},{"id":"https://openalex.org/C95627345","wikidata":"https://www.wikidata.org/wiki/Q1138961","display_name":"Quotient group","level":4,"score":0.4153999984264374},{"id":"https://openalex.org/C42058472","wikidata":"https://www.wikidata.org/wiki/Q810214","display_name":"Base (topology)","level":2,"score":0.40709999203681946},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.4018999934196472},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.39969998598098755},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3847000002861023},{"id":"https://openalex.org/C159876591","wikidata":"https://www.wikidata.org/wiki/Q662830","display_name":"Fundamental group","level":2,"score":0.3840999901294708},{"id":"https://openalex.org/C147441545","wikidata":"https://www.wikidata.org/wiki/Q220409","display_name":"Braid theory","level":3,"score":0.37630000710487366},{"id":"https://openalex.org/C149364088","wikidata":"https://www.wikidata.org/wiki/Q185917","display_name":"Translation (biology)","level":4,"score":0.32670000195503235},{"id":"https://openalex.org/C167204820","wikidata":"https://www.wikidata.org/wiki/Q912058","display_name":"Simply connected space","level":2,"score":0.3264000117778778},{"id":"https://openalex.org/C76739243","wikidata":"https://www.wikidata.org/wiki/Q16908484","display_name":"Maximal subgroup","level":4,"score":0.304500013589859},{"id":"https://openalex.org/C2778134712","wikidata":"https://www.wikidata.org/wiki/Q1047307","display_name":"Bundle","level":2,"score":0.2978000044822693},{"id":"https://openalex.org/C105515060","wikidata":"https://www.wikidata.org/wiki/Q743179","display_name":"Normal subgroup","level":3,"score":0.296999990940094},{"id":"https://openalex.org/C140865510","wikidata":"https://www.wikidata.org/wiki/Q522216","display_name":"Commutator subgroup","level":4,"score":0.29269999265670776},{"id":"https://openalex.org/C154610627","wikidata":"https://www.wikidata.org/wiki/Q382491","display_name":"Affine plane (incidence geometry)","level":3,"score":0.29260000586509705},{"id":"https://openalex.org/C201064014","wikidata":"https://www.wikidata.org/wiki/Q3956993","display_name":"Artin group","level":3,"score":0.2881999909877777},{"id":"https://openalex.org/C31498916","wikidata":"https://www.wikidata.org/wiki/Q583034","display_name":"Locally compact space","level":2,"score":0.28679999709129333},{"id":"https://openalex.org/C40753290","wikidata":"https://www.wikidata.org/wiki/Q1139111","display_name":"Quotient space (topology)","level":3,"score":0.2703999876976013},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.2621000111103058}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.20478","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20478","pdf_url":"https://arxiv.org/pdf/2610.20478","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.20478","doi":"https://doi.org/10.48550/arxiv.2610.20478","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.20478","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.20478","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.20478","pdf_url":"https://arxiv.org/pdf/2610.20478","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7222202387.pdf","grobid_xml":"https://content.openalex.org/works/W7222202387.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3116538170","https://openalex.org/W2962847444","https://openalex.org/W2212827489","https://openalex.org/W2123879911","https://openalex.org/W2226568402","https://openalex.org/W2550330308","https://openalex.org/W2143305397","https://openalex.org/W2728664650","https://openalex.org/W2479493876","https://openalex.org/W2331183417"],"abstract_inverted_index":{"For":[0],"$n\\geq3$,":[1],"let":[2],"$\\overline":[3,17,61],"B_n=B_n/[P_n,P_n]$":[4],"be":[5],"the":[6,10,38,109,116,126],"crystallographic":[7],"quotient":[8],"of":[9,40,59,86,97,108,125,130],"Artin":[11],"braid":[12],"group,":[13],"with":[14,53],"translation":[15],"subgroup":[16,58,83,93],"P_n=P_n/[P_n,P_n]\\cong\\mathbb":[18],"Z^N$,":[19],"where":[20,69],"$N=\\binom":[21],"n2$.":[22],"We":[23,104],"prove":[24],"that":[25,100],"every":[26,66],"compact":[27,50,131],"connected":[28],"flat":[29,51,132],"$3$-manifold":[30],"$M$,":[31,78],"considered":[32],"up":[33],"to":[34],"affine":[35],"equivalence,":[36],"is":[37,48,71],"base":[39],"a":[41,49,56,81,90],"geometric":[42],"torus":[43],"bundle":[44],"whose":[45],"total":[46],"space":[47],"manifold":[52],"fundamental":[54],"group":[55,112],"Bieberbach":[57,82],"some":[60],"B_n$.":[62],"More":[63],"precisely,":[64],"for":[65,115,122],"$n\\geq":[67],"n_M$,":[68],"$n_M\\in\\{3,4,5\\}$":[70],"an":[72],"explicit":[73],"optimal":[74],"threshold":[75],"depending":[76],"on":[77],"we":[79],"construct":[80],"$\u0393_n(M)\\leq\\overline":[84],"B_n$":[85],"dimension":[87],"$N$":[88],"and":[89],"complete":[91],"normal":[92],"$K_n(M)\\cong\\mathbb":[94],"Z^{N-3}$":[95],"consisting":[96],"translations":[98],"such":[99],"$$":[101,103],"\u0393_n(M)/K_n(M)\\cong\u03c0_1(M).":[102],"also":[105],"determine":[106],"which":[107],"associated":[110],"torus-bundle":[111],"extensions":[113],"split:":[114],"realizations":[117],"constructed":[118],"here,":[119],"splitting":[120],"occurs":[121],"exactly":[123],"four":[124],"ten":[127],"affine-equivalence":[128],"classes":[129],"$3$-manifolds.":[133]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-10-16T00:00:00"}
