{"id":"https://openalex.org/W7204939334","doi":"https://doi.org/10.48550/arxiv.2608.30875","title":"A Decomposition Theorem for Topological Semi-small Maps","display_name":"A Decomposition Theorem for Topological Semi-small Maps","publication_year":2026,"publication_date":"2026-08-31","ids":{"openalex":"https://openalex.org/W7204939334","doi":"https://doi.org/10.48550/arxiv.2608.30875"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2608.30875","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30875","pdf_url":"https://arxiv.org/pdf/2608.30875","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","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2608.30875","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5099038714","display_name":"Shahryar Ghaed Sharaf","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Sharaf, Shahryar Ghaed","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5099038714"],"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.8349999785423279,"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/T10061","display_name":"Algebraic Geometry and Number Theory","score":0.8349999785423279,"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.09220000356435776,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.04039999842643738,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/covering-spaces","display_name":"covering spaces","score":0.8109999895095825},{"id":"https://openalex.org/keywords/fiber-bundles","display_name":"fiber bundles","score":0.953000009059906},{"id":"https://openalex.org/keywords/decomposition-theorem","display_name":"decomposition theorem","score":0.7419999837875366},{"id":"https://openalex.org/keywords/gysin-sequence","display_name":"Gysin sequence","score":0.9850000143051147},{"id":"https://openalex.org/keywords/normal-bundle","display_name":"normal bundle","score":0.9129999876022339},{"id":"https://openalex.org/keywords/continuous-surjections","display_name":"continuous surjections","score":0.5600000023841858}],"concepts":[{"id":"https://openalex.org/C119238805","wikidata":"https://www.wikidata.org/wiki/Q229102","display_name":"Surjective function","level":2,"score":0.8224999904632568},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7876999974250793},{"id":"https://openalex.org/C81332173","wikidata":"https://www.wikidata.org/wiki/Q179899","display_name":"Topological space","level":2,"score":0.5016000270843506},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.4921000003814697},{"id":"https://openalex.org/C4042151","wikidata":"https://www.wikidata.org/wiki/Q215111","display_name":"Homomorphism","level":2,"score":0.48339998722076416},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.44839999079704285},{"id":"https://openalex.org/C2991919592","wikidata":"https://www.wikidata.org/wiki/Q170058","display_name":"Continuous map","level":2,"score":0.43639999628067017},{"id":"https://openalex.org/C4017995","wikidata":"https://www.wikidata.org/wiki/Q595298","display_name":"Sheaf","level":2,"score":0.43309998512268066},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.41999998688697815},{"id":"https://openalex.org/C146710177","wikidata":"https://www.wikidata.org/wiki/Q1154996","display_name":"Fiber bundle","level":3,"score":0.40639999508857727},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.38190001249313354},{"id":"https://openalex.org/C65496060","wikidata":"https://www.wikidata.org/wiki/Q2632649","display_name":"Open and closed maps","level":3,"score":0.3815999925136566},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.35510000586509705},{"id":"https://openalex.org/C24977911","wikidata":"https://www.wikidata.org/wiki/Q1491995","display_name":"Connected space","level":3,"score":0.3474000096321106},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.33869999647140503},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.33629998564720154},{"id":"https://openalex.org/C164953516","wikidata":"https://www.wikidata.org/wiki/Q320357","display_name":"Closed set","level":2,"score":0.3327000141143799},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.321399986743927},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.31940001249313354},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.30880001187324524},{"id":"https://openalex.org/C129089157","wikidata":"https://www.wikidata.org/wiki/Q621550","display_name":"General topology","level":3,"score":0.2953999936580658},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.2840999960899353},{"id":"https://openalex.org/C47776270","wikidata":"https://www.wikidata.org/wiki/Q1455249","display_name":"Topological vector space","level":3,"score":0.2825999855995178},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.2687000036239624},{"id":"https://openalex.org/C18648836","wikidata":"https://www.wikidata.org/wiki/Q381892","display_name":"Compact space","level":2,"score":0.2612999975681305},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.2606000006198883},{"id":"https://openalex.org/C167204820","wikidata":"https://www.wikidata.org/wiki/Q912058","display_name":"Simply connected space","level":2,"score":0.2583000063896179},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.25609999895095825},{"id":"https://openalex.org/C519885992","wikidata":"https://www.wikidata.org/wiki/Q161","display_name":"Fiber","level":2,"score":0.25459998846054077}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2608.30875","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30875","pdf_url":"https://arxiv.org/pdf/2608.30875","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.2608.30875","doi":"https://doi.org/10.48550/arxiv.2608.30875","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2608.30875","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2608.30875","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2608.30875","pdf_url":"https://arxiv.org/pdf/2608.30875","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7204939334.pdf","grobid_xml":"https://content.openalex.org/works/W7204939334.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4417143700"],"related_works":["https://openalex.org/W4417143700","https://openalex.org/W2460954636","https://openalex.org/W3194065547","https://openalex.org/W4377086718","https://openalex.org/W2794067441","https://openalex.org/W2055120883","https://openalex.org/W2014675282","https://openalex.org/W2025285312","https://openalex.org/W7131387911","https://openalex.org/W2070618124"],"abstract_inverted_index":{"A":[0],"continuous":[1],"surjection":[2],"$f:":[3],"X":[4],"\\rightarrow":[5],"Y$":[6,22],"is":[7,38,49],"an":[8,56],"almost":[9,57,94],"covering":[10,40,58],"map":[11,41,59,64],"if":[12],"there":[13],"exists":[14],"a":[15,24,39,50,61,161],"nowhere":[16],"dense":[17],"closed":[18],"set":[19],"$R":[20,27],"\\subset":[21,128,143],"with":[23],"finite":[25],"decomposition":[26],"=":[28],"\\bigsqcup_\u03b2":[29],"T_\u03b2$":[30,129],"such":[31],"that":[32,119],"the":[33,46,69,74,78,90,98,102,117,130,134,138,149,154,158],"restriction":[34,47],"$f\\vert_{f^{-1}(Y":[35],"\\setminus":[36],"R)}$":[37],"and,":[42],"for":[43,66,83,120],"each":[44,67,121,124],"$\u03b2$,":[45,68],"$f\\vert_{f^{-1}(T_\u03b2)}$":[48,72],"fiber":[51,70],"bundle.":[52],"We":[53,87],"refer":[54],"to":[55,109],"as":[60,82],"topological":[62,91,110],"semi-small":[63,85,111],"if,":[65],"of":[71,93,137,141,153],"and":[73,96,113,123],"space":[75],"$T_\u03b2$":[76],"satisfy":[77],"same":[79],"dimension":[80],"condition":[81],"algebraic":[84],"maps.":[86],"first":[88],"study":[89],"properties":[92],"coverings":[95],"develop":[97],"necessary":[99],"tools.":[100],"In":[101],"next":[103],"step,":[104],"we":[105],"restrict":[106],"our":[107],"attention":[108],"maps":[112],"prove":[114],"that,":[115],"under":[116],"assumption":[118],"$\u03b2$":[122],"trivializing":[125],"neighborhood":[126],"$U":[127],"connecting":[131],"homomorphisms":[132],"in":[133,146],"Gysin":[135],"sequence":[136],"normal":[139],"bundle":[140],"$f^{-1}(U)":[142],"X$":[144],"vanish":[145],"suitable":[147],"degrees,":[148],"derived":[150],"direct":[151],"image":[152],"constant":[155],"sheaf":[156],"on":[157],"domain":[159],"admits":[160],"decomposition.":[162]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-02T00:00:00"}