{"id":"https://openalex.org/W7213528100","doi":"https://doi.org/10.48550/arxiv.2609.18942","title":"A brief note on the Hopf and Kervaire invariant one problems","display_name":"A brief note on the Hopf and Kervaire invariant one problems","publication_year":2026,"publication_date":"2026-07-15","ids":{"openalex":"https://openalex.org/W7213528100","doi":"https://doi.org/10.48550/arxiv.2609.18942"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.18942","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18942","pdf_url":"https://arxiv.org/pdf/2609.18942","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/2609.18942","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5141067159","display_name":"J. P. May","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"May, J. P.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5141067159"],"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/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9965999722480774,"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"}},"topics":[{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9965999722480774,"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":0.0013000000035390258,"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/T10061","display_name":"Algebraic Geometry and Number Theory","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"}}],"keywords":[{"id":"https://openalex.org/keywords/homotopy-groups-of-spheres","display_name":"homotopy groups of spheres","score":0.7360000014305115},{"id":"https://openalex.org/keywords/exotic-spheres","display_name":"exotic spheres","score":0.574999988079071},{"id":"https://openalex.org/keywords/stable-homotopy-theory","display_name":"stable homotopy theory","score":0.7059999704360962}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.826200008392334},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.7247999906539917},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5264999866485596},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.37040001153945923},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3425999879837036},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.30160000920295715},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.2689000070095062},{"id":"https://openalex.org/C134530390","wikidata":"https://www.wikidata.org/wiki/Q2920801","display_name":"Invariant polynomial","level":4,"score":0.22910000383853912},{"id":"https://openalex.org/C29712632","wikidata":"https://www.wikidata.org/wiki/Q1627597","display_name":"Hopf algebra","level":3,"score":0.2159000039100647},{"id":"https://openalex.org/C78078846","wikidata":"https://www.wikidata.org/wiki/Q1443624","display_name":"Finite type invariant","level":5,"score":0.20100000500679016}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.18942","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18942","pdf_url":"https://arxiv.org/pdf/2609.18942","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.2609.18942","doi":"https://doi.org/10.48550/arxiv.2609.18942","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.18942","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:2609.18942","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.18942","pdf_url":"https://arxiv.org/pdf/2609.18942","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/W7213528100.pdf","grobid_xml":"https://content.openalex.org/works/W7213528100.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4405468701"],"related_works":["https://openalex.org/W2582288169","https://openalex.org/W1722536541","https://openalex.org/W2332130173","https://openalex.org/W106697381","https://openalex.org/W7100221584","https://openalex.org/W7095769181","https://openalex.org/W7098033388","https://openalex.org/W2032742908","https://openalex.org/W3206492815","https://openalex.org/W1529108554"],"abstract_inverted_index":{"This":[0],"expository":[1],"note":[2],"advertises":[3],"a":[4],"little":[5],"known":[6],"but":[7],"very":[8],"old":[9],"one-line":[10],"solution":[11],"of":[12,23],"the":[13,21,24,27],"Hopf":[14],"invariant":[15,29],"one":[16,30],"problem":[17],"and":[18],"speculates":[19],"on":[20],"(in)applicability":[22],"idea":[25],"to":[26],"Kervaire":[28],"problem.":[31]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-18T00:00:00"}