Publicação
On the algebraic approximation of Lusternik-Schnirelmann category
| Resumo: | Algebraic approximations have proved to be very useful in the investigation of Lusternik-Schnirelmann category. In this paper the L.-S. category and its approximations are studied from the point of view of abstract homotopy theory. We introduce three notions of L.-S. category for monoidal cofibration categories, i.e., cofibration categories with a suitably incorporated tensor product. We study the fundamental properties of the abstract invariants and discuss, in particular, their behaviour with respect to cone attachments and products. Besides the topological L.-S. category the abstract concepts cover classical algebraic approximations of the L.-S. category such as the Toomer invariant, rational category, and the A- and M-categories of Halperin and Lemaire. We also use the abstract theory to introduce a new algebraic approximation of L.-S. category. This invariant which we denote by $\ell$ is the first algebraic approximation of the L.-S. category which is not necessarily $\leq 1$ for spaces having the same Adams-Hilton model as a wedge of spheres. For a space $X$ the number $\ell (X)$ can be determined from an Anick model of $X$. Thanks to the general theory one knows \textit{a priori} that $\ell$ is a lower bound of the L.-S. category which satisfies the usual product inequality and increases by at most 1 when a cone is attached to a space. |
|---|---|
| Autores principais: | Kahl, Thomas |
| Assunto: | Lusternik-Schnirelmann category Hopf algebras up to homotopy Model categories |
| Ano: | 2003 |
| País: | Portugal |
| Tipo de documento: | artigo |
| Tipo de acesso: | acesso aberto |
| Instituição associada: | Universidade do Minho |
| Idioma: | inglês |
| Origem: | RepositóriUM - Universidade do Minho |
| _version_ | 1867439392980729856 |
|---|---|
| author | Kahl, Thomas |
| author_facet | Kahl, Thomas |
| author_role | author |
| contributor_name_str_mv | RepositóriUM - Universidade do Minho |
| country_str | PT |
| creators_json_txt | [{\"Person.name\":\"Kahl, Thomas\"}] |
| datacite.contributors.contributor.contributorName.fl_str_mv | RepositóriUM - Universidade do Minho |
| datacite.creators.creator.creatorName.fl_str_mv | Kahl, Thomas |
| datacite.date.Accepted.fl_str_mv | 2003-06-01T00:00:00Z |
| datacite.date.available.fl_str_mv | 2006-02-08T15:38:02Z |
| datacite.date.embargoed.fl_str_mv | 2006-02-08T15:38:02Z |
| datacite.rights.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| datacite.subjects.subject.fl_str_mv | Lusternik-Schnirelmann category Hopf algebras up to homotopy Model categories |
| datacite.titles.title.fl_str_mv | On the algebraic approximation of Lusternik-Schnirelmann category |
| dc.contributor.none.fl_str_mv | RepositóriUM - Universidade do Minho |
| dc.creator.none.fl_str_mv | Kahl, Thomas |
| dc.date.Accepted.fl_str_mv | 2003-06-01T00:00:00Z |
| dc.date.available.fl_str_mv | 2006-02-08T15:38:02Z |
| dc.date.embargoed.fl_str_mv | 2006-02-08T15:38:02Z |
| dc.format.none.fl_str_mv | application/pdf |
| dc.identifier.none.fl_str_mv | https://hdl.handle.net/1822/4382 |
| dc.language.none.fl_str_mv | eng |
| dc.publisher.none.fl_str_mv | Elsevier |
| dc.rights.none.fl_str_mv | http://purl.org/coar/access_right/c_abf2 |
| dc.subject.none.fl_str_mv | Lusternik-Schnirelmann category Hopf algebras up to homotopy Model categories |
| dc.title.fl_str_mv | On the algebraic approximation of Lusternik-Schnirelmann category |
| dc.type.none.fl_str_mv | http://purl.org/coar/resource_type/c_6501 |
| description | Algebraic approximations have proved to be very useful in the investigation of Lusternik-Schnirelmann category. In this paper the L.-S. category and its approximations are studied from the point of view of abstract homotopy theory. We introduce three notions of L.-S. category for monoidal cofibration categories, i.e., cofibration categories with a suitably incorporated tensor product. We study the fundamental properties of the abstract invariants and discuss, in particular, their behaviour with respect to cone attachments and products. Besides the topological L.-S. category the abstract concepts cover classical algebraic approximations of the L.-S. category such as the Toomer invariant, rational category, and the A- and M-categories of Halperin and Lemaire. We also use the abstract theory to introduce a new algebraic approximation of L.-S. category. This invariant which we denote by $\ell$ is the first algebraic approximation of the L.-S. category which is not necessarily $\leq 1$ for spaces having the same Adams-Hilton model as a wedge of spheres. For a space $X$ the number $\ell (X)$ can be determined from an Anick model of $X$. Thanks to the general theory one knows \textit{a priori} that $\ell$ is a lower bound of the L.-S. category which satisfies the usual product inequality and increases by at most 1 when a cone is attached to a space. |
| dirty | 0 |
| eu_rights_str_mv | openAccess |
| format | article |
| fulltext.url.fl_str_mv | https://repositorium.uminho.pt/bitstreams/98c6ddd5-b703-4257-add4-2bd29daded3e/download |
| id | rum_f2bf4fd116a9f8098d4cfe6e189e1cae |
| identifier.url.fl_str_mv | https://hdl.handle.net/1822/4382 |
| instacron_str | repositorium |
| institution | Universidade do Minho |
| instname_str | Universidade do Minho |
| language | eng |
| network_acronym_str | rum |
| network_name_str | RepositóriUM - Universidade do Minho |
| oai_identifier_str | oai:repositorium.uminho.pt:1822/4382 |
| organization_str_mv | urn:organizationAcronym:repositorium |
| person_str_mv | Kahl, Thomas |
| publishDate | 2003 |
| publisher.none.fl_str_mv | Elsevier |
| reponame_str | RepositóriUM - Universidade do Minho |
| repository_id_str | urn:repositoryAcronym:rum |
| service_str_mv | urn:repositoryAcronym:rum |
| spelling | engElsevierengAlgebraic approximations have proved to be very useful in the investigation of Lusternik-Schnirelmann category. In this paper the L.-S. category and its approximations are studied from the point of view of abstract homotopy theory. We introduce three notions of L.-S. category for monoidal cofibration categories, i.e., cofibration categories with a suitably incorporated tensor product. We study the fundamental properties of the abstract invariants and discuss, in particular, their behaviour with respect to cone attachments and products. Besides the topological L.-S. category the abstract concepts cover classical algebraic approximations of the L.-S. category such as the Toomer invariant, rational category, and the A- and M-categories of Halperin and Lemaire. We also use the abstract theory to introduce a new algebraic approximation of L.-S. category. This invariant which we denote by $\ell$ is the first algebraic approximation of the L.-S. category which is not necessarily $\leq 1$ for spaces having the same Adams-Hilton model as a wedge of spheres. For a space $X$ the number $\ell (X)$ can be determined from an Anick model of $X$. Thanks to the general theory one knows \textit{a priori} that $\ell$ is a lower bound of the L.-S. category which satisfies the usual product inequality and increases by at most 1 when a cone is attached to a space.application/pdfengOn the algebraic approximation of Lusternik-Schnirelmann categoryKahl, ThomasHostingInstitutionOrganizationalRepositóriUM - Universidade do Minhoe-mailmailto:repositorium@usdb.uminho.ptrepositorium@usdb.uminho.ptCITATION"Journal of Pure and Applied Algebra". ISSN 0022-4049. 181:2/3 (2003) 227-277.ISSNIsPartOf0022-4049DOIIsPartOf10.1016/S0022-4049(02)00306-72006-02-08T15:38:02Z2003-062003-06-01T00:00:00ZHandlehttps://hdl.handle.net/1822/4382http://purl.org/coar/access_right/c_abf2open accessLusternik-Schnirelmann categoryHopf algebras up to homotopyModel categories498954 bytesliteraturehttp://purl.org/coar/resource_type/c_6501journal articlehttp://purl.org/coar/access_right/c_abf2application/pdffulltexthttps://repositorium.uminho.pt/bitstreams/98c6ddd5-b703-4257-add4-2bd29daded3e/download |
| spellingShingle | On the algebraic approximation of Lusternik-Schnirelmann category Kahl, Thomas Lusternik-Schnirelmann category Hopf algebras up to homotopy Model categories |
| status | SINGLETON |
| subject.fl_str_mv | Lusternik-Schnirelmann category Hopf algebras up to homotopy Model categories |
| title | On the algebraic approximation of Lusternik-Schnirelmann category |
| title_full | On the algebraic approximation of Lusternik-Schnirelmann category |
| title_fullStr | On the algebraic approximation of Lusternik-Schnirelmann category |
| title_full_unstemmed | On the algebraic approximation of Lusternik-Schnirelmann category |
| title_short | On the algebraic approximation of Lusternik-Schnirelmann category |
| title_sort | On the algebraic approximation of Lusternik-Schnirelmann category |
| topic | Lusternik-Schnirelmann category Hopf algebras up to homotopy Model categories |
| topic_facet | Lusternik-Schnirelmann category Hopf algebras up to homotopy Model categories |
| url | https://hdl.handle.net/1822/4382 |
| visible | 1 |