Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2510.12671 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910219908415488 |
|---|---|
| author | Parent, Paul-Eugène Tanré, Daniel |
| author_facet | Parent, Paul-Eugène Tanré, Daniel |
| contents | Lusternik-Schnirelmann category (LS-category) of a topological space is the least integer $n$ such that there is a covering of $X$ by $n+1$ open sets, each of them being contractible in $X$. The cone length is the minimum number of cofibations necessary to get a space in the homotopy type of $X$, starting from a suspension and attaching suspensions. The LS-category of a space is always less than or equal to its cone length. Moreover, these two invariants differ by at most one. In 1981, J.-M. Lemaire and F. Sigrist conjectured that they are always equal for rational spaces. This conjecture is clearly true for spaces of LS-category 1 and, in 1986, Y. Félix and J-C. Thomas verify it for spaces of LS-category 2. But, in 1999, the general conjecture is invalidated by N. Dupont who built a rational space of cone-length 4 and LS-category 3. In this work, we provide examples of rational spaces of cone-length $(k+1)$ and LS-category $k$ for any $k>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_12671 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cone length and Lusternik-Schnirelmann category in rational homotopy Parent, Paul-Eugène Tanré, Daniel Algebraic Topology Lusternik-Schnirelmann category (LS-category) of a topological space is the least integer $n$ such that there is a covering of $X$ by $n+1$ open sets, each of them being contractible in $X$. The cone length is the minimum number of cofibations necessary to get a space in the homotopy type of $X$, starting from a suspension and attaching suspensions. The LS-category of a space is always less than or equal to its cone length. Moreover, these two invariants differ by at most one. In 1981, J.-M. Lemaire and F. Sigrist conjectured that they are always equal for rational spaces. This conjecture is clearly true for spaces of LS-category 1 and, in 1986, Y. Félix and J-C. Thomas verify it for spaces of LS-category 2. But, in 1999, the general conjecture is invalidated by N. Dupont who built a rational space of cone-length 4 and LS-category 3. In this work, we provide examples of rational spaces of cone-length $(k+1)$ and LS-category $k$ for any $k>2$. |
| title | Cone length and Lusternik-Schnirelmann category in rational homotopy |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2510.12671 |