About sectional category of the Ganea maps
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arXiv
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| Format: | Preprint |
| Publié: |
2015
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| _version_ | 1866913910501670912 |
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| author | Doeraene, Jean-Paul |
| author_facet | Doeraene, Jean-Paul |
| contents | We first compute the James' sectional category (secat) of the Ganea map g_k of any map f in terms of the sectional category of f: We show that secat(g_k) is the integer part of secat(f)/(k+1). Next we compute the relative category (relcat) of g_k. In order to do this, we introduce the relative category of order k (relcat_k) of a map and show that relcat(g_k) is the integer part of relcat_k(f)/(k+1). Then we establish some inequalities linking secat and relcat of any order: We show that secat(f) <= relcat_k(f) <= secat(f) + k + 1 and relcat_k(f) <= relcat_(k+1)(f) <= relcat_k(f) + 1. We give examples that show that these inequalities may be strict. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1507_07752 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | About sectional category of the Ganea maps Doeraene, Jean-Paul Algebraic Topology 55M30 We first compute the James' sectional category (secat) of the Ganea map g_k of any map f in terms of the sectional category of f: We show that secat(g_k) is the integer part of secat(f)/(k+1). Next we compute the relative category (relcat) of g_k. In order to do this, we introduce the relative category of order k (relcat_k) of a map and show that relcat(g_k) is the integer part of relcat_k(f)/(k+1). Then we establish some inequalities linking secat and relcat of any order: We show that secat(f) <= relcat_k(f) <= secat(f) + k + 1 and relcat_k(f) <= relcat_(k+1)(f) <= relcat_k(f) + 1. We give examples that show that these inequalities may be strict. |
| title | About sectional category of the Ganea maps |
| topic | Algebraic Topology 55M30 |
| url | https://arxiv.org/abs/1507.07752 |