On Topological Complexity of $(r,ρ(R))$-mild spaces
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911883798249472 |
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| author | Benzaki, Smail Rami, Youssef |
| author_facet | Benzaki, Smail Rami, Youssef |
| contents | In this paper, we first prove the existence of relative free models of morphisms (resp. relative commutative models) in the category of $DGA(R)$ (resp. $CDGA(R)$), where $R$ is a principal ideal domain containing $\frac{1}{2}$. Next, we restrict to the category of $(r,ρ(R))$-H-mild algebras and we introduce, following Carrasquel's characterization, $secat(-, R)$, the sectional category for surjective morphisms. We then apply this to the $n$-fold product of the commutative model of an $(r,ρ(R))$-mild CW-complex of finite type to introduce $TC_n(X,R)$, $mTC_n(X,R)$ and $HTC_n(X,R)$ which extend well known rational topological complexities. We do the same for $\operatorname{sc(-, \mathbb{Q})}$ to introduce analogous algebraic $\operatorname{sc(-,R)}$ in terms of their commutative models over $R$ and prove that it is an upper bound for $secat(-, R)$. This also yields, for any $(r,ρ(R))$-mild CW-complex, the algebraic $tc_n(X,R)$, $mtc_n(X,R)$ and $Htc_n(X,R)$ whose relation to the homology nilpotency is investigated. In the last section, in the same spirit, we introduce in $DGA(R)$, $secat(-, R)$, $\operatorname{sc(-,R)}$ and their topological correspondents. We then prove, in particular, that $ATC_n(X,R)\leq TC_n(X,R)$ and $Atc_n(X,R)\leq tc_n(X,R)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13257 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Topological Complexity of $(r,ρ(R))$-mild spaces Benzaki, Smail Rami, Youssef Algebraic Topology Primary 55P62, Secondary 55M30 In this paper, we first prove the existence of relative free models of morphisms (resp. relative commutative models) in the category of $DGA(R)$ (resp. $CDGA(R)$), where $R$ is a principal ideal domain containing $\frac{1}{2}$. Next, we restrict to the category of $(r,ρ(R))$-H-mild algebras and we introduce, following Carrasquel's characterization, $secat(-, R)$, the sectional category for surjective morphisms. We then apply this to the $n$-fold product of the commutative model of an $(r,ρ(R))$-mild CW-complex of finite type to introduce $TC_n(X,R)$, $mTC_n(X,R)$ and $HTC_n(X,R)$ which extend well known rational topological complexities. We do the same for $\operatorname{sc(-, \mathbb{Q})}$ to introduce analogous algebraic $\operatorname{sc(-,R)}$ in terms of their commutative models over $R$ and prove that it is an upper bound for $secat(-, R)$. This also yields, for any $(r,ρ(R))$-mild CW-complex, the algebraic $tc_n(X,R)$, $mtc_n(X,R)$ and $Htc_n(X,R)$ whose relation to the homology nilpotency is investigated. In the last section, in the same spirit, we introduce in $DGA(R)$, $secat(-, R)$, $\operatorname{sc(-,R)}$ and their topological correspondents. We then prove, in particular, that $ATC_n(X,R)\leq TC_n(X,R)$ and $Atc_n(X,R)\leq tc_n(X,R)$. |
| title | On Topological Complexity of $(r,ρ(R))$-mild spaces |
| topic | Algebraic Topology Primary 55P62, Secondary 55M30 |
| url | https://arxiv.org/abs/2405.13257 |