On Topological Complexity of $(r,\rho(R))$-mild spaces

TL;DR

This paper explores the topological complexity of $(r, ho(R))$-mild spaces by establishing algebraic models, defining sectional categories, and relating these to topological and homological invariants, extending known rational complexities.

Contribution

It introduces new algebraic models and invariants for $(r, ho(R))$-mild spaces, connecting algebraic and topological complexities, and generalizing rational topological complexity concepts.

Findings

Existence of relative free models in $DGA(R)$ and $CDGA(R)$.

Introduction of algebraic invariants $tc_n$, $mtc_n$, $Htc_n$ for $(r, ho(R))$-mild CW-complexes.

Proved inequalities relating algebraic and topological complexities, such as $ATC_n(X,R) \\leq TC_n(X,R)$.

Abstract

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 $(r,\rho(R))$-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 $(r,\rho(R))$-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 $(r,\rho(R))$-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 $Atc_n(X,R)\leq tc_n(X,R)$.