Related papers: Sectional number of a morphism
In this paper, we systematically study the $m$-dimensional sectional category of a fibration, introduced by Schwarz, as an approximating invariant for the sectional category. We develop the basic theory of this invariant, establish its…
We explore the notion of sectional number of a group homomorphism, leading to a generalization of the covering number of a group, and present several characterizations when the sectional number is finite, providing estimates for computing…
Schwarz genus $\mathsf{g}(\xi)$ of a fibration $\xi\colon E\to B$ is defined as the minimal integer $n$, such that there exists a cover of $B$ by $n$ open sets that admit partial section to $\xi$. Many important concepts, including…
The concept of relative sectional category expands upon classical sectional category theory by incorporating the pullback of a fibration along a map. Our paper aims not only to explore this extension but also to thoroughly investigate its…
We give simple upper bounds for rational sectional category and use them to compute invariants of the type of Farber's topological complexity of rational spaces. In particular we show that the sectional category of formal morphisms reaches…
Let $X$ be a $G$-space. In this paper, we introduce the notion of sectional category with respect to $G$. As a result, we obtain $G$-homotopy invariants: the LS category with respect to $G$, the sequential topological complexity with…
We formulate the concept of minimal fibration in the context of fibrations in the model category $\mathbf{S}^\mathcal{C}$ of $\mathcal{C}$-diagrams of simplicial sets, for a small index category $\mathcal{C}$. When $\mathcal{C}$ is an…
Based on a Whitehead-type characterization of the sectional category we develop the notion of weak sectional category. This is a new lower bound of the sectional category, which is inspired by the notion of weak category in the sense of…
By analogy with the invariant Q-category defined by Scheerer, Stanley and Tanr\'e, we introduce the notions of Q-sectional category and Q-topological complexity. We establish several properties of these invariants. We also obtain a formula…
We study a generalization of the Svarc genus of a fiber map. For an arbitrary collection E of spaces and a map f:X-->Y, we define a numerical invariant, the E-sectional category of f, in terms of open covers of Y. We obtain several basic…
We develop a theory of generalized Hopf invariants in the setting of sectional category. In particular we show how Hopf invariants for a product of fibrations can be identified as shuffle joins of Hopf invariants for the factors. Our…
The sectional category of a subgroup inclusion $H \hookrightarrow G$ can be defined as the sectional category of the corresponding map between Eilenberg--MacLane spaces. We extend a characterization of topological complexity of aspherical…
The \textit{slope} of a fibered $3$-folds $f:X \to B$ is a relative numerical invariant defined by $\lambda(f) := K_{f}^{3}/\mathrm{deg}(f_{\ast}\omega_{f})$, where $K_{f}$ is the relative canonical divisor and $\omega_{f}$ is the relative…
In the paper we introduce the notions of a singular fibration and a singular Seifert fibration. These notions are natural generalizations of the notion of a locally trivial fibration to the category of stratified pseudomanifolds. For…
In this paper we characterize the sectional category of subgroup inclusions and the $r^{th}$-sequential topological complexity of aspherical spaces of a group G in terms of the A-genus in the sense of Clapp-Puppe and Bartsch for a suitable…
For a Hausdorff space $X$, we exhibit an unexpected connection between the sectional number of the Fadell-Neuwirth fibration $\pi_{2,1}^X:F(X,2)\to X$, and the fixed point property (FPP) for self-maps on $X$. Explicitly, we demonstrate that…
Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op} \rightsquigarrow {\bf Cat}$, while the special case of discrete fibrations encode presheaves $B^{op} \to {\bf Set}$. A two-sided…
We introduce the concept of injective category number $\text{IC}(f)$ for a continuous map $f\colon X\to~Y$, and present fundamental results concerning this numerical invariant. The value $\text{IC}(f)$ quantifies the \aspas{complexity} or…
We construct an explicit semifree model for the fiber join of two fibrations p: E --> B and p': E' --> B from semifree models of p and p'. Using this model, we introduce a lower bound of the sectional category of a fibration p which can be…
We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction on its CW skeleta. The k-th term in the categorical sequence…