Parametrized homotopic distance

TL;DR

This paper introduces the parametrized homotopic distance in fibrewise topology, establishing bounds, properties, and relations with other invariants, extending classical concepts to a fibrewise setting with applications to fibrewise H-spaces.

Contribution

It extends the classical homotopic distance to the fibrewise context, providing bounds, properties, and relations with fibrewise sectional category and topological complexity, including a pointed version.

Findings

Establishes cohomological lower bounds for parametrized homotopic distance.

Shows the equality of parametrized topological complexity and fibrewise Lusternik-Schnirelman category for fibrewise H-spaces.

Provides sharp estimates for fibrewise H-spaces arising as sphere bundles with fibre S^7.

Abstract

We introduce the concept of parametrized homotopic distance, extending the classical notion of homotopic distance to the fibrewise setting. We establish its correspondence with the fibrewise sectional category of a specific fibrewise fibration and derive cohomological lower bounds and connectivity upper bounds under mild conditions. We also analyze the behavior of parametrized homotopic distance under compositions and products of fibrewise maps, along with its interaction with the triangle inequality. We establish several sufficient conditions for fibrewise $H$-spaces to admit a fibrewise division map and prove that their parametrized topological complexity equals their fibrewise unpointed Lusternik-Schnirelman category, extending Lupton and Scherer's theorem to the fibrewise setting. Additionally, we give sharp estimates for the parametrized topological complexity of a class fibrewise $H$-spaces which arises as sphere bundles with fibre $S^7$. Furthermore, we estimate the parametrized homotopic distance of fibre-preserving, fibrewise maps between fibrewise fibrations, in terms of the parametrized homotopic distance of the induced fibrewise maps between individual fibres, as well as the fibrewise unpointed Lusternik-Schnirelman category of the base space. Finally, we define and study a pointed version of parametrized homotopic distance, establishing cohomological bounds and identifying key conditions for its equivalence with the unpointed version, thus providing a finer classification of fibrewise homotopy invariants.