Upper and lower bounds of the (co)chain type level of a space

TL;DR

This paper investigates bounds on the (co)chain type level of spaces, providing upper bounds for pull-back fibrations, examples for specific spaces, and relations to algebraic invariants like E-category and cohomology dimensions.

Contribution

It establishes new upper bounds for the cochain type level of total spaces in pull-back fibrations and relates chain type levels to algebraic invariants such as E-category.

Findings

Upper bounds for cochain type levels of pull-back fibrations.

Examples of levels for path spaces, Borel constructions, and specific spaces.

Chain type level of homotopy fibre exceeds E-category and relates to cohomology dimensions.

Abstract

We establish an upper bound for the cochain type level of the total space of a pull-back fibration. It explains to us why the numerical invariant for a principal bundle over the sphere are less than or equal to two. Moreover computational examples of the levels of path spaces and Borel constructions, including biquotient spaces and Davis-Januszkiewicz spaces, are presented. We also show that the chain type level of the homotopy fibre of a map is greater than the E-category in the sense of Kahl, which is an algebraic approximation of the Lusternik-Schnirelmann category of the map. The inequality fits between the grade and the projective dimension of the cohomology of the homotopy fibre.