The principal fibration sequence and the second cohomotopy set

TL;DR

This paper explores the structure of homotopy classes related to principal fibrations, classifying maps, and loop spaces, providing new insights into the algebraic and topological properties of these mappings.

Contribution

It introduces a novel description of the set of lifts to the free loop space and characterizes isotropy subgroups for various maps in principal fibrations.

Findings

The set of homotopy classes of lifts to the free loop space forms a group.

The set of lifts of a map to the total space is identified with a cokernel of a natural homomorphism.

Enumeration of homotopy classes of maps from a 4-complex to S^2 is provided.

Abstract

Let $p:E -> B$ be a principal fibration with classifying map $w:B -> C$. It is well-known that the group $[X,\Omega C]$ acts on $[X,E]$ with orbit space the image of $p_#$, where $p_#: [X,E] -> [X,B]$. The isotropy subgroup of the map of $X$ to the base point of $E$ is also well-known to be the image of $[X, \Omega B]$. The isotropy subgroups for other maps $e:X -> E$ can definitely change as $e$ does. The set of homotopy classes of lifts of $f$ to the free loop space on $B$ is a group. If $f$ has a lift to $E$, the set $p_#^{-1}(f)$ is identified with the cokernel of a natural homomorphism from this group of lifts to $[X, \Omega C]$. As an example, $[X,S^2]$ is enumerated for $X$ a 4-complex.