Spherical Milnor Spaces II: Projective Quotients and Higher Topological Structures

TL;DR

This paper develops a spherical variant of Milnor's classifying construction within diffeological geometry, exploring quotient spaces, twisted structures, and their relation to higher topological and geometric objects.

Contribution

It introduces a new spherical construction for diffeological groups, analyzing quotient spaces and their applications to higher bundles and cohomological obstructions.

Findings

Constructs a contractible diffeological space with group actions.

Defines a projective model and double quotient space for twisted structures.

Connects the framework to non-abelian gerbes and higher bundles.

Abstract

We introduce a spherical variant of Milnor's classifying construction for diffeological groups, based on quadratic normalization of barycentric coordinates. This construction gives rise to a contractible diffeological space endowed with commuting actions of a group $G$ and of $\mathbb{Z}_2$, leading naturally to a hierarchy of quotient spaces. We investigate the topological and geometric properties of these quotients, including a projective model and a double quotient space which encodes twisted and higher structures. In particular, we show that this framework provides a natural setting for the study of principal bundles with $\mathbb{Z}_2$-twists, and leads to obstruction classes in low-degree cohomology. The construction is further related to non-abelian gerbes and higher bundles, providing a bridge between diffeological geometry, classifying space theory, and higher topological structures.