Classifying spaces for homogeneous manifolds and their related Lie isometry deformations

TL;DR

This paper introduces a new topological framework using classifying spaces for homogeneous manifolds and their Lie isometry deformations, providing insights into their algebraic relationships and geometric classifications relevant to cosmology.

Contribution

It develops a novel topology on classifying spaces of Lie algebras, including the concept of transitions as limits of deformations, and explicitly constructs these spaces for low-dimensional cases.

Findings

Classifying spaces encode algebraic relationships between Lie algebras.

Transitions generalize Inönü-Wigner contractions and affect local geometry.

Explicit classification of 3-dimensional homogeneous spaces and their isometry types.

Abstract

Among plenty of applications, low-dimensional homogeneous spaces appear in cosmological models as both, classical factor spaces of multidimensional geometry and minisuperspaces in canonical quantization. Here a new tool to restrict their continuous deformations is presented: Classifying spaces for homogeneous manifolds and their related Lie isometry deformations. The adjoint representation of n-dimensional real Lie algebras induces a natural topology on their classifying space, which encodes the natural algebraic relationship between different Lie algebras therein. For n>1 this topology is not Hausdorffian. Even more it satisfies only the separation axiom T_0, but not T_1, i.e. there is a constant sequence which has a limit different from the members of the sequence. Such a limit is called a transition. Recently it was found that transitions are the natural generalization and transitive completion of the well-known In\"on\"u-Wigner contractions. For n<5 the relational classifying spaces are constructed explicitly. Calculating their characteristic scalar invariants via triad representations of the characteristic isometry, local homogeneous Riemannian 3-spaces are classified in their natural geometrical relations to each other. Their classifying space is a composition of pieces with different isometry types. Although it is Hausdorffian, different topological transitions to the same limit may induce locally non-Euclidean regions (e.g. at Bianchi tppes VII_0).