Real homogenous spaces, Galois cohomology, and Reeder puzzles

TL;DR

This paper develops a method to determine the number of connected components of real points of homogeneous spaces G/H for certain algebraic groups, using solutions to generalized Reeder puzzles, linking algebraic and topological properties.

Contribution

It introduces a novel approach combining Galois cohomology and Reeder puzzles to analyze the topology of real homogeneous spaces.

Findings

Provides a systematic method to count connected components of X(R)

Connects algebraic group theory with combinatorial puzzles

Offers new tools for studying real algebraic homogeneous spaces

Abstract

Let G be a simply connected absolutely simple algebraic group defined over the field of real numbers R. Let H be a simply connected semisimple R-subgroup of G. We consider the homogeneous space X=G/H. We ask: How many connected components has X(R)? We give a method of answering this question. Our method is based on our solutions of generalized Reeder puzzles.