Real homogeneous spaces, Galois cohomology, and Reeder puzzles.

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. [ABSTRACT FROM AUTHOR]