Title: Arithmetic purity of strong approximation for homogeneous spaces,
Abstract: We prove that any open subset $U$ of a semi-simple simply connected
quasi-split linear algebraic group $G$ with ${codim} (G\setminus U, G)\geq 2$
over a number field satisfies strong approximation by establishing a fibration
of $G$ over a toric variety. We also prove a similar result of strong
approximation with Brauer-Manin obstruction for a partial equivariant smooth
compactification of a homogeneous space where all invertible functions are
constant and the semi-simple part of the linear algebraic group is quasi-split.
Some semi-abelian varieties of any given dimension where the complements of a
rational point do not satisfy strong approximation with Brauer-Manin
obstruction are given.