Flasque quasi-resolutions of algebraic varieties

Flasque resolutions play an important role in understanding birational properties of algebraic tori. For instance, Colliot-Thélène and Sansuc have used them to compute $R$-equivalence classes of algebraic tori. We extend this notion to a larger class of algebraic varieties, including homogeneous spaces. This leads to a lower bound on the number of $R$-equivalence classes of homogeneous spaces, which is a slightly stronger version of a theorem of Colliot-Thélène and Kunyavskii.