  Flasque resolutions play an important role in understanding birational
properties of algebraic tori. For instance, Colliot-Th\'{e}l\`{e}ne 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-Th\'el\`ene and Kunyavskii.
