Let \(F\hookrightarrow M{\mathop {\rightarrow }\limits ^{q\,}}M/F\) be a fibration, where M is a compact Lie group, \(F\subset M\) is a closed subgroup and M/F is the homogeneous space of left cosets. Let W be an orientable, connected, closed manifold of the same dimension as M. Given maps \(f,g:W\rightarrow M/F\) that lift through q, we establish a comparison between the ranks of the \(\check{\textrm{C}}\) ech cohomology groups of the fiber F and those of the coincidence set \(\textrm{Coin}(f,g)\) . As a consequence, we show that for any maps \(f_1\simeq f\) and \(g_1\simeq g\) , the coincidence set \(\textrm{Coin}(f_1,g_1)\) cannot be a proper subset of any embedded copy of F within W, unless it is empty. We then apply this result to describe minimal coincidence sets for any pairs of maps from either \(S^3\) or \(\mathbb {R}\textrm{P}^3\) into either \(S^2\) or \(\mathbb {R}\textrm{P}^2\) .
