On the coincidence set of certain maps into homogeneous spaces of compact Lie groups

Let F M → q\,M/F F ↪ M → q M / F be a fibration, where M is a compact Lie group, F⊂ M F ⊂ 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→ M/F f , g : W → M / F that lift through q , we establish a comparison between the ranks of the C C ˇ ech cohomology groups of the fiber F and those of the coincidence set Coin(f,g) Coin ( f , g ) . As a consequence, we show that for any maps f₁ f f 1 ≃ f and g₁ g g 1 ≃ g , the coincidence set Coin(f₁,g₁) 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 S³ S 3 or RP³ R P 3 into either S² S 2 or RP² R P 2