What question did this study set out to answer?

The aim is to compare the ranks of cohomology groups associated with maps from a manifold into a homogeneous space.

April 12, 2026Journal of Fixed Point Theory and Applications0 citationsOpen Access

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

MFMarcio Colombo FenilleUniversidade Federal de Uberlândia DGDaciberg Lima GonçalvesBrazilian Society of Computational and Applied Mathematics

Key Points

The aim is to compare the ranks of cohomology groups associated with maps from a manifold into a homogeneous space.
Defined a fibration with a compact Lie group and its closed subgroup.
Investigated maps from an orientable, connected manifold into the homogeneous space.
Examined ranks of Cech cohomology groups of the fiber and the coincidence set.
Established that certain maps result in coincidence sets with specific topological properties.
Demonstrated under what conditions coincidence sets cannot be proper subsets of embedded occurrences of the fiber.
Described characteristics of minimal coincidence sets for mappings involving S3 or RP3.

Abstract

Abstract 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

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