Abstract
For a fibre preserving map ϕ: E → E on a fibration (E, π, B), we construct a grading preserving map T(ϕ, π) between H*(E) and H*(B) that generalizes the Lefschetz number. If T(ϕ, π) is an isomorphism between H 0(E) and H 0(B), then π restricts to a surjective local diffeomorphism on each connected component of the fixed point set of ϕ under a transversality condition. This yields a characterization for the bundle H → G → G/H to be trivial when π 1 (G/H) = 0.
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Manoharan, P. Lefschetz fixed point theory on fibrations. manuscripta math. 125, 127–137 (2008). https://doi.org/10.1007/s00229-007-0145-8
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DOI: https://doi.org/10.1007/s00229-007-0145-8