Illinois Journal of Mathematics

Lefschetz Theory on Fibre bundles via Gysin homomorphism

Palanivel Manoharan

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Abstract

For a pair of fibre preserving continuous functions $f,g:E_{1}\rightarrow E_{2}$ between two compact smooth fibre bundles over $B$, we construct a transfer map $T(f,g):H^{*}(B)\rightarrow H^{*}(B)$ that generalizes Lefschetz number $\lambda_{f,g}$ of the pair of maps. If the pair $(f,g)$ is smooth satisfying a transversality condition and $T(f,g)$ is non-zero, then there is a surjective submersion from any connected component of $\{x\mid f(x)=g(x)\}$ to $B$. This yields a necessary and sufficient condition for a principal $G$-bundle over a simply connected compact manifold to be trivial and we also get a necessary condition for every smooth map from $S^{2n+1}$ to $S^{1}$ for all $n\geq1$.

Article information

Source
Illinois J. Math., Volume 57, Number 2 (2013), 595-602.

Dates
First available in Project Euclid: 19 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1408453596

Digital Object Identifier
doi:10.1215/ijm/1408453596

Mathematical Reviews number (MathSciNet)
MR3263047

Zentralblatt MATH identifier
1297.58003

Subjects
Primary: 58A10: Differential forms 58A12: de Rham theory [See also 14Fxx]
Secondary: 32C37: Duality theorems

Citation

Manoharan, Palanivel. Lefschetz Theory on Fibre bundles via Gysin homomorphism. Illinois J. Math. 57 (2013), no. 2, 595--602. doi:10.1215/ijm/1408453596. https://projecteuclid.org/euclid.ijm/1408453596


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References

  • R. Bott and L. Tu, Differential forms in algebraic topology, Springer, Heidelberg, 1982.
  • M. Karoubi and C. Leruste, Algebraic topology via differential geometry, London Math. Soc. Lecture Note Series, vol. 99, Cambridge University Press, Cambridge, 1987.
  • S. Lefschetz, On the fixed point formula, Ann. of Math. (2) 38 (1937), no. 4, 819–822.
  • P. Manoharan, (2008) Lefschetz fixed point theory on fibrations, Manuscripta Math. 125, 127–137.