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Computing the Gysin Map Using Fixed Points

Part of the Progress in Mathematics book series (PM,volume 321)

Abstract

The Gysin map of a map between compact oriented manifolds is the map in cohomology induced by the push-forward map in homology. In enumerative algebraic geometry, formulas for the Gysin map of a flag bundle play a vital role. These formulas are usually proven by algebraic or combinatorial means. This article shows how the localization formula in equivariant cohomology provides a systematic method for calculating the Gysin homomorphism in the ordinary cohomology of a fiber bundle. As examples, we recover classical pushforward formulas for generalized flag bundles. Our method extends the classical formulas to fiber bundles with equivariantly formal fibers.

Keywords

  • Atiyah–Bott–Berline–Vergne localization formula
  • equivariant localization formula
  • pushforward
  • Gysin map
  • equivariant cohomology
  • Lagrange–Sylvester symmetrizer
  • Jacobi symmetrizer

Mathematics Subject Classification (2010). Primary: 55R10, 55N25, 14C17; Secondary: 14M17

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Correspondence to Loring W. Tu .

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Tu, L.W. (2017). Computing the Gysin Map Using Fixed Points. In: Mourtada, H., Sarıoğlu, C., Soulé, C., Zeytin, A. (eds) Algebraic Geometry and Number Theory . Progress in Mathematics, vol 321. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-47779-4_5

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