Computing topological invariants using fixed points. (English) Zbl 1382.55010
Lin, Chang-Shou (ed.) et al., Proceedings of the sixth international congress of Chinese mathematicians, ICCM 2013, Taipei, Taiwan, July 14–19, 2013. Volume II. Somerville, MA: International Press; Beijing: Higher Education Press (ISBN 978-1-57146-349-4/pbk; 978-1-57146-350-0/set). Advanced Lectures in Mathematics (ALM) 37, 285-298 (2017).
The equivariant localization formula of Atiyah-Bott and Berline-Vergne converts the integral of an equivariantly closed form into a finite sum over the fixed points of the action. This paper describes how this can be exploited in the case of a homogeneous space \(G/H\) where a maximal torus for \(H\) is also one for \(G\) to compute characteristic numbers. The approach is also exploited to derive a formula for the Gysin map attached to a fibre bundle. Because the article is a survey proofs are outlined, details can be found in the references.
For the entire collection see [Zbl 1375.00096].
For the entire collection see [Zbl 1375.00096].
Reviewer: Jonathan Hodgson (Swarthmore)
MSC:
| 55R10 | Fiber bundles in algebraic topology |
| 55N25 | Homology with local coefficients, equivariant cohomology |
| 14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |
| 14M17 | Homogeneous spaces and generalizations |
| 57T15 | Homology and cohomology of homogeneous spaces of Lie groups |
| 55-02 | Research exposition (monographs, survey articles) pertaining to algebraic topology |
