Surjectivity for Hamiltonian $G$-spaces in $K$-theory
- Autores: Megumi Harada, Gregory D. Landweber
- Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 12, 2007, págs. 6001-6025
- Idioma: inglés
- DOI: 10.1090/s0002-9947-07-04164-5
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Resumen
Let be a compact connected Lie group, and a Hamiltonian -space with proper moment map . We give a surjectivity result which expresses the -theory of the symplectic quotient in terms of the equivariant -theory of the original manifold , under certain technical conditions on . This result is a natural -theoretic analogue of the Kirwan surjectivity theorem in symplectic geometry. The main technical tool is the -theoretic Atiyah-Bott lemma, which plays a fundamental role in the symplectic geometry of Hamiltonian -spaces. We discuss this lemma in detail and highlight the differences between the -theory and rational cohomology versions of this lemma.
We also introduce a -theoretic version of equivariant formality and prove that when the fundamental group of is torsion-free, every compact Hamiltonian -space is equivariantly formal. Under these conditions, the forgetful map is surjective, and thus every complex vector bundle admits a stable equivariant structure. Furthermore, by considering complex line bundles, we show that every integral cohomology class in admits an equivariant extension in .
