Spectral Sequences

Victor W. Guillemin, Shlomo Sternberg and Jochen Brüning
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Victor W. Guillemin: Massachusetts Institute of Technology, Department of Mathematics
Shlomo Sternberg: Harvard University, Department of Mathematics
Jochen Brüning: Humboldt-Universität Berlin, Institut für Mathematik Mathematisch-Naturwissenschaftliche Fakultät II

Chapter Chapter 6 in Supersymmetry and Equivariant de Rham Theory, 1999, pp 61-76 from Springer

Abstract: Abstract We begin this chapter with a review of the theory of spectral sequences in the special context of double complexes. We then apply these results to equivariant cohomology: We will show that if a G⋆ morphism between two G⋆ modules induces an isomorphism on cohomology it induces an isomorphism on equivariant cohomology. Given a G⋆ module A, we will discuss the structure of H G (A) as an S(g⋆)G-module, and show that if the spectral sequence associated with A collapses at its E1 stage then H G (A) is free as an S(g⋆)G-module. Finally, we will prove an abelianization theorem which says that $${H_G}\left( A \right) = {H_T}{\left( A \right)^W}$$ where T is a Cartan subgroup (maximal torus) of G and W its Weyl group.

Keywords: Spectral Sequence; Weyl Group; Maximal Torus; Cartan Subgroup; Equivariant Cohomology (search for similar items in EconPapers)
Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-03992-2_6

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DOI: 10.1007/978-3-662-03992-2_6

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