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Spectral Sequences

In: Supersymmetry and Equivariant de Rham Theory

Author

Listed:
  • 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)

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.

Suggested Citation

  • Victor W. Guillemin & Shlomo Sternberg & Jochen Brüning, 1999. "Spectral Sequences," Springer Books, in: Supersymmetry and Equivariant de Rham Theory, chapter 0, pages 61-76, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03992-2_6
    DOI: 10.1007/978-3-662-03992-2_6
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