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The Abstract Localization Theorem

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

In this chapter we will examine the localization theorem from a more abstract perspective and explain why such a theorem “has to be true”. As in Section 10.9 we will assume that the group G is a compact connected Abelian Lie group; i.e., an n dimensional torus. The main result of this chapter is a theorem of Borel and Hsiang which asserts that, for a compact G-manifold, M, the restriction map, H G (M) → 4 H G (M G ) is injective “modulo torsion”. From this we will deduce a theorem of Chang and Skjelbred which describes the image of this map when M is “equivariantly formal”. For this we will need the equivariant versions of some standard results about de Rham co-homology and some elementary commutative algebra. We will go over these prerequisites in Sections 11.1–11.3.

Suggested Citation

  • Victor W. Guillemin & Shlomo Sternberg & Jochen Brüning, 1999. "The Abstract Localization Theorem," Springer Books, in: Supersymmetry and Equivariant de Rham Theory, chapter 0, pages 173-188, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03992-2_11
    DOI: 10.1007/978-3-662-03992-2_11
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