The Abstract Localization Theorem

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 11 in Supersymmetry and Equivariant de Rham Theory, 1999, pp 173-188 from Springer

Abstract: 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.

Keywords: Exact Sequence; Short Exact Sequence; Tubular Neighborhood; Torsion Module; 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_11

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DOI: 10.1007/978-3-662-03992-2_11

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