The Abstract Localization Theorem

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.