Localization Theorem and Symplectic Torus Actions

Abstract

These notes are intended to be a brief introduction to the cohomology theory of transformation groups with applications to symplectic and Hamiltonian torus actions. More or less without exception — see, e.g., Theorem 1.6.5 — we consider only torus actions, and, often only circle actions. The main theorem of the subject is the Localization Theorem (Theorem 1.3.7), and, except in the more general statement and proof of this theorem, we use only cohomology with coefficients in a field of characteristic zero. Also, to further simplify the presentation, we prove the Localization Theorem only in the compact case, discussing other cases in remarks. As in the original Smith theory, the cohomology theory is useful too for studying elementary abelian p-group actions; but that will not be done here: see [4] for a more comprehensive introduction. For an introduction to compact Lie group actions in general, see [14], [28] or [56].