Cohomology Theory of Topological Transformation Groups

ISBN-13: 9783642660542 / Angielski / Miękka / 2011 / 166 str.

W. y. Hsiang

Opis

Historically, applications of algebraic topology to the study of topological transformation groups were originated in the work of L. E. 1. Brouwer on periodic transformations and, a little later, in the beautiful fixed point theorem ofP. A. Smith for prime periodic maps on homology spheres. Upon comparing the fixed point theorem of Smith with its predecessors, the fixed point theorems of Brouwer and Lefschetz, one finds that it is possible, at least for the case of homology spheres, to upgrade the conclusion of mere existence (or non-existence) to the actual determination of the homology type of the fixed point set, if the map is assumed to be prime periodic. The pioneer result of P. A. Smith clearly suggests a fruitful general direction of studying topological transformation groups in the framework of algebraic topology. Naturally, the immediate problems following the Smith fixed point theorem are to generalize it both in the direction of replacing the homology spheres by spaces of more general topological types and in the direction of replacing the group tl by more general compact groups.

Spis treści

I. Generalities on Compact Lie Groups and G-Spaces.- § 1. General Properties of Compact Topological Groups.- § 2. Generalities of Fibre Bundles and Free G-Spaces.- § 3. The Existence of Slice and its Consequences on General G-Space.- §4. General Theory of Compact Connected Lie Groups.- II. Structural and Classification Theory of Compact Lie Groups and Their Representations.- § 1. Orbit Structure of the Adjoint Action.- § 2. Classification of Compact Connected Lie Groups.- § 3. Classification of Irreducible Representations.- III. An Equivariant Cohomology Theory Related to Fibre Bundle Theory.- § 1. The Construction of HG*(X) and its Formal Properties.- § 2. Localization Theorem of Borel-Atiyah-Segal Type.- IV The Orbit Structure of a G-Space X and the Ideal Theoretical Invariants of HG*(X).- § 1. Some Basic Fixed Point Theorems.- § 2. Torsions of Equivariant Cohomology and F-Varieties of G-Spaces.- § 3. A Splitting Theorem for Poincaré Duality Spaces.- V. The Splitting Principle and the Geometric Weight System of Topological Transformation Groups on Acyclic Cohomology Manifolds or Cohomology Spheres.- § 1. The Splitting Principle and the Geometric Weight System for Actions on Acyclic Cohomology Manifolds.- § 2. Geometric Weight System and Orbit Structure.- § 3. Classification of Principal Orbit Types for Actions of Simple Compact Lie Groups on Acyclic Cohomology Manifolds.- §4. Classification of Connected Principal Orbit Types for Actions of (General) Compact Connected Lie Groups on Acyclic Cohomology Manifolds.- § 5. A Basis Fixed Point Theorem.- § 6. Low Dimensional Topological Representations of Compact Connected Lie Groups.- § 7. Concluding Remarks Related to Geometric Weight System.- VI. The Splitting Theorems and the Geometric Weight System of Topological Transformation Groups on Cohomology Projective Spaces.- § 1. Transformation Groups on Cohomology Complex Projective Spaces.- § 2. Transformation Groups on Cohomology Quaternionic Projective Spaces.- § 3. Structure Theorems for Actions of ?p-Tori on ?p-Cohomology Projective Spaces (p Odd Primes).- §4. Structure Theorems for Actions of ?2-Tori on ?2-Cohomology Projective Spaces.- VII Transformation Groups on Compact Homogeneous Spaces.- § 1. Topological Transformation Groups on Spaces of the Rational Homotopy Type of Product of Odd Spheres.- §2. Degree of Symmetry of Compact Homogeneous Spaces.- References.