nLab Transformation Groups

This entry compiles material related to the textbook

• Tammo tom Dieck,

  
  

  Transformation Groups

  
  

  de Gruyter 1987

  doi:10.1515/9783110858372

on transformation groups/topological G-spaces, equivariant homotopy theory and equivariant cohomology.

Beware the existence of a book of similar title, by the same author and largely on the same subjects:

• Tammo tom Dieck, Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766, Springer 1979 (doi:10.1007/BFb0085965)

See also:

• Glen Bredon, Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596

  , pdf)

Chapter I Foundations

1. Basic notions

2. General remarks. Examples

3. Elementary properties

4. Functorial properties

5. Differentiable manifolds. Tubes and slices

   • equivariant differential topology

   • slice theorem

6. Families of subgroups

7. Equivariant maps

   • equivariant function

8. Bundles

   • equivariant bundle

9. Vector bundles

   • equivariant vector bundle

10. Orbit categories, fundamental groups, and coverings

• orbit category

  • equivariant set of connected components

  • equivariant fundamental groupoid

  • equivariant universal covering space

1. Elementary algebra of transformation groups

Chapter II Algebraic Topology

1. Equivariant CW-complexes

   • G-CW complex

2. Maps between complexes

3. Obstruction theory

4. The classification theorem of Hopf

   • equivariant Hopf degree theorem

5. Maps between complex representation spheres

6. Stable homotopy. Homology. Cohomology

   • G-spectrum

   • equivariant stable homotopy theory

   • equivariant cohomology

7. Homology with families

8. The Burnside ring and stable homotopy

   • equivariant stable Cohomotopy

9. Bredon homology and Mackey functors

   • Bredon cohomology, Mackey functor

10. Homotopy representations

Chapter III Localization

1. Equivariant bundle cohomology

2. Cohomology of some classifying spaces

3. Localization

4. Applications of localization

5. Borel-Smith functions

6. Further results for cyclic groups. Applications

Chapter IV The Burnside Ring

1. Additive invariants

2. The Burnside ring

   • Burnside ring

3. The space of subgroups

4. Prime ideals

5. Congruences

6. Finiteness theorems

7. Idempotent elements

8. Induction categories

9. Induction theory

10. The Burnside ring and localization

category: reference