Cohomological Methods in Transformation Groups
In the large and thriving field of compact transformation groups an important role has long been played by cohomological methods. This book aims to give a contemporary account of such methods, in particular the applications of ordinary cohomology theory and rational homotopy theory with principal emphasis on actions of tori and elementary abelian p-groups on finite-dimensional spaces. For example, spectral sequences are not used in Chapter 1, where the approach is by means of cochain complexes; and much of the basic theory of cochain complexes needed for this chapter is outlined in an appendix. For simplicity, emphasis is put on G-CW-complexes; the refinements needed to treat more general finite-dimensional (or finitistic) G-spaces are often discussed separately. Subsequent chapters give systematic treatments of the Localization Theorem, applications of rational homotopy theory, equivariant Tate cohomology and actions on Poincaré duality spaces. Many shorter and more specialized topics are included also. Chapter 2 contains a summary of the main definitions and results from Sullivan's version of rational homotopy theory which are used in the book.
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Chapter 1 Equivariant cohomology of GCWcomplexes and the Borel construction
Chapter 2 Summary of some aspects of rational homotopy theory
Chapter 3 Localization
Chapter 4 General results on torus and ptor us actions
Chapter 5 Actions on Poincare duality spaces
Appendix A Commutative alge bar
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A-module acts trivially Alexander-Spanier algebra assume Borel Bredon CGDA cochain complex coefﬁcients commutative commutative ring component condition LT connected cup-product deﬁned Deﬁnition Deﬁnition Let degree denote Diagram differential dimk dimQ elements equivariant equivariant cohomology exact sequence example exterior algebra ﬁbration ﬁeld of characteristic ﬁltration ﬁnite ﬁnite group ﬁnite-dimensional G-CW-complex ﬁnitistic ﬁrst ﬁxed point set FMCOT follows functor G acts graded Gysin homomorphism Halperin Hence HG(X homogeneous homology homomorphism homotopy equivalence inclusion induced integer isomorphism Lemma Leray-Serre spectral sequence Let f Let G Localization Theorem manifold minimal Hirsch-Brown model minimal model module morphism Noetherian non-zero notation orbit p-torus paracompact G-space Poincaré algebra Poincaré duality polynomial prime ideal Proof rational homotopy Remark resp ring satisﬁes satisfying condition Section space Spec(R submodule subtorus Suppose surjective Theorem Theorem Let topological topological space