Equivariant Homotopy and Cohomology Theory

Contents......Page page000005.djvu
Introduction......Page page000011.djvu
1. Some basic definitions and adjunctions......Page page000021.djvu
2. Analogues for based G-spaces......Page page000022.djvu
3. G-CW complexes......Page page000023.djvu
4. Ordinary homology and cohomology theories......Page page000026.djvu
5. Obstruction theory......Page page000028.djvu
6. Universal coefficient spectral sequences......Page page000029.djvu
1. Eilenberg-Mac Lane G-spaces and Postnikov systems......Page page000031.djvu
2. Summary: localizations of spaces......Page page000032.djvu
3. Localizations of G-spaces......Page page000033.djvu
4. Summary: completions of spaces......Page page000034.djvu
5. Completions of G-spaces......Page page000036.djvu
1. Summary: the theory of minimal models......Page page000037.djvu
2. Equivariant minimal models......Page page000039.djvu
3. Rational equivariant Hopf spaces......Page page000041.djvu
1. Smith theory via Bredon cohomology......Page page000043.djvu
2. Borel cohomology, localization, and Smith theory......Page page000045.djvu
1. Coends and geometric realization......Page page000049.djvu
2. Homotopy colimits and limits......Page page000051.djvu
3. Elmendorf\'s theorem on diagrams of fixed point spaces......Page page000053.djvu
4. Eilenberg-MacLane G-spaces and universal F-spaces......Page page000055.djvu
1. Elementary homotopy theory of diagrams......Page page000057.djvu
2. Homotopy groups......Page page000059.djvu
3. Cellular theory......Page page000060.djvu
4. The homology and cohomology theory of diagrams......Page page000062.djvu
5. The closed model structure on U^J......Page page000063.djvu
6. Another proof of Elmendorf\'s theorem......Page page000066.djvu
1. The definition of equivariant bundles......Page page000069.djvu
2. The classification of equivariant bundles......Page page000070.djvu
3. Some examples of classifying spaces......Page page000072.djvu
1. Statements of versions of the Sullivan conjecture......Page page000077.djvu
2. Algebraic preliminaries: Lannes\' functors T and Fix......Page page000080.djvu
3. Lannes\' generalization of the Sullivan conjecture......Page page000081.djvu
4. Sketch proof of Lannes\' theorem......Page page000083.djvu
5. Maps between classifying spaces......Page page000085.djvu
1. G-spheres in homotopy theory......Page page000089.djvu
2. G-Universes and stable G-maps......Page page000091.djvu
3. Euler characteristic and transfer G-maps......Page page000093.djvu
4. Mackey functors and coMackey functors......Page page000094.djvu
5. RO(G)-graded homology and cohomology......Page page000096.djvu
6. The Conner conjecture......Page page000098.djvu
1. Motivation for cellular theories based on representations......Page page000099.djvu
2. G-CW(V) complexes......Page page000100.djvu
3. Homotopy theory of G-CW(V) complexes......Page page000102.djvu
4. Ordinary RO(G)-graded homology and cohomology......Page page000104.djvu
1. Background on the classical theorems......Page page000107.djvu
2. Formulation of the problem and counterexamples......Page page000108.djvu
3. An oversimplified description of the results......Page page000111.djvu
4. The statements of the theorems......Page page000113.djvu
5. Sketch proofs of the theorems......Page page000116.djvu
1. An introductory overview......Page page000121.djvu
2. Prespectra and spectra......Page page000123.djvu
3. Smash products......Page page000125.djvu
4. Function spectra......Page page000127.djvu
6. Spheres and homotopy groups......Page page000129.djvu
7. G-CW spectra......Page page000132.djvu
8. Stability of the stable category......Page page000135.djvu
9. Getting into the stable category......Page page000136.djvu
1. Axioms for RO(G)-graded cohomology theories......Page page000139.djvu
2. Representing RO(G)-graded theories by G-spectra......Page page000141.djvu
3. Brown\'s theorem and RO(G)-graded cohomology......Page page000144.djvu
4. Equivariant Eilenberg-Mac Lane spectra......Page page000146.djvu
5. Ring G-spectra and products......Page page000149.djvu
1. The definition and basic properties of KG-theory......Page page000153.djvu
2. Bundles over a point: the representation ring......Page page000154.djvu
3. Equivariant Bott periodicity......Page page000156.djvu
4. Equivariant K-theory spectra......Page page000157.djvu
5. The Atiyah-Segal completion theorem......Page page000159.djvu
6. The generalization to families......Page page000161.djvu
1. A review of nonequivariant cobordism......Page page000163.djvu
2. Equivariant cobordism and Thom spectra......Page page000165.djvu
3. Computations: the use of families......Page page000168.djvu
4. Special cases: odd order groups and Z/2......Page page000171.djvu
1. Fixed point spectra and orbit spectra......Page page000173.djvu
2. Split G-spectra and free G-spectra......Page page000175.djvu
3. Geometric fixed point spectra......Page page000176.djvu
4. Change of groups and the Wirthmüller isomorphism......Page page000177.djvu
5. Quotient groups and the Adams isomorphism......Page page000179.djvu
6. The construction of G/N-spectra from G-spectra......Page page000181.djvu
7. Spanier-Whitehead duality......Page page000183.djvu
8. V-duality of G-spaces and Atiyah duality......Page page000185.djvu
9. Poincaré duality......Page page000186.djvu
1. Generalized Euler characteristics and transfer maps......Page page000189.djvu
2. The Burnside ring A(G) and the zero stem pi_0^G(S)......Page page000192.djvu
3. Prime ideals of the Burnside ring......Page page000193.djvu
4. Idempotent elements of the Burnside ring......Page page000195.djvu
5. Localizations of the Burnside ring......Page page000196.djvu
6. Localization of equivariant homology and cohomology......Page page000198.djvu
1. The transfer and a dimension-shifting variant......Page page000201.djvu
2. Basic properties of transfer maps......Page page000203.djvu
3. Smash products and Euler characteristics......Page page000205.djvu
4. The double coset formula and its applications......Page page000207.djvu
5. Transitivity of the transfer......Page page000211.djvu
1. The splitting of equivariant stable homotopy groups......Page page000213.djvu
2. Generalizations of the splitting theorems......Page page000216.djvu
3. Equivalent definitions of Mackey functors......Page page000217.djvu
4. Induction theorems......Page page000219.djvu
5. Splittings of rational G-spectra for finite groups G......Page page000222.djvu
1. The statement in terms of completions of G-spectra......Page page000225.djvu
2. A calculational reformulation......Page page000227.djvu
3. A generalization and the reduction to finite p-groups......Page page000229.djvu
4. The proof of the Segal conjecture for finite p-groups......Page page000231.djvu
5. Approximations of singular subspaces of G-spaces......Page page000233.djvu
6. An inverse limit of Adams spectral sequences......Page page000235.djvu
7. Further generalizations; maps between classifying spaces......Page page000237.djvu
1. Definitions and basic properties......Page page000241.djvu
2. Ordinary theories; Atiyah-Hirzebruch spectral sequences......Page page000244.djvu
3. Cohomotopy, periodicity, and root invariants......Page page000246.djvu
4. The generalization to families......Page page000247.djvu
5. Equivariant K-theory......Page page000250.djvu
6. Further calculations and applications......Page page000252.djvu
1. Introduction......Page page000257.djvu
2. The category GS(U\'; U)......Page page000260.djvu
3. Smash products and function spectra......Page page000261.djvu
4. The object M alpha in GS(U\'; U)......Page page000262.djvu
5. Twisted half-smash products and function spectra......Page page000265.djvu
6. Homotopical properties of alpha x E and F[alpha, E\')......Page page000267.djvu
1. The category of S-modules......Page page000271.djvu
2. Categories of R-modules......Page page000273.djvu
3. The algebraic theory of R-modules......Page page000275.djvu
4. The homotopical theory of R-ring spectra......Page page000277.djvu
5. Categories of R-algebra......Page page000281.djvu
6. Bousfield localizations of R-modules and algebras......Page page000284.djvu
7. Topological Hochschild homology and cohomology......Page page000287.djvu
8. The construction of THH via the standard complex......Page page000289.djvu
1. The category of L-spectra......Page page000293.djvu
2. Aoo and Eoo ring spectra and S-algebras......Page page000296.djvu
3. Alternative perspectives on equivariance......Page page000298.djvu
4. The construction of equivariant algebras and modules......Page page000301.djvu
5. Comparisons of categories of L-G-spectra......Page page000305.djvu
1. Introduction......Page page000309.djvu
2. Local and Cech cohomology in algebra......Page page000310.djvu
3. Brave new versions of local and Cech cohomology......Page page000311.djvu
4. Localization theorems in equivariant homology......Page page000312.djvu
5. Completions, completion theorems, and local homology......Page page000315.djvu
6. A proof and generalization of the localization theorem......Page page000317.djvu
7. The application to K-theory......Page page000320.djvu
8. Local Tate cohomology......Page page000321.djvu
1. The localization theorem for stable complex bordism......Page page000325.djvu
2. An outline of the proof......Page page000326.djvu
3. The norm map and its properties......Page page000328.djvu
4. The idea behind the construciton of norm maps......Page page000330.djvu
5. Global F*-functors with smash product......Page page000332.djvu
6. The definition of the norm map......Page page000335.djvu
7. The splitting of MU_G as an algebra......Page page000336.djvu
1. Statement of the theorem and preliminary results......Page page000337.djvu
2. Gysin sequences......Page page000339.djvu
3. Iterated completions......Page page000340.djvu
1. Notations and terminology......Page page000343.djvu
2. Stably almost complex structures and bordism......Page page000344.djvu
3. Tangential structures......Page page000346.djvu
4. Calculational tools......Page page000349.djvu
5. Statements of the main results......Page page000352.djvu
6. Preliminary lemmas and families in GxS^1......Page page000353.djvu
7. On the families F_i in GxS^1......Page page000354.djvu
8. Passing from G to GxS^1 and GxZ_k......Page page000359.djvu
Bibliography......Page page000363.djvu
Index......Page page000371.djvu