Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture

Unstable Modules over the Steenrod Algebra and Sullivan\'s Fixed Point Set Conjecture......Page 1
Contents......Page 7
Introduction......Page 11
Part 1. The algebraic structure of the category U and the functor TV......Page 24
1.1. The Steenrod algebra......Page 25
1.2. Generators for the Steenrod algebra......Page 26
1.3. The instability condition......Page 28
1.4. Unstable A-algebras......Page 29
1.5. Notation and basic example......Page 30
1.6. Free objects in the category U......Page 31
1.7. Instability and the Adem relations......Page 34
1.8. The category U is locally noetherian......Page 37
1.9. Proof of Proposition 1.6.3 and of Corollary 1.8.5......Page 39
1.10. Appendix on Milnor\'s dual A* of the Steenrod algebra and on Milnor\'s derivations......Page 40
2.1. Generalities......Page 43
2.2. A representability statement......Page 45
2.3. Brown-Gitler modules......Page 47
2.4. The bigraded module J**......Page 49
2.5. The relation between J** and Milnor\'s algebra A*......Page 53
2.6. Carlsson\'s modules K(i) and reduced stable A-modules......Page 54
2.7. Carlsson\'s bigraded algebra K**......Page 58
2.8. Tensor products of injective unstable A-modules......Page 60
2.9. The unstable modules K(i) and binary trees......Page 63
3.1. U-Injectivity of H*V ⊗ J(n)......Page 68
3.2. Lannes\' functor TV......Page 70
3.3. First examples of TV-computations......Page 74
3.4. Commutation of TV and Φ......Page 77
3.5. Commutation of TV with tensor products......Page 80
3.6. The case of an odd prime......Page 83
3.7. Comment and exercise......Page 85
3.8. The functor TV and unstable algebras......Page 87
3.9. Further examples of TV-computation......Page 93
3.10. The formula for TVH*BG......Page 96
3.11. The classification theorem for injective unstable A-modules......Page 97
3.12. Reduced indecomposable U-injectives......Page 100
3.13. The general case......Page 102
3.14. Applications......Page 105
Part 2. Deeper algebraic structure......Page 109
4.1. The structure of the set L......Page 110
4.2. Results on the Poincaré series of indecomposable reduced U-injectives......Page 112
4.3. The decomposition of the Carlsson modules K(i)......Page 117
4.4. Information about the A-module structure of the reduced indecomposable U-injectives......Page 118
5.1. The category U/Nil and the functor f¯......Page 120
5.2. Analytic functor......Page 123
5.3. Injective objects in the category Fω......Page 125
5.4. Proof of Theorem 5.2.6......Page 130
5.5. The functor pn:U/Nil→ModF2[Sn] and the filtration on U/Nil......Page 131
5.6. Simple objects of U/Nil......Page 136
5.7. Proof of Formula 4.3.1......Page 139
5.8. Comments on the Weyl `correspondence\'......Page 141
5.9. The case of an odd prime in the preceding sections......Page 142
5.10. The Grothendieck ring of U......Page 143
6.1. The categories Nill and the quotient categories Nill/Nill+1......Page 144
6.2. The category B of locally finite unstable A-modules and the categories N¯ill......Page 149
6.3. Localization away from Nill......Page 152
6.4. The categories Nill and the functors Tor......Page 154
Part 3. The Sullivan conjecture and the cohomology of mapping spaces......Page 158
7.1. Simplicial resolutions in the categories K and Alg......Page 159
7.2. André-Quillen cohomology of unstable A-algebras......Page 164
7.3. A connectivity result for André-Quillen cohomology......Page 167
7.4. Proof of Proposition 7.3.3......Page 169
7.5. Miller\'s spectral sequence......Page 174
7.6. A change of rings theorem......Page 175
7.7. Derivations......Page 177
7.8. Derived functors of derivations and Lannes\' theorem......Page 179
8.2. Bousfield-Kan\'s and Bousfield\'s theorems......Page 182
8.3. Resolutions of spaces and mapping spaces......Page 185
8.4. Proof of Theorem 8.1.1......Page 187
8.5. Comments about the connected components of the total space......Page 191
8.6. H. Miller\'s theorem and a converse......Page 192
8.7. Applications of the Eilenberg-Moore spectral sequence......Page 195
8.8. A topological characterization of spaces X such that H*X ∈ N¯ilk......Page 201
9.1. The Sullivan conjecture......Page 202
9.2. A comparison theorem......Page 204
9.3. The case of the Borel construction......Page 205
9.4. Proof of the Sullivan conjecture......Page 207
9.5. The case of the action of a finite p-group π......Page 209
9.6. The space map(BV,BG)......Page 210
9.7. The cohomology of mapping spaces with source BV......Page 212
9.8. A special case......Page 215
9.9. The Eilenberg-Moore spectral sequence......Page 217
9.10. A lemma of Bousfield......Page 220
References......Page 222
Index of notation......Page 231
Index......Page 232