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
Contents
Introduction
Part 1. The algebraic structure of the category U and the functor TV
	1. Recollections concerning the Steenrod algebra and unstable A-modules
		1.1. The Steenrod algebra
		1.2. Generators for the Steenrod algebra
		1.3. The instability condition
		1.4. Unstable A-algebras
		1.5. Notation and basic example
		1.6. Free objects in the category U
		1.7. Instability and the Adem relations
		1.8. The category U is locally noetherian
		1.9. Proof of Proposition 1.6.3 and of Corollary 1.8.5
		1.10. Appendix on Milnor's dual A* of the Steenrod algebra and on Milnor's derivations
	2. Algebraic Brown-Gitler technology
		2.1. Generalities
		2.2. A representability statement
		2.3. Brown-Gitler modules
		2.4. The bigraded module J**
		2.5. The relation between J** and Milnor's algebra A*
		2.6. Carlsson's modules K(i) and reduced stable A-modules
		2.7. Carlsson's bigraded algebra K**
		2.8. Tensor products of injective unstable A-modules
		2.9. The unstable modules K(i) and binary trees
	3. U-Injectivity of the mod p cohomology of elementary abelian p-groups and Lannes' functor TV
		3.1. U-Injectivity of H*V . J(n)
		3.2. Lannes' functor TV
		3.3. First examples of TV-computations
		3.4. Commutation of TV and .
		3.5. Commutation of TV with tensor products
		3.6. The case of an odd prime
		3.7. Comment and exercise
		3.8. The functor TV and unstable algebras
		3.9. Further examples of TV-computation
		3.10. The formula for TVH*BG
		3.11. The classification theorem for injective unstable A-modules
		3.12. Reduced indecomposable U-injectives
		3.13. The general case
		3.14. Applications
Part 2. Deeper algebraic structure
	4. The structure of indecomposable reduced U-injectives
		4.1. The structure of the set L
		4.2. Results on the Poincaré series of indecomposable reduced U-injectives
		4.3. The decomposition of the Carlsson modules K(i)
		4.4. Information about the A-module structure of the reduced indecomposable U-injectives
	5. The category U/Nil, analytic functors, and representations of the symmetric groups
		5.1. The category U/Nil and the functor f¯
		5.2. Analytic functor
		5.3. Injective objects in the category F.
		5.4. Proof of Theorem 5.2.6
		5.5. The functor pn:U/Nil>ModF2[Sn] and the filtration on U/Nil
		5.6. Simple objects of U/Nil
		5.7. Proof of Formula 4.3.1
		5.8. Comments on the Weyl `correspondence'
		5.9. The case of an odd prime in the preceding sections
		5.10. The Grothendieck ring of U
	6. Subcategories of U
		6.1. The categories Nill and the quotient categories Nill/Nill+1
		6.2. The category B of locally finite unstable A-modules and the categories N¯ill
		6.3. Localization away from Nill
		6.4. The categories Nill and the functors Tor
Part 3. The Sullivan conjecture and the cohomology of mapping spaces
	7. Non-Abelian homological algebra and André-Quillen cohomology
		7.1. Simplicial resolutions in the categories K and Alg
		7.2. André-Quillen cohomology of unstable A-algebras
		7.3. A connectivity result for André-Quillen cohomology
		7.4. Proof of Proposition 7.3.3
		7.5. Miller's spectral sequence
		7.6. A change of rings theorem
		7.7. Derivations
		7.8. Derived functors of derivations and Lannes' theorem
	8. On homotopy classes of maps from BV
		8.1. Miller's conjecture
		8.2. Bousfield-Kan's and Bousfield's theorems
		8.3. Resolutions of spaces and mapping spaces
		8.4. Proof of Theorem 8.1.1
		8.5. Comments about the connected components of the total space
		8.6. H. Miller's theorem and a converse
		8.7. Applications of the Eilenberg-Moore spectral sequence
		8.8. A topological characterization of spaces X such that H*X . N.ilk
	9. The generalized Sullivan conjecture and the cohomology of mapping spaces
		9.1. The Sullivan conjecture
		9.2. A comparison theorem
		9.3. The case of the Borel construction
		9.4. Proof of the Sullivan conjecture
		9.5. The case of the action of a finite p-group .
		9.6. The space map(BV,BG)
		9.7. The cohomology of mapping spaces with source BV
		9.8. A special case
		9.9. The Eilenberg-Moore spectral sequence
		9.10. A lemma of Bousfield
References
Index of notation
Index