Activities Spring 2014

Tuesday activities take place  between 15:15-17:00. When at KTH, then in room 3418, and when at SU, then in room 306. Divergence from this is marked with a "*" and completed by, if needed, neccessary information.  

April 8, 15:15-17:00, room 306 SU. Tilman continues.

April 1, 15:15-17:00 room 3418 KTH.  We will continue talking  about  André-Qullen cohomology. 

March 18, 15:15-17:00, room 3418 KTH.  Sebastian Öberg will tell us about  André-Qullen cohomology. 

March 11, 15:15-17:00, room 306 SU. Tilman speakes about the division functor

March 4, 15:15-17:00, room 3418 KTH. João will continue discussing unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture.  

February 18, 15:15-17:00, room 3418 KTH. Magnus and  João will continue discussing unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture.  

February 11, 15:15-17:00, room 306 SU. Anssi's talk is  be based on the second chapter of Lionel Schwartz's book "Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture"

February 4, 15:15-17:00, room 3418 KTH. Tilman Bauer: Introduction to unstable modules, Lannes’s T-functor, and the Sullivan ConjectureAbstract: The weakest version of the Sullivan Conjecture states that every continuous map from BG to X, where X is a finite cell complex, is homotopic to a constant map. Here BG is the classifying space of a finite group G (e.g. infinite projective space for G=Z/2Z). The Sullivan Conjecture was proven by G. Carlsson for Z/2Z and by H. Miller in general in the 1980s. I will give an overview of this very conceptual proof and of the theory and methods that go into it, featuring:

* Homological algebra of unstable modules over the Steenrod algebra and the fact that H^*(BZ/pZ;Z/p) is an injective object in this category

* Lannes’s division functors and in particular his functor T: H^*(X) -> H^*(map(BZ/pZ,X))

* André-Quillen cohomology of commutative algebras

* p-completion of spaces and the unstable Adams spectral sequence

This overview talk will kick of a new topic in the weekly topology reading seminar. Everybody, expecially advanced students, is invited to participate.

Here is a suggestion from Tilman of how we might structure the talks of the seminar on Lannes’s functor T and the Sullivan conjecture. There is lots of literature out there, but I’ve collected some sources that I think are particularly good. The schedule just a suggestion, we can deviate or rearrange or add or take away at any point. Anssi is talking next Tuesday and after that we need speakers.

1-2. Unstable modules over the Steenrod algebra and their properties. Projectives and injectives. [14, Chapters 1-2, 3.1]
3. [optional] Kuhn’s generic representations and an alternative view on the injectivity on H^*(BV) [6-9,11]
4. Lannes’s division functors, the functor T and its properties, sample computations [14, Chapters 3.2-3.7]
5. Derived functors of non-additive functors and André-Quillen homology [2]; [14, Chapters 7.1-2]; [10, Chapter 1]; ...
6. The functor T and unstable algebras [14, Chapters 3.8-10], [9, 2.3-2.6]
7-8. p-completion of spaces and the Bousfield-Kan spectral sequence [3]
9-10. On maps out of BV [10, 3.1-3.2]; [13], [5], [1], [14, Chapter 8]
11. Sullivan’s conjecture [4], [12], [12a]
12. The generalized Sullivan conjecture [14, Chapter 9]; [10, Chapter 4]

[1] Aguadé, Broto, Saumell: The functor T and the cohomology of mapping spaces. Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001), 1–20, Progr. Math., 215

[2] Bousfield, A. K.: Appendix of "Nice homology coalgebras." Trans. Amer. Math. Soc. 148 1970 473–489.

[3] Bousfield, A. K.: Kan, D. M.: Homotopy limits, completions and localizations. Lecture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972. v+348 pp.

[4] Carlsson, Gunnar: Equivariant stable homotopy and Sullivan's conjecture. Invent. Math. 103(1991), no. 3, 497–525.

[5] Dror Farjoun, E.; Smith, J.: A geometric interpretation of Lannes' functor T. International Conference on Homotopy Theory (Marseille-Luminy, 1988). Astérisque No. 191 (1990), 6, 87–95.

[6] Kuhn, Nicholas J.: Generic representation theory and Lannes' T-functor. Adams Memorial Symposium on Algebraic Topology, 2 (Manchester, 1990), 235–262, London Math. Soc. Lecture Note Ser., 176, Cambridge Univ. Press, Cambridge, 1992.

[7] Kuhn, Nicholas J.: Generic representations of the finite general linear groups and the Steenrod algebra. I. Amer. J. Math. 116 (1994), no. 2, 327–360.

[8] Kuhn, Nicholas J.: Generic representations of the finite general linear groups and the Steenrod algebra. II. K-Theory 8 (1994), no. 4, 395–428.

[9] Kuhn, Nicholas J.: Generic representations of the finite general linear groups and the Steenrod algebra. III. K-Theory 9 (1995), no. 3, 273–303.

[10] Lannes, Jean: Sur les espaces fonctionnels dont la source est le classifiant d'un p-groupe abélien élémentaire. Inst. Hautes Études Sci. Publ. Math. No. 75 (1992), 135–244.

[11] Lurie, Jacob: Lecture notes, MIT, http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/

[12] Miller, Haynes: The Sullivan conjecture on maps from classifying spaces. Ann. of Math. (2) 120(1984), no. 1, 39–87.

[12a] Correction to: "The Sullivan conjecture on maps from classifying spaces’' Ann. of Math. (2) 121 (1985), no. 3, 605–609.

[13] Morel, Fabian: Quelques remarques sur la cohomologie modulo p continue des pro-p-espaces et les résultats de J. Lannes concernant les espaces fonctionnels hom(BV,X). Ann. Sci. École Norm. Sup., 26 (1993), 309–360.

[14] Schwartz, Lionel: Unstable modules over the Steenrod algebra and Sullivan's fixed point set conjecture. Chicago Lectures in Mathematics. University of Chicago Press, 1994.