# 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.