Organizers: Tilman Bauer
and André Henriques
Introductory seminar
to Stable homotopy theory
The seminar takes place each Wednesday afternoon, and lasts twice 45 minutes.
We meet at 17:15 sometimes in Utrecht (Wiskundegebouw 611) and sometimes at the VU (room R231, science building).

Oct 1^{st}: Jacobien (meet at UU)
Spectra.
The Freudenthal suspension theorem.
Ωspectra.
The homotopy category.
Homotopy groups of spectra.
[Switzer, R. Algebraic topology, homotopy and homology.
Reprint of the 1975 original. Classics in Mathematics. Springer, Berlin, 2002]

Oct 8^{th}: Floris (meet at UU)
Fibers and cofibers of maps between spectra. The long exact sequence in homotopy.
The smash product. Homology groups for spectra.
Chapter 3 of [Adams, J. F.
Stable homotopy and generalised homology
Reprint of the 1974 original. Chicago Lectures in Mathematics]

Oct 15^{th}: Dave (meet at VU)
Examples of spectra:
The spectra HZ and HZ/2, representing ordinary cohomology.
The spectra ku, KU, ko, KO, representing the various flavours of topological Ktheory.
The sphere spectrum. Suspension spectra.
Chapter 3 of [Adams, J. F.
Stable homotopy and generalised homology
Reprint of the 1974 original. Chicago Lectures in Mathematics]

Oct 22^{nd}: Camilo (meet at UU)
Stable and unstable cohomology operations.
Steenrod squares, via the construction X → (X × X)_{hZ/2}.
Examples: RP^{n}, CP^{n}, SU(3).
Chapter 4.L of
[Hatcher, A. Algebraic topology Cambridge University Press, Cambridge, 2002.], available
here.

Oct 29^{th}: Alvise (meet at UU)
The Adem relations. The relation between the Bockstein and the operation Sq^{1}.
Description of the Steenrod algebra in low degrees.
Chapter 4.L of
[Hatcher, A. Algebraic topology Cambridge University Press, Cambridge, 2002.], available
here.
 Nov 5^{th}: No talk (people are away)

Nov 12^{th}: Benoît (meet at VU)
Projective and injective modules, resolutions, fundamental lemma of homological algebra.
Definition of Tor and Ext. Mention (probably without proof) the symmetry of Tor and the fact that we can injectively resolve in the second variable,
or projectively in the first, for computing Ext. Examples: Tor and Ext for finitely generated abelian groups.
[Weibel, C.A. An introduction to homological algebra.
Cambridge Studies in Advanced Mathematics. 38. Cambridge: Cambridge Univ. Press]
[Bauer, T. Homologische Algebra und Gruppenkohomologie. Draft, 2004.], available
here.
[Cartan, Henri; Eilenberg, Samuel Homological algebra. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics]

Nov 19^{th}: Dmitry (meet at UU)
The product on Ext (both composition and Yoneda construction). Ext over graded connected kalgebras, where k is a field.
Examples: Ext for an exterior algebra, a polynomial algebra, a truncated polynomial algebra. Ext over A(1). Introduce the pictorial notation for minimal resolutions.
[Weibel, C.A. An introduction to homological algebra.
Cambridge Studies in Advanced Mathematics. 38. Cambridge: Cambridge Univ. Press]
[Bauer, T. Homologische Algebra und Gruppenkohomologie. Draft, 2004.], available
here.
[Cartan, Henri; Eilenberg, Samuel Homological algebra. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton Landmarks in Mathematics]

Nov 26^{th}: Gil (meet at UU)
Exact couples and unrolled exact couples. Derived exact couples; proof that they're again exact.
Example: filtered chain complex. E^{n}term and its bigrading, E ^{oo}term, target of a spectral sequence, notion of convergence.
[McCleary, J. A user's guide to spectral sequences. 2nd ed. Cambridge Studies in Advanced Mathematics. 58. Cambridge: Cambridge University Press]
[Hatcher, A. Spectral sequences in algebraic topology. Book draft.], available
here.
[Boardman, J.M. Conditionally convergent spectral sequences. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 4984,
Contemp. Math., 239, 1999.]

Dec 3^{rd}: Dietrich (meet at VU)
Example: Bockstein spectral sequence, examples (with differentials!). Products in spectral sequences, examples of multiplicative extensions.
Time permitting: spectral sequences from towers of spectra.
[McCleary, J. A user's guide to spectral sequences. 2nd ed. Cambridge Studies in Advanced Mathematics. 58. Cambridge: Cambridge University Press] (new edition!)
[Kochman, S. O. Bordism, stable homotopy and Adams spectral sequences. Fields Institute Monographs, 7. American Mathematical Society, 1996]

Dec 10^{th} . . .
Adams resolutions, construction of the Adams spectral sequence. Change of rings; the Adams spectral sequence for ko using A(1).
For the ASS for the sphere, compute the 0line and the 1line.
[McCleary, J. A user's guide to spectral sequences. 2nd ed. Cambridge Studies in Advanced Mathematics. 58. Cambridge: Cambridge University Press] (new edition!)
[Kochman, S. O. Bordism, stable homotopy and Adams spectral sequences. Fields Institute Monographs, 7. American Mathematical Society, 1996]
Appendix of
[Ravenel, Douglas C. Complex cobordism and stable homotopy groups of spheres. Pure and Applied Mathematics, 121. Academic Press, 1986]

Dec 17^{th} . . .
Beginning computations if π_{n}(S) by hand (will get you up to dim 5 or so). The May SS and computing with it.
[McCleary, J. A user's guide to spectral sequences. 2nd ed. Cambridge Studies in Advanced Mathematics. 58. Cambridge: Cambridge University Press] (new edition!)
[Kochman, S. O. Bordism, stable homotopy and Adams spectral sequences. Fields Institute Monographs, 7. American Mathematical Society, 1996]

Jan ?^{th} . . .
The Adams differential. Hopf invariant 1 using the Adams SS. Massey products and their relation to differentials.