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Here are my current notes (last updated May 4th).
- John
Due to a conference in Regensburg, the Easter holidays and Ib Madsen's retirement conference, I will replace the April 7th and April 21st classes with lectures at the following times:
- Tuesday April 11th at 10.15-12.00 in room B738, and
- Wednesday April 19th at 14.15-16.00 in room B738.
We will resume the regular schedule on April 28th and May 5th. The exam will be on May 12th.
- John
Some references on the stable homotopy category:
- Adams: Stable homotopy and generalised homology (1974)
- Margolis: Spectra and the Steenrod algebra (1983)
- May: The additivity of traces in triangulated categories (2001)
- Whitehead: Generalized homology theories (1962)
The student representative this term is Alice Hedenlund.
The lectures have been moved to Fridays 10.15-14.00 in room B738.
- John
- Mandell-May: Equivariant orthogonal spectra and S-modules (2002)
- Stolz: Equivariant structure on smash powers of commutative ring spectra (2011)
- Hausmann: G-symmetric spectra, semistability and the multiplicative norm (preprint, 2014)
- Hill-Hopkins-Ravenel: On the nonexistence of elements of Kervaire invariant one (2016)
- Schwede:...
- Elmendorf-Kriz-Mandell-May: Rings, modules, and algebras in stable homotopy theory (1997)
- Hovey-Shipley-Smith: Symmetric spectra (1999)
- Mandell-May-Schwede-Shipley: Model categories of diagram spectra (2000)
- Schwede: Symmetric spectra ( in preparation, 2012)
Some references on point set topology:
- May-Ponto: More concise algebraic topology (2012)
- McCord: Classifying spaces and infinite symmetric products (1969)
- Steenrod: A convenient category of topological spaces (1967)
- Strickland: The category of CGWH spaces (notes, 2009)
- Str?m: The homotopy category is a homotopy category (1972)
Some references on topological K-theory:
- Atiyah-Hirzebruch: Vector bundles and homogeneous spaces (1961)
- Milnor: Morse theory (1963)
- Atiyah: K-theory (1967)
- Hatcher: Vector Bundles & K-Theory https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html
Some resources on (co-)bordism:
- Thom: Quelques propriétés globales des variétés différentiables (1954)
- Stong: Notes on cobordism theory (1968)
- Milnor-Stasheff: Characteristic classes (1974)
- Hopkins: The Kervaire invariant problem (2016)
I plan to lecture about foundations and applications of stable homotopy theory.
As motivation, we may study examples of generalized (co)homology theories, such as bordism [Pontryagin, Thom] and $K$-theory [Bott, Atiyah-Hirzebruch]. Such functors become representable in the stable homotopy category [Boardman, Adams], which is triangulated and closed symmetric monoidal. In the 1990s, this structure was found to arise as the homotopy category of a model category, in several ways, including $S$-modules, symmetric spectra and orthogonal spectra [Elmendorf-Kriz-Mandell-May, Hovey-Shipley-Smith, Lydakis, May-Mandell-Schwede-Shipley, ...]. We will follow a book project by Stefan Schwede, titled "Symmetric Spectra", as an introduction to these results. Thereafter we will turn to the equivariant theory [Adams, Lewis-May-Steinberger,...