V5D1 - Advanced Topics in Topology: Elliptic cohomology and topological modular forms (SoSe 2017)

Time: Wednesday and Friday 14-16
Room: N0.007

No lecture on June 23

Lecture Notes
Exercises 1 (posted May 10)
Exercises 2 (posted May 31)
Exercises 3 (posted July 5)

Elliptic homology is a family of generalized homology theories based on elliptic curves. Their historic origin lies in the theory of elliptic genera. A genus is a ring homomorphism from a bordism ring (i.e. manifolds with extra structure up to cobordism) into some other ring. Classic examples are the Euler characteristic mod 2, Hirzebruch's L-genus (giving the signature formula) and Hirzebruch's A-hat genus. The last two genera are actually special values of of the original example of an elliptic genus, based on an addition formula for elliptic integrals. Elliptic homology was introduced by Landweber, Ravenel and Stong to give a more conceptual definition of an elliptic genus that also applies to families. It is constructed via the Landweber exact functor theorem from complex bordism.

In many respects elliptic homology is analogous to complex K-theory (for example, the Conner-Floyd theorem states that the latter can also be constructed from complex bordism). But for many applications the more powerful (but more difficult to handle) theory of real K-theory is more useful. The spectrum of topological modular forms (TMF) is analogously a more powerful and difficult version of elliptic homology. It is constructed as the global sections of a sheaf of E ring spectra on the moduli stack of elliptic curves. It will allow us to construct the (refined) Witten genus and also to have applications to periodic families in the stable homotopy groups of spheres.

As this description probably makes clear, this topic lies at the crossroad between algebraic topology and algebraic geometry (with a few drops of manifold theory): Elliptic curves and algebraic stacks are part of algebraic geometry, while homology theories and E ring spectra are part of algebraic topology. How much I will assume as background will depend on the audience. I will try to make it accessible for students in the current topology cycle (attending AlgTop II next semester) who have also a bit of background in algebraic geometry (like AlgGeo I). More precisely, you should at least know what a scheme, what a (ring) spectrum and what bordism is -- knowledge of topics like E ring spectra (e.g. in the model of commutative orthogonal ring spectra), model or ∞-categories, complex oriented cohomology theories and flat morphisms (in algebraic geometry) would be useful extras, but are not strictly necessary.

If you want to take an exam, please contact me in one of the first lectures.

Literature

  • Douglas, Francis, Henriques, Hill: Topological modular forms
  • Hopkins: Algebraic topology and modular forms
  • Lurie: Survey of Elliptic Cohomology
  • Hirzebruch, Berger, Jung: Manifolds and modular forms
  • Mathew: The homology of tmf
  • Bauer: Computation of the homotopy of the spectrum tmf
  • Lurie: Lectures on Chromatic Homotopy Theory
  • Landweber: Elliptic cohomology and modular forms
  • Goerss: Topological modular forms


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