Seminar

In 1986 Peter May made the following conjecture:

Suppose that R is a ring spectrum with power operations (e.g., an E∞ ring spectrum/ commutative S-algebra). Then the torsion elements in the kernel of the integral Hurewicz homomorphism π∗ R → H∗(R;𝕫) are nilpotent.

If R is the sphere spectrum, this is Nishida's nilpotence theorem. If we strengthen the condition on the integral homology to a condition about the complex bordism of R, then this is a special case of the nilpotence theorem of Devinatz, Hopkins, and Smith.

The proof is short and simple, using only results that have been around since the late 90's. As a corollary we obtain results on the non-existence of commutative S-algebra structures on various quotients of MU. For example MU / (pi) or ku / (pi v) for i > 0. We also obtain new results about the behavior of the Adams spectral sequence for Thom and THH spectra.

This project is joint with Akhil Mathew and Niko Naumann.

I will fill any remaining time with some fun results about ring spectra with power operations.