Computing the Morava K–theory of real Grassmannians using chromatic fixed point theory

Kuhn, Nicholas J¹ ; Lloyd, Christopher J R²

¹ Department of Mathematics, University of Virginia, Charlottesville, VA, United States
² Arlington, VA, United States

Algebraic and Geometric Topology, Tome 24 (2024) no. 2, pp. 919-950

Résumé

We study K(n)∗(Gr ⁡ d(ℝm)), the 2–local Morava K–theories of the real Grassmannians, about which very little has been previously computed. We conjecture that the Atiyah–Hirzebruch spectral sequences computing these all collapse after the first possible nonzero differential d2n+1−1, and give much evidence that this is the case.

We use a novel method to show that higher differentials can’t occur: we get a lower bound on the size of K(n)∗(Gr ⁡ d(ℝm)) by constructing a C4–action on our Grassmannians and then applying the chromatic fixed point theory of the authors’ previous paper. In essence, we bound the size of K(n)∗(Gr ⁡ d(ℝm)) by computing K(n − 1)∗(Gr ⁡ d(ℝm)C4).

Meanwhile, the size of E2n+1 is given by Qn–homology, where Qn is Milnor’s n th primitive mod 2 cohomology operation. Whenever we are able to calculate this Qn–homology, we have found that the size of E2n+1 agrees with our lower bound for the size of K(n)∗(Gr ⁡ d(ℝm)). We have two general families where we prove this: m ≤ 2n+1 and all d, and d = 2 and all m and n. Computer calculations have allowed us to check many other examples with larger values of d.

Détail

DOI : 10.2140/agt.2024.24.919

Keywords: Morava K–theory, Grassmannians, chromatic fixed point theory