Computing the Morava K-theory of real Grassmanians using chromatic fixed point theory

Abstract

We study K(n)*(Gr(d,m)) for all n - the 2-local Morava K-theories of the real Grassmanian Gr(d,m) of d-planes in R^m, about which very little has been previously computed. We conjecture that the Atiyah-Hirzebruch Spectral Sequences computing these all collapse after the first possible non-zero differential, and give much evidence that this is the case. Computational patterns for all n seem similar to the known calculation of H*(Gr(d,m);Q), the n=0 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,m)) by constructing an action of C = the cyclic group of order 4, on our Grassmanians, and then applying the chromatic fixed point theory of the authors' previous paper. In essence, we bound the size of K(n)*(Gr(d,m)) from below by computing K(n-1)*(Gr(d,m)^C). Meanwhile, the AHSS after the first differential is determined by Q_n-homology, where Q_n is Milnor's nth primitive operation in mod 2 cohomology. Whenever we are able to calculate this, we have found that it agrees with our lower bound for the size of K(n)*(Gr(d,m)). We have two general families where we prove this: m at most 2^{n+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.