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.