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

We study K.n/ .Gr d .R m //, 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 d 2 nC1 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 .R m // by constructing a C 4 -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 .R m // by computing K.n 1/ .Gr d .R m / C 4 /.Meanwhile, the size of E 2 nC1 is given by Q n -homology, where Q n is Milnor's n th primitive mod 2 cohomology operation.Whenever we are able to calculate this Q n -homology, we have found that the size of E 2 nC1 agrees with our lower bound for the size of K.n/ .Gr d .R m //.We have two general families where we prove this: m Ä 2 nC1 and all d , and d D 2 and all m and n.Computer calculations have allowed us to check many other examples with larger values of d .55M35, 55N20; 55P91, 57S17