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

Nicholas J Kuhn and Christopher J R Lloyd

Algebraic & Geometric Topology 24 (2024) 919–950
Abstract

We study K(n)(Gr d(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 d2n+11, 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(m)) 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(m)) by computing K(n 1)(Gr d(m)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(m)). 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.

Keywords
Morava K–theory, Grassmannians, chromatic fixed point theory
Mathematical Subject Classification
Primary: 55M35, 55N20
Secondary: 55P91, 57S17
References
Publication
Received: 16 November 2021
Revised: 14 July 2022
Accepted: 8 August 2022
Published: 12 April 2024
Authors
Nicholas J Kuhn
Department of Mathematics
University of Virginia
Charlottesville, VA
United States
Christopher J R Lloyd
Arlington, VA
United States

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