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

Kuhn, Nicholas J; Lloyd, Christopher J R

doi:10.2140/agt.2024.24.919

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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

DOI: 10.2140/agt.2024.24.919

Abstract

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 $d_{2^{n + 1} - 1}$ , 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 $K {(n)}^{*} ({Gr}_{d} (ℝ^{m}))$ 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 $E_{2^{n + 1}}$ 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 $E_{2^{n + 1}}$ agrees with our lower bound for the size of $K {(n)}^{*} ({Gr}_{d} (ℝ^{m}))$ . We have two general families where we prove this: $m \leq 2^{n + 1}$ 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 $d$ .

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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Volume 24, issue 2 (2024)