Download this article
 Download this article For screen
For printing
Recent Issues

Volume 24
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
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

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.

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

Open Access made possible by participating institutions via Subscribe to Open.