Periodicity and finite complexity in higher real $K$-theories

TL;DR

This paper proves periodicity and finiteness properties of higher real K-theories at all heights and primes, advancing the understanding of their structure and computational aspects.

Contribution

It establishes new periodicity results for higher real K-theories and analyzes the $RO(G)$-periodicity lattice of Lubin--Tate theory, providing foundational insights.

Findings

Proved periodicity results for higher real K-theories at all heights and primes.

Analyzed the $RO(G)$-periodicity lattice of Lubin--Tate theory.

Established explicit finiteness results for $RO(G)$-graded homotopy groups.

Abstract

In this paper, we establish periodicity results for higher real $K$-theories at all heights and for all finite subgroups of the Morava stabilizer group at the prime 2. We further analyze the $RO(G)$-periodicity lattice of the height-$h$ Lubin--Tate theory, proving new $RO(G)$-graded periodicities and explicit finiteness results for the $RO(G)$-graded homotopy groups of $E_h$. Together, these results provide a foundation for both the structural and computational study of higher real $K$-theories.