On the Tate spectrum of tmf at the prime 2

TL;DR

This paper analyzes the Tate spectrum of tmf at prime 2, providing a decomposition that supports the idea that the Tate spectrum of v_n periodic cohomology theories is v_n torsion, with implications for understanding root invariants.

Contribution

It offers a new splitting of the Tate spectrum of tmf into suspensions of kO, reinforcing the conjecture about v_n torsion in Tate spectra.

Findings

Tate spectrum of tmf splits into suspensions of kO.

Supports the idea that Tate spectra of v_n periodic theories are v_n torsion.

Provides computational evidence aligning with the root invariant slogan.

Abstract

Computations involving the root invariant prompted Mahowald and Shick to develop the slogan: "the root invariant of v_n periodic homotopy is v_n torsion." While neither a proof, nor a precise statement, of this slogan appears in the literature, numerous authors have offered computational evidence in support of its fundamental idea. The root invariant is closely related to Mahowald's inverse limit description of the Tate spectrum, and computations have shown the Tate spectrum of v_n periodic cohomology theories to be v_n torsion. The purpose of this paper is to split the Tate spectrum of tmf as a wedge of suspensions of kO, providing yet another example in support of the slogan to the existing literature.