Computation of the homotopy of the spectrum tmf

TL;DR

This paper provides a comprehensive computation of the homotopy groups of the tmf spectrum, utilizing spectral sequences and algebraic methods to determine the structure at primes 2 and 3, advancing understanding in algebraic topology.

Contribution

It offers the first complete calculation of the homotopy ring of tmf, including differentials in the elliptic Adams-Novikov spectral sequence, extending prior partial results.

Findings

Homotopy ring of tmf computed at primes 2 and 3

All differentials in the elliptic Adams-Novikov spectral sequence determined

Results confirm previous unpublished computations by Hopkins and Mahowald

Abstract

This paper contains a complete computation of the homotopy ring of the spectrum of topological modular forms constructed by Hopkins and Miller. The computation is done away from 6, and at the (interesting) primes 2 and 3 separately, and in each of the latter two cases, a sequence of algebraic Bockstein spectral sequences is used to compute the E_2 term of the elliptic Adams-Novikov spectral sequence from the elliptic curve Hopf algebroid. In a further step, all the differentials in the latter spectral sequence are determined. The result of this computation is originally due to Hopkins and Mahowald (unpublished).