Tilman Bauer

Computation of the homotopy of the spectrum tmf

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 E2 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 The result of this computation is originally due to Hopkins and Mahowald (unpublished). 55N34; 55T15

Finite loop spaces are manifolds

One of the motivating questions for surgery theory was whether every finite H:space is homotopy equivalent to a Lie group. This question was answered in the negative by Hilton and Roitberg s discovery of some counterexamples [18]. However, the problem remained whether every finite H-space is homotopy equivalent to a closed, smooth manifold. This question is still open, but in case the H-space admits a classifying space we have the following theorem.

A finite loop space not rationally equivalent to a compact Lie group

We construct a connected finite loop space of rank 66 and dimension 1254 whose rational cohomology is not isomorphic as a graded vector space to the rational cohomology of any compact Lie group, hence providing a counterexample to a classical conjecture. Aided by machine calculation we verify that our counterexample is minimal, i.e., that any finite loop space of rank less than 66 is in fact rationally equivalent to a compact Lie group, extending the classical known bound of 5.