### On the Whitehead spectrum of the circle

The seminal work of Waldhausen, Farrell and Jones, Igusa,
and Weiss and Williams shows that the homotopy groups in low degrees
of the space of homeomorphisms of a closed Riemannian manifold of
negative sectional curvature can be expressed as a functor of the
fundamental group of the manifold. To determine this functor, however,
it remains to determine the homotopy groups of the topological
Whitehead spectrum of the circle. The cyclotomic trace of
Bokstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead
to an expression for these homotopy groups in terms of the equivariant
homotopy groups of the homotopy fiber of the map from the topological
Hochschild **T**-spectrum of the sphere spectrum to that of the
ring of integers induced by the Hurewicz map. We evaluate the latter
homotopy groups, and hence, the homotopy groups of the topological
Whitehead spectrum of the circle in low degrees. The result extends
earlier work by Anderson and Hsiang and by Igusa and complements recent
work by Grunewald, Klein, and Macko.
Lars Hesselholt
<larsh@math.nagoya-u.ac.jp>