  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 B
okstedt, 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.
