On the Whitehead spectrum of the circle

TL;DR

This paper computes the low-degree homotopy groups of the topological Whitehead spectrum of the circle, which are crucial for understanding the homotopy groups of homeomorphism spaces of negatively curved manifolds.

Contribution

It provides explicit calculations of these homotopy groups using advanced tools like the cyclotomic trace and equivariant homotopy theory, extending previous results.

Findings

Homotopy groups of the Whitehead spectrum of the circle are explicitly determined in low degrees.

Results connect Whitehead spectrum computations to the algebraic K-theory of spheres and integers.

Extends and complements earlier work by Anderson, Hsiang, Igusa, and recent research by Grunewald, Klein, and Macko.

Abstract

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.