The smooth Whitehead spectrum of a point at odd regular primes

Abstract

Let p be an odd regular prime, and assume that the Lichtenbaum-Quillen conjecture holds for K (Z[1/p]) at p. Then the p-primary homotopy type of the smooth Whitehead spectrum W h( * ) is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S 1 -transfer map t : ΣCP ∞ → S . The homotopy groups of W h( * ) are determined in a range of degrees, and the cohomology of W h( * ) is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.

The Circle Transfer and Cobordism Categories

“…The circle transfer in its general form is an infinite loop map trf : QΣY hS 1 → QY, where Y hS 1 = ES 1 × S 1 Y is the homotopy quotient. An important special case is when Y = LX + is the free loop space on some space X plus a disjoint basepoint; the resulting circle transfer in this case is then (1) QΣ(LX hS 1 ) + → QLX + , and this map is central to the construction of topological cyclic homology and the trace map that relates it with Waldhausen's A(X) and the smooth Whitehead space; see e.g., [BHM93] and [Rog03]. Taking Y to a be a single point, the circle transfer is a map QΣBS 1 + → QS 0 that has been studied by stable homotopy theorists; in particular its image in the stable homotopy groups of spheres has been examined by many authors, such as [Muk82,Muk94], [Ima91], [Mil82], and [Bak88, BCG + 88].…”

Cubical and cosimplicial descent

“…See also [, Chapter 7]. In the case of the sphere spectrum, is ‐adically equivalent to , so calculations of are possible in a moderate range of degrees (see also more recent work of Blumberg–Mandell at irregular primes). Nonetheless, complete calculations are at least as hard as those for , hence appear to be out of reach.…”

The Cyclotomic Trace for S-Algebras

“…It also has geometric significance. As an example, the cyclotomic trace is crucial to J. Rognes' recent results [17,18] identifying the homotopy type of A-theory and the Whitehead spectrum of a point, giving new insight into the diffeomorphisms of high dimensional disks.…”