The smooth Whitehead spectrum of a point at odd regular primes

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 $t : Σ ℂ P^{\infty} \to S$ . 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.

Keywords

algebraic $K$-theory, topological cyclic homology, Lichtenbaum–Quillen conjecture, transfer, $h$-cobordism, concordance, pseudoisotopy

Mathematical Subject Classification 2000

Primary: 19D10

Secondary: 19F27, 55P42, 55Q52, 57R50, 57R80

References

Forward citations

Publication

Received: 30 November 2001

Revised: 7 February 2003

Accepted: 13 March 2003

Published: 14 March 2003

Proposed: Haynes Miller

Seconded: Gunnar Carlsson, Thomas Goodwillie

Authors

John Rognes
Department of Mathematics University of Oslo N–0316 Oslo Norway

Volume 7, issue 1 (2003)

John Rognes