#### Geometry & Topology, Vol. 7 (2003)
Paper no. 4, pages 155--184.

## The smooth Whitehead spectrum of a point at odd regular primes

### John Rognes

**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 Wh(*) 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: SigmaCP^infty--> S. The homotopy groups
of Wh(*) are determined in a range of degrees, and the cohomology of
Wh(*) 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

**AMS subject classification**.
Primary: 19D10.
Secondary: 19F27, 55P42, 55Q52, 57R50, 57R80.

**DOI:** 10.2140/gt.2003.7.155

**E-print:** `arXiv:math.AT/0304384`

Submitted to GT on 30 November 2001.
(Revised 7 February 2003.)
Paper accepted 13 March 2003.
Paper published 14 March 2003.

Notes on file formats
John Rognes

Department of Mathematics, University of Oslo

N--0316 Oslo, Norway

Email: rognes@math.uio.no

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