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The smooth Whitehead spectrum of a point at odd regular primes

John Rognes

Geometry & Topology 7 (2003) 155–184

arXiv: math.AT/0304384


Let p be an odd regular prime, and assume that the Lichtenbaum–Quillen conjecture holds for K([1p]) 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 S1-transfer map t: ΣP 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.

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
Forward citations
Received: 30 November 2001
Revised: 7 February 2003
Accepted: 13 March 2003
Published: 14 March 2003
Proposed: Haynes Miller
Seconded: Gunnar Carlsson, Thomas Goodwillie
John Rognes
Department of Mathematics
University of Oslo
N–0316 Oslo