#### Volume 7, issue 1 (2003)

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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
##### Abstract

Let $p$ be an odd regular prime, and assume that the Lichtenbaum–Quillen conjecture holds for $K\left(ℤ\left[1∕p\right]\right)$ at $p$. Then the $p$–primary homotopy type of the smooth Whitehead spectrum $Wh\left(\ast \right)$ 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 $t:\Sigma ℂ{P}^{\infty }\to S$. The homotopy groups of $Wh\left(\ast \right)$ are determined in a range of degrees, and the cohomology of $Wh\left(\ast \right)$ 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