The smooth Whitehead spectrum of a point at odd regular primes

Rognes, John¹

¹ Department of Mathematics, University of Oslo, N–0316 Oslo, Norway

Geometry & topology, Tome 7 (2003) no. 1, pp. 155-184.

Résumé

Let $p$ be an odd regular prime, and assume that the Lichtenbaum–Quillen conjecture holds for $K\left(\mathbb{Z}\left[1∕p\right]\right)$ 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 $t:\Sigma ℂP^{\infty }\to S$. 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.

Détail

DOI : 10.2140/gt.2003.7.155

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

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