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 : ΣCP ∞ → 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.