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: Σ ℂ P∞→ 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.