Let be an odd regular prime, and assume that the Lichtenbaum–Quillen conjecture holds for at . Then the –primary homotopy type of the smooth Whitehead spectrum 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 -transfer map . The homotopy groups of are determined in a range of degrees, and the cohomology of is expressed as an -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.
"The smooth Whitehead spectrum of a point at odd regular primes." Geom. Topol. 7 (1) 155 - 184, 2003. https://doi.org/10.2140/gt.2003.7.155