The smooth Whitehead spectrum of a point at odd regular primes

Abstract

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.