Note

Classification of E_n-Maps from Z to Pic(S)

Author: Wei Yang · Added: July 23, 2026

Article date: July 22, 2026.

Abstract

Regard the ordinary abelian group $(\mathbb{Z},+)$ as a discrete grouplike $\mathbb{E}_\infty$ -space, and let $\operatorname{Pic}(\mathbb{S})$ be the Picard space of the sphere spectrum. For every finite $n \geq 1$ , we determine the full derived mapping space of unital $\mathbb{E}_n$ -maps $\mathbb{Z} \to \operatorname{Pic}(\mathbb{S})$ . The answer is a canonical homotopy fiber obtained from the truncation fiber sequence of the Picard spectrum.

The possible degrees are all integers for $n=1$ , the even integers for $n=2$ , and only zero for $n \geq 3$ . The existence of a degree-two $\mathbb{E}_2$ -map is due to Lurie. Nonzero degree for $n \geq 3$ is excluded by a Thom-spectrum argument and the odd-primary Dyer—Lashof obstruction of Rognes—Sagave—Schlichtkrull. We also treat $\mathbb{E}_\infty$ -maps. The classification is at the level of mapping spaces; the residual degree-zero information is expressed as $\operatorname{gl}_1(\mathbb{S})$ -cohomology of Eilenberg—Mac Lane spaces.