nLab 2-periodic sphere spectrum

Idea

The 2-periodic ring spectrum-version of the sphere spectrum. It is formed by taking the Thom spectrum of the $E_2$ map

$$\mathbb{Z} \;\simeq\; \Omega^2 B U(1) \longrightarrow \Omega^2 B U \longrightarrow B U \times \mathbb{Z} \,,$$

where the last map is specified by Bott periodicity (Lurie, Rotation Invariance, Remark 3.5.13). It cannot be equipped with an $E_{\infty}$ structure.

As a spectrum it is given by the direct sum (wedge sum) $\bigoplus_{n \in \mathbb{Z}} \mathbb{S}^{-2n}$ of all even-degree suspensions of the plain sphere spectrum.

References

• Jacob Lurie, Rotation Invariance in Algebraic K-Theory, (pdf)

• Mohammed Abouzaid, Andrew J. Blumberg, A.2.3 in: Arnold Conjecture and Morava K-theory (arXiv:2103.01507).