nLab
2-periodic sphere spectrum

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

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

Ω 2BU(1)Ω 2BUBU×, \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 E_{\infty} structure.

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

References

Last revised on September 10, 2021 at 10:20:05. See the history of this page for a list of all contributions to it.