nLab
periodic cohomology theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

An even 2-periodic cohomology theory or just periodic cohomology theory for short is an even multiplicative cohomology theory EE with a Bott element βE 2(*)\beta \in E^2({*}) which is invertible (under multiplication in the cohomology ring of the point) so that multiplication by it induces an isomorphism

()β:E *(*)E *+2(*). (-)\cdot \beta : E^*({*}) \simeq E^{*+2}({*}) \,.

Via the Brown representability theorem this corresponds to a periodic ring spectrum.

Compare with the notion of weakly periodic cohomology theory.

More generally one considers 2n2n-periodic cohomology theories

Properties

Periodicity of the \infty-category of \infty-modules

For EE an E-∞ ring representing a periodic cohomology (a periodic ring spectrum) double suspension/looping on any EE-∞-module NN is equivalent to the identity

Ω 2NNΣ 2N. \Omega^2 N \simeq N \simeq \Sigma^2 N \,.

This equivalence ought to be coherent to yield a /2\mathbb{Z}/2\mathbb{Z} ∞-action on the (∞,1)-category of (∞,1)-modules EModE Mod (MO discussion).

References

Lecture notes include

Revised on November 1, 2014 22:26:06 by Urs Schreiber (141.0.9.61)