nLab periodic cohomology theory

Definition

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

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

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 $2n$-periodic cohomology theories

Properties

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

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

$$\Omega^2 N \simeq N \simeq \Sigma^2 N \,.$$

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

Landweber exact functor theorem

There is an analogue of the Landweber exact functor theorem for even 2-periodic cohomology theories, with MU replaced by MP (Hovey-Strickland 99, theorem 2.8, Lurie lecture 18, prop. 11).

Examples

• KU

• MP

• elliptic cohomology

• Morava E-theory

Related entries

• Bott periodicity

• periodic ring spectrum

• A Survey of Elliptic Cohomology - cohomology theories

References

The concept of even 2-periodic multiplicative cohomology theories originates with

• Matthew Ando, Michael Hopkins, Neil Strickland, Elliptic spectra, the Witten genus, and the theorem of the cube, Inventiones Mathematicae, 146:595–687, 2001, DOI 10.1007/s002220100175

The analogue of the Landweber exact functor theorem for even 2-periodic cohomology is discussed in

• Mark Hovey, Neil Strickland, theorem 2.8 of Morava K-theories and localisation Mem. Amer. Math. Soc., 139(666):viii+100, 1999.

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010 , Lecture 18 Even periodic cohomology theories (pdf)

Review includes

• section 5.1 by Markus Land in these “TMF seminar” notes: pdf

• Akhil Mathew, Lennart Meier, section 2.1 of Affineness and chromatic homotopy theory, J. Topol. 8 (2015), no. 2, 476–528 (arXiv:1311.0514)