nLab even cohomology theory

Idea

A multiplicative cohomology theory $E$ is called even if its cohomology ring is trivial in all odd degrees:

$$E^{2k+1}(X) = 0 \,.$$

Properties

For an even cohomology theory $E$:

• There is an isomorphism of rings

$$E^{\ast}(\mathbb{C}P^n) \cong E^{\ast}(S^0)[x]/\langle x^{n+1} \rangle,$$

where $|x|=2$.

• $E^{\ast}(\mathbb{C}P^{\infty} \times \ldots \times \mathbb{C}P^{\infty}) \cong E^{\ast}(S^0)[[x_1, \ldots, x_k]]$.

Remark

Any even cohomology theory is complex orientable; a choice of complex orientation gives an isomorphism

$$E^{\ast}(\mathbb{C}P^{\infty}) \cong E^{\ast}(S^0)[[x]].$$

See here at complex oriented cohomology theory (review also in Mehrle 18, 1.1).

Related entries

• even periodic ring spectrum

• A Survey of Elliptic Cohomology - cohomology theories

References

• David Mehrle, Chromatic homotopy theory: Journey to the frontier, Graduate workshop notes, Boulder 16-17 May 2018, (pdf, pdf)