Contents

Contents

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).

References

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

Last revised on September 23, 2021 at 05:20:07. See the history of this page for a list of all contributions to it.