nLab
even cohomology theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebraic topology

Contents

Idea

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

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

Properties

For an even cohomology theory EE:

  • There is an isomorphism of rings
E *(P n)E *(S 0)[x]/x n+1, E^{\ast}(\mathbb{C}P^n) \cong E^{\ast}(S^0)[x]/\langle x^{n+1} \rangle,

where |x|=2|x|=2.

  • E *(P ××P )E *(S 0)[[x 1,,x k]]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 *(P )E *(S 0)[[x]].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.