# nLab equivariant complex oriented cohomology theory

Contents

cohomology

### Theorems

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Equivariant complex oriented cohomology theory is the generalization of complex oriented cohomology theory to equivariant cohomology.

In generalization of topological K-theory as the prototypical example of a complex oriented cohomology theory, its generalization to equivariant K-theory is equivariantly complex oriented.

## Examples

### Equivariant complex K-theory

###### Proposition

(equivariant K-theory of projective G-space)

For $G$ an abelian compact Lie group, let

$\underset{i}{\oplus} \, \mathbf{1}_{V_i} \;\; \in \;\; G Representations_{\mathbb{C}}^{fin}$

The $G$-equivariant K-theory ring $K_G(-)$ of the corresponding projective G-space $P(-)$ is the following quotient ring of the polynomial ring in one variable $L$ over the complex representation ring $R(G)$ of $G$:

(1)$K_G \Big( P \big( \underset{i}{\oplus} \, \mathbf{1}_{V_i} \big) \Big) \;\; \simeq \;\; R(G) \big[ L \big] \big/ \underset{i}{\prod} \big( 1 - 1_{{}_{V_i}} L \big) \,,$

where

###### Corollary

(equivariant complex orientation of equivariant K-theory)

For $G$ an abelian compact Lie group and $\mathbf{1}_V \,\in\, G Representations_{\mathbb{C}}^{fin}$ a complex 1-dimensional linear representation, the corresponding representation sphere is the projective G-space $S^{\mathbf{1}_V} \,\simeq\, P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$ (this Prop.) and so, by Prop. ,

\begin{aligned} \widetilde K_G \big( S^{\mathbf{1}_V} \big) & \simeq\, K_G \big( P( \mathbf{1}_V \oplus \mathbf{1} ) ; \, \underset{ \simeq \, pt }{ \underbrace{ P( \mathbf{1} ) } } \big) \\ & \simeq ker \Big( R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \longrightarrow \underset{ \simeq \, R(G) }{ \underbrace{ R(G)\big[L\big] \big/ (1 - L) } } \Big) \\ & \simeq (1 - L) \cdot R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \end{aligned}

is generated by the Bott element $(1 - L)$ over $P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$. By the nature of the tautological equivariant line bundle, this Bott element is the restriction of that on infinite complex projective G-space $P\big(\mathcal{U}_G\big)$. The latter is thereby exhibited as an equivariant complex orientation in equivariant complex K-theory.

### Equivariant complex cobordism

For an abelian compact Lie group $G$, equivariant complex cobordism theory $MU_G$ is an equivariant complex oriented cohomology theory (Greenlees 01, Sec. 13).

Much as in the non-equivariant case (see at universal complex orientation on MU), $MU_G$ is universal in that there is a bijection between equivariant complex orientations (in degree 2) on some cohomology theory $E_G$ and homotopy ring homomorphisms of $G$-spectra $MU_G \to E_G$ (Cole-Greenlees-Kriz 02, Theorem 1.2).

For the analogous statement on the equivariant Lazard ring see Greenlees 01a, Greenlees 01, Theorem 13.1, Cole-Greenlees-Kriz 02, Theorem 1.3.