nLab equivariant complex oriented cohomology theory





Special and general types

Special notions


Extra structure



Representation theory



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.


Equivariant complex K-theory

equivariant complex K-theory is an equivariant complex oriented cohomology theory (Greenlees 01, Section 10):


(equivariant K-theory of projective G-space)

For GG an abelian compact Lie group, let

i1 V iGRepresentations fin \underset{i}{\oplus} \, \mathbf{1}_{V_i} \;\; \in \;\; G Representations_{\mathbb{C}}^{fin}

be a finite-dimensional direct sum of complex 1-dimensional linear representations.

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

(1)K G(P(i1 V i))R(G)[L]/i(11 V iL), 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) \,,



(equivariant complex orientation of equivariant K-theory)

For GG an abelian compact Lie group and 1 VGRepresentations fin\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 1 VP(1 V1)S^{\mathbf{1}_V} \,\simeq\, P\big( \mathbf{1}_V \oplus \mathbf{1} \big) (this Prop.) and so, by Prop. ,

K˜ G(S 1 V) K G(P(1 V1);P(1)pt) ker(R(G)[L]/(11 VL)(1L)R(G)[L]/(1L)R(G)) (1L)R(G)[L]/(11 VL)(1L) \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 (1L)(1 - L) over P(1 V1)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(𝒰 G)P\big(\mathcal{U}_G\big). The latter is thereby exhibited as an equivariant complex orientation in equivariant complex K-theory.

(Greenlees 01, p. 248 (24 of 39))

Equivariant complex cobordism

For an abelian compact Lie group GG, equivariant complex cobordism theory MU GMU_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 GMU_G is universal in that there is a bijection between equivariant complex orientations (in degree 2) on some cohomology theory E GE_G and homotopy ring homomorphisms of GG-spectra MU GE GMU_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.


See also:

Last revised on November 25, 2020 at 10:13:42. See the history of this page for a list of all contributions to it.