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equivariant complex oriented cohomology theory

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Cohomology

cohomology

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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

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

Proposition

(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) \,,

where

Corollary

(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.

References

See also:

Last revised on November 25, 2020 at 05:13:42. See the history of this page for a list of all contributions to it.