nLab equivariant K-theory of projective G-space

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Representation theory

Contents

Idea

For GG an abelian compact Lie group, the equivariant K-theory ring of projective G-spaces over a direct sum of complex 1-dimensinal linear representations is (2) the quotient ring of the polynomial ring in the tautological equivariant line bundle LL by the ideals generated by virtual differences 11 V iL1 - 1_{{}_{V_i}} L between its external tensor product with each of these 1d representations and the trivial line bundle; see Prop. below.

This is the generalization to equivariant K-theory of the formula

(1)K(P n)[L]/(1L) n+1 K\big( \mathbb{C}P^n \big) \;\simeq\; \mathbb{Z}\big[ L \big] \big/ \big( 1 - L \big)^{n+1}

(from the fundamental product theorem in topological K-theory) for the complex topological K-theory ring of complex projective space, where LL is the class of the tautological line bundle and 1L1- L the “Bott element”.

In generalization of how (1) exhibits complex orientation in topological complex K-theory, so the equivariant version (2) exhibits equivariant complex orientation of equivariant complex K-theory.

Statement

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:

(2)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

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

Applications

Complex orientation of equivariant complex K-theory

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 complex orientation in equivariant complex K-theory.

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

References

  • John Greenlees, Section 10 of: Equivariant formal group laws and complex oriented cohomology theories, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 (euclid:hha/1139840255)

Last revised on November 12, 2020 at 08:45:28. See the history of this page for a list of all contributions to it.