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projective G-space

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Context

Geometry

Representation theory

Contents

Idea

A projective GG-space is a projective topological G-space.

Definition

Let GG be a finite group (or maybe a compact Lie group) and let VV be a GG-linear representation over some topological ground field kk.

Then the corresponding projective GG-space is the quotient space of the complement of the origin in (the Euclidean space underlying) VV by the given action of the group of units of kk (from the kk-vector space-structure on VV):

kP(V):=(V{0})/k × k P(V) \;:=\; \big( V \setminus \{0\} \big) / k^\times

and equipped with the residual GG-action on VV (which passes to the quotient space since it commutes with the kk-action, by linearity).

Examples

Ordinary projective spaces

If G=1G = 1 is the trivial group, then

kP(k n+1)=kP n k P(k^{n+1}) \;=\; k P^n

is ordinary (non-equivariant) projective space of dimension nn over kk.

Representation spheres

Proposition

(1-dimensional representation spheres are projective G-spaces)

If 1 VGRepresentations k\mathbf{1}_V \,\in\, G Representations_k is 1-dimensional over the given ground field kk, stereographic projection identifies the representation sphere of VV with the projective G-space over kk of 1 V1\mathbf{1}_V \oplus \mathbf{1}:

V cpt kP(1 V1) v {[v,1] | vV [1,0] | v= \array{ V^{cpt} & \longrightarrow & k P \big( \mathbf{1}_V \oplus \mathbf{1} \big) \\ v &\mapsto& \left\{ \array{ [v,1] &\vert& v \in V \\ [1,0] &\vert& v = \infty } \right. }

(e.g. Atiyah 68, Sec. 4, Greenlees 01, 9.C)

Prop. underlies the concept of equivariant complex oriented cohomology theory.

Infinite projective space

Definition

(infinite complex projective G-space)

For GG an abelian compact Lie group, let

(1)𝒰 Gk1 VR(G)1 V \mathcal{U}_G \;\coloneqq\; \underset{k \in \mathbb{N}}{\bigoplus} \underset{\mathbf{1}_V \in R(G)}{\bigoplus} \mathbf{1}_V

be the G-universe being the infinite direct sum of all complex 1-dimensional linear representations of GG, regarded as a topological G-space with toplogy the colimit of its finite-dimensional linear subspaces.

Then the infinite complex projective G-space is the colimit

P(𝒰 G)limV𝒰 Gdim(V)<P(V) P\big( \mathcal{U}_G \big) \;\coloneqq\; \underset{ \underset{ { V \subset \mathcal{U}_G } \atop { dim(V) \lt \infty } }{\longrightarrow} }{\lim} P\big( V \big)

of the projective G-spaces for all the finite-dimensional GG-linear representations inside the G-universe (1).

(e.g. Greenlees 01, Sec. 9.2)

Properties

Equivariant complex K-theory

Proposition

(equivariant K-theory of projective G-space)

For GG an abelian group 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:

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))

References

Discussion in the context of Bott periodicity in equivariant K-theory:

Discussion in the context of equivariant complex oriented cohomology theory:

  • John Greenlees, Section 9.A 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 27, 2020 at 08:23:59. See the history of this page for a list of all contributions to it.