nLab projective G-space

Idea

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

Definition

Let $G$ be a finite group (or maybe a compact Lie group) and let $V$ be a $G$-linear representation over some topological ground field $k$.

Then the corresponding projective $G$-space is the quotient space of the complement of the origin in (the Euclidean space underlying) $V$ by the given action of the group of units of $k$ (from the $k$-vector space-structure on $V$):

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

and equipped with the residual $G$-action on $V$ (which passes to the quotient space since it commutes with the $k$-action, by linearity).

Examples

Ordinary projective spaces

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

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

is ordinary (non-equivariant) projective space of dimension $n$ over $k$.

Representation spheres

Proposition

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

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

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

(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 $G$ an abelian compact Lie group, let

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

be the G-universe being the infinite direct sum of all complex 1-dimensional linear representations of $G$, 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

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

of the projective G-spaces for all the finite-dimensional $G$-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 $G$ an abelian group compact Lie group, let

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

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

The $G$-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 $G$:

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

where

• $L \,=\, \big[ \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} }\big]$ is the K-theory class of the tautological equivariant line bundle on the given projective G-space

• $1_{{}_{V_i}} L \;=\; \big[ \mathbf{1}_{V_i} \boxtimes \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} } \big]$ is the class of its external tensor product of equivariant vector bundles with the given linear representation.

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

Related concepts

• tautological equivariant line bundle

References

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

• Michael Atiyah, Bott periodicity and the index of elliptic operators, The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 113–140 (doi:10.1093/qmath/19.1.113)

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)