nLab infinite complex projective G-space

Idea

For $G$ a suitable equivariance group, the infinite projective G-space is the projective G-space corresponding to the “complete infinite $G$-representation”, namely to a given G-universe.

Over the ground field of complex numbers this is the generalization to topological G-spaces/$G$-equivariant homotopy theory of the infinite complex projective space, hence the classifying space for complex line bundles, now classifying equivariant complex line bundles.

Definition

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)

Related concepts

• equivariant complex oriented cohomology theory

References

• 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)