nLab
G-universe

Contents

Contents

Idea

In equivariant stable homotopy theory over a compact Lie group GG, a GG-universe is a GG-representation that contains “all” representations of GG of sorts.

This is used in one definition of G-spectra via looping and delooping by representation spheres.

Definition

A G-universe in this context is (e.g. Greenlees-May, p. 10) an infinite dimensional real inner product space equipped with a linear GG-action that is the direct sum of countably many copies of a given set of (finite dimensional) representations of GG, at least containing the trivial representation on \mathbb{R} (so that UU contains at least a copy of \mathbb{R}^\infty).

Applications

Infinite complex projective GG-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)

This concept is at the heart of equivariant complex oriented cohomology theory.

References

Last revised on November 12, 2020 at 08:41:53. See the history of this page for a list of all contributions to it.