G-universe

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

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

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

**(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)

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

- John Greenlees, Peter May,
*Equivariant stable homotopy theory*(pdf)

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