# nLab fundamental theorem of dg-algebraic equivariant rational homotopy theory

Contents

and

## Sullivan models

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

The fundamental theorem of equivariant rational homotopy theory modeled by equivariant dgc-algebras.

## Preliminaries

Let $G$ be a finite group.

Write

###### Definition

(simply connected and finite equivariant rational homotopy types)

Write

(1)$Ho \big( G SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 2} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( G SimplicialSets_{Qu} \big)$

for the full subcategory of the homotopy category of the model structure on equivariant simplicial sets on those equivariant homotopy types $X$ which over each $G/H \in G Orbits$ are

• connected: $\pi_0(X^H) = \ast$

• simply connected: $\pi_1(X^H) = 1$ is the trivial group;

• rationally of finite type: $dim_{\mathbb{Q}}\big( H^n(X^H;,\mathbb{Q}) \big) \lt \infty$ for all $n \in \mathbb{N}$.

and

(2)$Ho \big( G SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 2} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( SimplicialSets_{Qu} \big)$

for the futher full subcategory on those equivariant homotopy types that are already rational.

Similarly, write

(3)$Ho \big( G dgcAlgebras^{\geq 0}_{\mathbb{Q}} \big)_{fin}^{\geq 1} \overset{ \phantom{AAA} }{\hookrightarrow} Ho \big( G dgcAlgebras^{\geq 0}_{\mathbb{Q}} \big)$

for the full subcategory of the homotopy category of the projective model structure on equivariant connective dgc-algebras on those equivariant dgc-algebras $A$ which for each $G/H \in G Orbits$ are

• connected: $H^0(A^H) \simeq \mathbb{Q}$

• simply connected: $H^1(A^H) \simeq 0$

• finite type: $dim_{\mathbb{Q}}\big( H^n(A^H) \big) \lt \infty$ for all $n \in \mathbb{N}$.

## Statement

###### Proposition

(fundamental theorem of equivariant dg-algebraic rational homotopy theory)

$Ho \left( \big( G dgcAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right) \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot} Ho \big( G SimplicialSets_{Qu} \big)$

of the Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras (whose left adjoint is the equivariant PL de Rham complex-functor) has the following properties:

• on connected, simply connected, rationally finite equivariant homotopy types $X$ (1) the derived adjunction unit is equivariant rationalization

$\array{ Ho \big( G SimplicialSets_{Qu} \big)^{fin_{\mathbb{Q}}}_{\geq 1, nil} & \overset{ }{\longrightarrow} & Ho \big( G SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil} \\ X &\mapsto& \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X) }$
$X \underoverset {\eta_X^{der}} {rationalization} {\longrightarrow} \mathbb{R}\exp \circ \Omega^\bullet_{PLdR}(X)$
• on the full subcategories of connected, simply connected, and finite rational homotopy types from Def. it restricts to an equivalence of categories:

$Ho \left( \big( G dgcAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \right)^{\geq 1}_{fin} \underoverset { \underset {\;\;\; \mathbb{R} exp \;\;\;} {\longrightarrow} } { \overset {\;\;\; \mathbb{L} \Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\simeq} Ho \big( G SimplicialSets_{Qu} \big)^{\mathbb{Q}, fin_{\mathbb{Q}}}_{\geq 1, nil}$