nLab fundamental theorem of dg-algebraic equivariant rational homotopy theory

Idea

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

Preliminaries

Let $G$ be a finite group.

Write

• $\big(G dgcAlgebras^{\geq 0}_{k} \big)^{op}_{proj}$ for the opposite of the projective model structure on equivariant connective dgc-algebras;

• $G SimplicialSets_{Qu}$ for the model structure on equivariant simplicial sets, namely the global projective model structure on functors from the opposite of the orbit category to the classical model structure on simplicial sets).

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 $n \in \mathbb{N}$.

(Scull 08, p. 12, 14)

Statement

Proposition

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

The derived adjunction

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

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:

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

(Scull 08, Theorem 5.5)

Related concepts

• fundamental theorem of dg-algebraic rational homotopy theory

References

• Marek Golasiński, Equivariant rational homotopy theory as a closed model category, Journal of Pure and Applied Algebra Volume 133, Issue 3, 30 December 1998, Pages 271-287 (doi:10.1016/S0022-4049(97)00127-8)

  (for Hamiltonian groups)

• Laura Scull, A model category structure for equivariant algebraic models, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (doi:10.1090/S0002-9947-07-04421-2)