nLab equivariant PL de Rham complex

Definition

Let $G$ be a finite group.

Definition

(equivariant PL de Rham complex )

Let $S \in G SimplicialSets$ be a simplicial set equipped with $G$-action, for instance the singular simplicial set of a topological G-space.

The equivariant PL de Rham complex of $S$ is the equivariant dgc-algebra given as the functor from the orbit category of $G$ to the category of dgc-algebras

$$\array{ G Orbits & \overset{ \Omega^\bullet_{PLdR} \big( Maps(-,X)^G \big) }{\longrightarrow} & dgcAlgebras \\ G/H &\mapsto& \Omega^\bullet_{PLdR} \big( X^H \big) }$$

which to a coset space $G/H$ assigns the PL de Rham complex of the $H$-fixed locus $X^H \subset X$.

Properties

Proposition

(equivariant PL de Rham complex is degreewise injective in dual vector G-spaces)

The dual vector G-space underlying the equivariant PL de Rham complex (Def. )

$$\array{ G Orbits & \overset{ \Omega^\bullet_{PLdR} \big( Maps(-,X)^G \big) }{\longrightarrow} & dgcAlgebras &\overset{}{\longrightarrow}& VectorSpaces_{\mathbb{Q}} }$$

is degreewise an injective object.

(Triantafillou 82, Prop. 4.3)

Corollary

Any equivariant PL de Rham complex (Def. ) is a fibrant object in the model structure on equivariant connective dgc-algebras.

(also Scull 08, Lemma 5.2)

In fact:

Proposition

(Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras)

Let $G$ be a finite group.

The $G$-equivariant PL de Rham complex-construction is the left adjoint in a Quillen adjunction between

• the opposite of the projective model structure on equivariant connective dgc-algebras

• the model structure on equivariant simplicial sets

  (i.e.: the global projective model structure on functors from the opposite of the orbit category to the classical model structure on simplicial sets)

$$\big( G dgcAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \underoverset { \underset {\;\;\; exp \;\;\;} {\longrightarrow} } { \overset {\;\;\;\Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot_{\mathrlap{Qu}}} G SimplicialSets_{Qu}$$

(Scull 08, Prop. 5.1)

• equivariant PL de Rham theorem

Related concepts

• equivariant rational homotopy theory

References

• Georgia Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) (jstor:1999119)

• 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)