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equivariant PL de Rham complex

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Context

Representation theory

Algebra

Rational homotopy theory

Contents

Definition

Let GG be a finite group.

Definition

(equivariant PL de Rham complex )

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

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

GOrbits Ω PLdR (Maps(,X) G) dgcAlgebras G/H Ω PLdR (X H) \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/HG/H assigns the PL de Rham complex of the HH-fixed locus X HXX^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. )

GOrbits Ω PLdR (Maps(,X) G) dgcAlgebras VectorSpaces \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 GG be a finite group.

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

(GdgcAlgebras k 0) proj op QuexpΩ PLdR GSimplicialSets Qu \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)


References

Last revised on September 25, 2020 at 15:38:38. See the history of this page for a list of all contributions to it.