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fundamental theorem of dg-algebraic equivariant rational homotopy theory

Contents

Context

Rational homotopy theory

Representation theory

Contents

Idea

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

Preliminaries

Let GG be a finite group.

Write

Definition

(simply connected and finite equivariant rational homotopy types)

Write

(1)Ho(GSimplicialSets Qu) 2 fin AAAHo(GSimplicialSets Qu) 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 XX which over each G/HGOrbitsG/H \in G Orbits are

  • connected: π 0(X H)=*\pi_0(X^H) = \ast

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

  • rational finite type: dim (H n(X H;,))<dim_{\mathbb{Q}}\big( H^n(X^H;,\mathbb{Q}) \big) \lt \infty for all nn \in \mathbb{N}.

and

(2)Ho(GSimplicialSets Qu) 2 ,fin AAAHo(SimplicialSets Qu) 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(GdgcAlgebras 0) fin 1AAAHo(GdgcAlgebras 0) 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 AA which for each G/HGOrbitsG/H \in G Orbits are

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

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

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

(Scull 08, p. 12, 14)

Statement

Proposition

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

The derived adjunction

Ho((GdgcAlgebras k 0) proj op)exp𝕃Ω PLdR Ho(GSimplicialSets Qu) 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 XX (1) the derived adjunction unit is equivariant rationalization

    Ho(GSimplicialSets Qu) 1,nil fin Ho(GSimplicialSets Qu) 1,nil ,fin X expΩ PLdR (X) \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) }
    Xη X derrationalizationexpΩ 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((GdgcAlgebras k 0) proj op) fin 1exp𝕃Ω PLdR Ho(GSimplicialSets Qu) 1,nil ,fin 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)

References

Last revised on October 3, 2020 at 13:26:36. See the history of this page for a list of all contributions to it.