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Sullivan model of a classifying space

Contents

Contents

Statement

Proposition

Let GG be a finite dimensional simply connected compact Lie group, with Lie algebra denoted 𝔤\mathfrak{g}

Then the Sullivan minimal model of the classifying space BGB G is the graded algebra

inv (𝔤)Sym(𝔤 *[2]) G inv^\bullet(\mathfrak{g}) \; \coloneqq \; Sym \big( \mathfrak{g}^\ast[2] \big)^G

of invariant polynomials on 𝔤\mathfrak{g}, equipped with vanishing differential:

(inv (𝔤),d=0)CE(𝔩BG). \big( inv^\bullet(\mathfrak{g}), d = 0 \big) \overset{\simeq}{\longrightarrow} CE \big( \mathfrak{l} B G \big) \,.
Proof

That the rational cohomology of BGB G is given by inv(𝔤)inv(\mathfrak{g})

inv (𝔤)H (BG,k) inv^\bullet(\mathfrak{g}) \;\simeq\; H^\bullet \big( B G, k \big)

is due to Chern 50, (11), Bott 73, p. 239 (5 of 15).

(Here kk is the ground field of characteristic zero).

That the rational cohomology algebra of BGB G, with trivial differential, is the minimal Sullivan model for BGB G is discussed for instance in (Félix-Oprea-Tanré 08).

Examples of Sullivan models in rational homotopy theory:

References

Last revised on August 27, 2020 at 11:15:07. See the history of this page for a list of all contributions to it.