nLab Sullivan model of a classifying space





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) \,.

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 1950 (11); Bott 1973, 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 20 08.

Examples of Sullivan models in rational homotopy theory:


Last revised on November 26, 2023 at 15:05:53. See the history of this page for a list of all contributions to it.