nLab Sullivan model of a classifying space

Statement

Proposition

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

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

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

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

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

Proof

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

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

is due to Chern 1950 (11); Bott 1973, p. 239 (5 of 15).

(Here $k$ is the ground field of characteristic zero).

That the rational cohomology algebra of $B G$, with trivial differential, is the minimal Sullivan model for $B G$ is discussed for instance in Félix, Oprea & Tanré 20 08.

Related concepts

• Chern-Weil homomorphism

Examples of Sullivan models in rational homotopy theory:

• Sullivan model of an n-sphere

• Sullivan model of a spherical fibration

• Sullivan model of complex projective space

• Sullivan model of a classifying space

• Sullivan model of mapping space

• Sullivan model of free loop space

• Sullivan model of a suspension

• Sullivan model of a finite G-quotient

References

• Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf, full proceedings vol 2 pdf)

• Raoul Bott, On the Chern-Weil homomorphism and the continuous cohomology of Lie-groups, Advances in Mathematics Volume 11, Issue 3, December 1973, Pages 289-303 (doi:10.1016/0001-8708(73)90012-1)

• Yves Félix, John Oprea, Daniel Tanré, Algebraic models in geometry, Oxford University Press 2008 (pdf, ISBN:9780199206520)