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34). 11, using the existence of a universal bundle for a Lie group (see page 40). 45 Suppose α ∈ 2I (G). 30, if we can show that α = 0, then this will imply H 2 (G; R) = 0. Expressing the L-invariance of α for a left invariant vector ﬁeld X gives: 0 = L(X)α = (i(X)d + di(X))α = i(X)dα + di(X)α = di(X)α, since α is closed. Hence, i(X)α is a closed 1-form. Since G is semisimple, we have H 1 (G; R) = 0, so i(X)α is an exact 1-form. That is, there exists a smooth function f : G → R such that i(X)α = df .

DLg (Yk ))(gx) = i(X)ω(Y1 , . . , Yk )(x). 6 Invariant forms Hence, i(X)ω is left invariant. The veriﬁcation of the other statements is similar. The previous result justiﬁes the following deﬁnition. 27 Let G be a Lie group and M be a left G-manifold. The invariant cohomology of M is the homology of the cochain complex ( L (M), d). We denote it by HL∗ (M). The main result is the following theorem. 28 Let G be a compact connected Lie group and M be a compact left G-manifold. Then HL∗ (M) ∼ = H ∗ (M; R).

Xr ) → H. Fact 2: The morphism φ is an isomorphism. By construction, φ is surjective so we are reduced to establishing its injectivity. Observe that the restriction of φ to ∧ (x1 ) is injective. We argue by induction and suppose that its restriction to ∧ x1 , . . , xk−1 is injective. Let a ∈ ∧ (x1 , . . , xk ) be such that φ(a) = 0. We decompose a into a = a1 + xk a2 , with a1 and a2 in ∧ x1 , . . , xk−1 . We denote by φ k the following composition ∧ (x1 , . . , xk ) φ /H µ∗ /H ⊗ H qk ⊗id / H ⊗ H.