Exercise 4, § 6

Let X be a differentiable manifold of class $C^{\infty}$. In this and the following exercises, we shall denote simply by H(X) the graded $\mathbf{R}$-space $H(\Omega (X))$.

a) Show that $\Omega (X)$ is an associative and anti-commutative graded differential algebra (Algèbre, Chap. X, p. 183, Exerc. 18). Deduce that H(X) has a natural associative and anti-commutative graded algebra structure.

b) If X is connected and of dimension $p$, then $H^i(X) = 0$ for $i > p$ and dim$_{\mathbf{R}}H^0(X) = 1$. Moreover, if X is compact, the space $H^p(X)$ is of dimension one (use Differentiable and Analytic Manifolds, Results, 11.2.4).

c) Let Y be another manifold of class $C^{\infty}$, and $f: X\rightarrow Y$ a map of class $C^{\infty}$. The map $f^*:\Omega (Y)\rightarrow \Omega (X)$ is a morphism of complexes (Differentiable and Analytic Manifolds, Results, 8.3.5); show that $H(f^*) : H(Y)\rightarrow H(X)$ is a morphism of algebras.

d) Assume that we are given a law of operation of G on X of class $C^{\infty}$. Show that the subcomplex $\Omega (X)^G$ of invariant forms is a subalgebra of $\Omega (X)$. Deduce that the map $H(i) : H(\Omega (X)^G)\rightarrow H(X)$ defined in Th. 2 is an isomorphism of graded algebras.

e) Show that (Alt($\mathfrak{g}$))$^G$ is a subalgebra of Alt($\mathfrak{g}$), and that the graded algebra H(G) is isomorphic to it.