$H_d^k(n\text{-torus})$ isomorphic to $\Lambda^k(\mathbb{R}^n)$.

Question

Let $M^n = \mathbb{R}^n/\mathbb{Z}^n$ be the $n$-torus. Show that $H_d^k(M^n)$ is isomorphic to $\Lambda^k(\mathbb{R}^n)$.

My thoughts on the problem are as follows.

1. The map $G_{ty}(x) = x + ty$, $0 \le 1 \le 1$ descends to $M^n$, to give a homotopy formula,$$G_y^* \varphi - \varphi = dP_y\varphi + Pyd\varphi.$$
2. Maybe I should average this with respect to $y \in M^n$, or in a period rectangle?

Unfortunately I'm not sure how to continue. Could anybody help me finish?

Answer 1

Any time you have a compact Lie group $G$ acting on a manifold $M$, there is an isomorphism $H^k_d (M) \cong H^k_G (M)$, where the latter is computed from the deRham complex of $G$-invariant forms on $M$. (Yes, you average over the $G$ action.) Now $H^1_{S^1}(S^1) \cong \Lambda^1(\Bbb R) \cong \Bbb R$, and similarly, letting $\Bbb T = (S^1)^n$, $H^k_{\Bbb T}(\Bbb T) \cong \Lambda^k(\Bbb R^n)$.