2 Deleted a sentence and changed initial sentence.Added remark on orientation and simple-connectedness

I think that at an elementary level in Algebraic/Differential Topology the most spectacular

An interesting application of De Rham's theorem is to show that certain differential manifolds are not diffeomorphic. Here are two examples.

1) For $n$ even the sphere $S^n$ and real projective space $\mathbb P^n(\mathbb R)$ are not diffeomorphic since $H^n(S^n) \simeq \mathbb R$ while $H^n(\mathbb P^n(\mathbb R))=0$. Ah, you say, but I can see that with the concepts of orientation or the fundamental group $\pi_1$: I don't need your Swiss's stuff! Fair enough: these are reasonable elementary alternatives.

2) Fix $N\geq 2$ and delete $k$ points from $\mathbb R ^N$: call $X_k$ the resulting manifold. Then for $k\neq l$, the manifolds $X_k$ and $X_l$ are not diffeomorphic since $dim_{\mathbb R} H^{N-1}(X_k )=k\neq l=dim_{\mathbb R} H^{N-1}(X_l )$ .

What However they are both orientable, and simply connected for $N\geq 3$. So the elementary tools of example 1) do you say now group fundamentalist Swissophobe?not apply.

1

I think that at an elementary level in Algebraic/Differential Topology the most spectacular application of De Rham's theorem is to show that certain differential manifolds are not diffeomorphic. Here are two examples.

1) For $n$ even the sphere $S^n$ and real projective space $\mathbb P^n(\mathbb R)$ are not diffeomorphic since $H^n(S^n) \simeq \mathbb R$ while $H^n(\mathbb P^n(\mathbb R))=0$. Ah, you say, I can see that with orientation or the fundamental group $\pi_1$: I don't need your Swiss's stuff! Fair enough.

2) Fix $N\geq 2$ and delete $k$ points from $\mathbb R ^N$: call $X_k$ the resulting manifold. Then for $k\neq l$, the manifolds $X_k$ and $X_l$ are not diffeomorphic since $dim_{\mathbb R} H^{N-1}(X_k )=k\neq l=dim_{\mathbb R} H^{N-1}(X_l )$ .

What do you say now group fundamentalist Swissophobe?