I just looked up the Atiyah-Singer theorem and by ignoring technical details I had the impression that it tells us that any elliptic operator on a compact manifold satisfies
Analytical index = Topological index.
Now, if I pick the Laplacian on such a manifold, then this is an elliptic operator. But clearly the analytical index of the Laplacian is always zero( operator is self-adjoint.)
So I just proved Topological index=0 for any compact manifold, which is clearly nonsense. What did I do wrong?