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Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...

Every continuous vector bundle on a closed smooth manifold $M$ has a smooth structure. On the other hand, every vector bundle $E$ is the image of a trivial bundle $M\times\mathbb{C}^n$ under some ...

Let $D$ be a first-order self-adjoint elliptic operator on a closed Riemannian manifold $M$. Then $D$ has discrete spectrum in $\mathbb{R}$, and there is an orthonormal basis for $L^2(M)$ consisting ...