nLab exterior bundle

Definition

For $X$ a smooth manifold its exterior bundle is the vector bundle $\wedge^\bullet T^\ast X$ which is the direct sum (Whitney sum) of the bundle of differential n-forms for all $n \in \mathbb{N}$.

Properties

If $(X,g)$ is a Riemannian manifold, then the exterior bundle supports a canonical Dirac operator, namely the Kähler-Dirac operator $d + d^\dagger$. The corresponding Fredholm operator $(d+ d^\ast)(1 + (d + d^\ast)^2)^{-1/2}$ constitutes a canonical class in the K-homology of $X$.

Related concepts

• exterior algebra

• supersymmetric quantum mechanics