Interesting vector bundles other than the tangent and cotangent bundle

Question

Given a real/complex topological/smooth manifold $M$, there are two natural vector bundles that we are interested in :

1. Tangent bundle $\pi: TM\rightarrow M$
2. Cotangent bundle $\pi^*:(TM)^*\rightarrow M$

I would like to know if there are any natural/interesting vector bundles that we would come across if we go further in Algebraic/Differential Geometry.

I am almost sure that this question is not well posed, but I am not able to make this any better.

Answer 1

Let Vect be the category of finite dimensional spaces over $\mathbb R$ and let $\mathcal F$ be a functor from that category into itself, like for example $V\mapsto V^*$ (dual) or $V\mapsto \wedge^3V$ (third exterior power). Suppose that $\mathcal F$ is smooth, i.e. that the maps $L(V,W)\to L(\mathcal FV,\mathcal FW)$ are all smooth.

Then given a smooth vector bundle $E$ on an arbitrary smooth manifold $M$ we get in a functorial way a vector bundle $\mathcal FE$ such that for the fibre at $m\in M$ of $\mathcal FE$ we have $(\mathcal FE)_m=\mathcal F(E_m)$.
The proof can be found (in the topological case) in Atiyah's K-Theory.
This construction can be generalized to functors in several variables, like $(V,W)\mapsto V\otimes_\mathbb R W$.

The above procedure allows you, starting from the tangent bundle $T_M=T$ of $M$, to construct infinitely many bundles: for example $T\otimes T^*\otimes \wedge^3 T $ .
Of course on a given manifold many such bundles will be isomorphic: for example on $\mathbb R^n$ all vector bundles are trivial so that all the bundles described above are isomorphic as soon as they have the same rank.

Answer 2

If $M$ admits a spin structure (i.e. $w_1(M) = 0$ and $w_2(M) = 0$), then one can form the spinor bundle $\mathbb{S} \to M$ which is a complex vector bundle.

A spin structure on $M$ is a choice of principal $\operatorname{Spin}(n)$-bundle which is an equivariant lift of the oriented orthonormal frame bundle (a principal $SO(n)$-bundle). The spinor bundle is the complex vector bundle associated to this principal $\operatorname{Spin}(n)$-bundle via a complex representation of $\operatorname{Spin}(n)$ which does not factor through $SO(n)$.

The construction of this bundle is the beginning of spin geometry. A reference for this material is Lawson and Michelsohn's Spin Geometry.