Vector bundles (and thus their associated principal bundles) can be examined using characteristic classes. For a given vector bundle ${(E,M,\mathbb{K}^{n})}$ these are elements in the cohomology groups of the base space ${c(E)\in H^{\ast}(M;R)}$, for some commutative unital ring ${R}$, which commute with the pullback of any ${f\colon N\rightarrow M}$:

$\displaystyle c\left(f^{*}\left(E\right)\right)=f^{*}\left(c\left(E\right)\right)$

In the second term, the pullback by ${f}$ means that ${f^{*}\left(c\left(E\right)\right)\in H^{\text{*}}(N;R)}$. Since a trivial vector bundle ${M\times\mathbb{K}^{n}}$ is the pullback of ${(E,0,\mathbb{K}^{n})}$ by ${f\colon M\rightarrow0}$ (where ${0}$ is the space with a single point), we have ${c\left(M\times\mathbb{K}^{n}\right)=c\left(f^{*}\left(E\right)\right)=f^{*}\left(c\left(E\right)\right)=0}$ (where ${0}$ is the ring zero). Therefore the characteristic classes of a trivial bundle vanish, or in other words a characteristic class acts as an obstruction to a bundle being trivial. However there exist non-trivial bundles whose characteristic classes also all vanish. Similarly, if two vector bundles with the same base space are isomorphic, then they are related by the identity pullback; thus a necessary (but not sufficient) condition for isomorphism is identical characteristic classes. All characteristic classes can be determined via the cohomology classes of the classifying spaces ${BO(n)}$ and ${BU(n)}$, since e.g. for real vector bundles any ${(E,M,\mathbb{R}^{n})}$ is the pullback of ${BO(n)}$ by some ${f}$, so that we have ${c\left(E\right)=c\left(f^{*}\left(BO(n)\right)\right)=f^{*}\left(c\left(BO(n)\right)\right)}$.

For a real vector bundle ${(E,M,\mathbb{R}^{n})}$ there are three kinds of characteristic classes (none of which we will define here): the Stiefel-Whitney classes ${w_{i}(E)\in H^{i}(M;\mathbb{Z}_{2})}$, the Pontryagin classes ${p_{i}(E)\in H^{4i}(M;\mathbb{Z})}$, and if the bundle is oriented the Euler class ${e(E)\in H^{n}(M;\mathbb{Z})}$. For complex vector bundles, there are the Chern classes ${c_{i}(E)\in H^{2i}(M;\mathbb{Z})}$. The characteristic class of a manifold ${M}$ is defined to be that of its tangent bundle, e.g.

$\displaystyle w_{i}(M)\equiv w_{i}(TM).$

If ${M}$ is a compact orientable four-dimensional manifold, then it is parallelizable iff ${w_{2}(M)=p_{1}(M)=e(M)=0}$.

A non-zero Stiefel-Whitney class ${w_{i}(E)}$ acts as an obstruction to the existence of ${(n-i+1)}$ everywhere linearly independent sections of ${E}$. Therefore, if such section do exist, then ${w_{j}(E)}$ vanishes for ${j\geq i}$; in particular, a non-zero ${w_{n}(E)}$ means there are no non-vanishing global sections. It can be shown that ${w_{1}(E)=0}$ iff ${E}$ is orientable, so that ${M}$ is orientable iff ${w_{1}(M)=0}$.

Spin structures exist on an oriented ${M}$ iff ${w_{2}(M)=0}$; if spin structures do exist, then their equivalency classes have a one-to-one correspondence with the elements of ${H^{1}(M,\mathbb{Z}_{2})}$. Inequivalent spin structures have either inequivalent spin frame bundles or inequivalent bundle maps; in four dimensions, there is only one spin frame bundle up to isomorphism, so that different spin structures correspond to different bundle maps (i.e. different spin connections).

Spin^c structures exist on an oriented ${M}$ if spin structures exist, but also in some cases where they do not; for example if ${M}$ is simply connected and compact. If spin^c structures do exist, then their equivalency classes have a one-to-one correspondence with the elements of ${H^{2}(M,\mathbb{Z})}$, and in four dimensions, unlike the case for spin structures, inequivalent spin^c structures can have inequivalent spin frame bundles.