Classifying spaces and universal bundles

A numerable trivializing cover admits a locally finite partition of unity subordinate to it. Paracompact Hausdorff bases supply this property for principal bundles. For a Lie group $G$, a universal numerable principal bundle $EG\to BG$ can be constructed with contractible total space, for example by a join construction. Its defining consequence is the bijection below between unbased homotopy classes of maps and isomorphism classes of principal bundles. Both the group and the base category are part of the statement.

For a map $f:X\to BG$, its pullback consists of pairs $(x,p)$ with $f(x)=\pi(p)$. The right action acts on the second component. A homotopy of maps yields a bundle over $X\times[0,1]$ whose endpoint restrictions are isomorphic under these hypotheses. Conversely, local bundle data and a partition of unity construct a classifying map, unique up to homotopy. We use that theorem rather than pretending a classifying map is a uniquely defined coordinate function.

For real lines one may use $BO(1)=\mathbb{RP}^{\infty}$, with the tautological line as universal associated bundle. For complex lines, $BU(1)=\mathbb{CP}^{\infty}$ is a model of $K(\mathbb Z,2)$. Over a CW complex, isomorphism classes of complex line bundles therefore correspond to integral second cohomology. The first Chern class realizes this correspondence. Real line bundles similarly correspond to first cohomology with mod-two coefficients through their first Stiefel–Whitney class. These are strong classification statements for lines, not general statements about all ranks.

The tautological complex line has first Chern class minus the positive hyperplane generator under the usual complex orientation. Its dual has the positive generator. Either generator identifies the infinite projective space with an Eilenberg–Mac Lane target, but keeping the sign fixed matters in numerical pairings. For higher-rank unitary bundles the infinite complex Grassmannian is a classifying model. Its multiple universal Chern classes provide useful invariants without guaranteeing that those classes completely distinguish every bundle over every base.

A universal cohomology class $u\in H^d(BG;R)$ assigns $f^*u$ to a bundle classified by $f$. Homotopy invariance makes the assignment independent of the classifying-map choice, and pullback functoriality supplies naturality. This explains why characteristic classes behave uniformly across unrelated bases. A reduction of structure group to a subgroup can often be expressed as a lift of the classifying map through the corresponding map of classifying spaces. The existence of a lift is a separate obstruction problem.

Contractibility of the base trivializes a numerable bundle, but it does not identify all connections on that trivial bundle. Even a circle, on which every complex line bundle is topologically trivial, supports a circle of inequivalent flat line connections. Conversely, a nontrivial characteristic class can obstruct a global frame before any connection is selected. Classification and connection moduli thus answer different questions: one concerns the gluing of fibers, and the other concerns comparison rules on those fibers.