Principal Bundles and Vector Bundles

Principal bundles and vector bundles are the two most important classes of fiber bundles, each with their own rich theory and deep connections to geometry and physics.

Principal Bundles

Definition

A principal $G$ -bundle is a fiber bundle $p : P \to B$ equipped with a free right action of a topological group $G$ on $P$ such that $B = P/G$ and $p$ is the orbit map. The fiber over each point is homeomorphic to $G$ , and the local trivializations are $G$ -equivariant: $\varphi : p^{-1}(U) \to U \times G$ satisfies $\varphi(pg) = \varphi(p) \cdot g$ for the right $G$ -action on $U \times G$ by right multiplication.

Definition

Let $p : P \to B$ be a principal $G$ -bundle and $F$ a left $G$ -space. The associated bundle is $P \times_G F = (P \times F) / \sim$ where $(pg, f) \sim (p, gf)$ for $g \in G$ . The projection $P \times_G F \to B$ given by $[p, f] \mapsto p(p)$ makes this a fiber bundle with fiber $F$ .

Every vector bundle can be viewed as a bundle associated to its frame bundle, and conversely, principal bundles give rise to vector bundles via representations.

Vector Bundles

Definition

A rank- $n$ real vector bundle is a fiber bundle $\xi : E \to B$ where each fiber $p^{-1}(b)$ has the structure of an $n$ -dimensional real vector space, and the local trivializations $\varphi : p^{-1}(U) \to U \times \mathbb{R}^n$ are fiberwise linear. The transition functions $g_{\alpha\beta} : U_\alpha \cap U_\beta \to GL(n, \mathbb{R})$ record how trivializations change on overlaps.

ExampleTangent and normal bundles

The tangent bundle $TM$ of a smooth $n$ -manifold $M$ has fiber $T_pM \cong \mathbb{R}^n$ at each point. If $M \hookrightarrow \mathbb{R}^N$ is embedded, the normal bundle $\nu_M$ has fiber equal to the orthogonal complement of $T_pM$ in $\mathbb{R}^N$ , and $TM \oplus \nu_M \cong M \times \mathbb{R}^N$ (the trivial bundle).

Operations on Vector Bundles

Theorem10.3K-Theory and Vector Bundles

The set $\operatorname{Vect}_n(B)$ of isomorphism classes of rank- $n$ real vector bundles over $B$ is in bijection with $[B, BO(n)]$ , the set of homotopy classes of maps into the classifying space $BO(n) = \varinjlim_{k \to \infty} Gr_n(\mathbb{R}^{n+k})$ , where $Gr_n$ is the Grassmannian. The universal bundle is the tautological bundle $\gamma^n \to BO(n)$ .

RemarkWhitney sum and K-theory

Vector bundles over $B$ admit direct sum $\oplus$ , tensor product $\otimes$ , and dual operations, making them an algebraic object. The K-theory $K(B) = K^0(B)$ is the Grothendieck group of the monoid $(\bigsqcup_n \operatorname{Vect}_n(B), \oplus)$ . Bott periodicity shows $K$ -theory is a generalized cohomology theory with period $2$ (complex) or $8$ (real).