03.05.02 · modern-geometry / bundles

Vector bundle

shipped3 tiersLean: none

Anchor (Master): Milnor-Stasheff §2; Husemoller §3; Atiyah K-Theory §1.1

Intuition [Beginner]

A vector bundle is a space made by attaching a vector space to every point of another space. The base tells you where you are. The attached vector space tells you what vectors are available at that point.

The simplest example is a product: every point carries the same copy of a vector space in the same way. More interesting examples twist, so nearby fibers can be compared locally but not always aligned globally.

This idea lets geometry turn pointwise linear algebra into global structure. Tangent vectors, spinors, and gauge fields all live in vector bundles.

Visual [Beginner]

The green shape is the base. Above each point is a small vector space. In the red window, the bundle looks like a product.

The fibers can be added and scaled one point at a time. The bundle records how those vector spaces vary across the base.

Worked example [Beginner]

Take a circle as the base. Attach a copy of the number line above every point.

If the lines are aligned the same way all around the circle, the result is a cylinder-like product bundle. If the line flips once before returning to the starting point, the result is the line-bundle version of a Mobius strip.

In both cases, each individual fiber is a line. The difference lies in how the lines are glued as the base point moves.

What this tells us: a vector bundle can have simple fibers while still carrying global twisting.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $B$ be a topological space and let $K$ be $R$ or $C$ . A rank- $n$ vector bundle over $B$ is a continuous surjection

π : E \to B

such that each fiber $E_{b} = π^{- 1} (b)$ is an $n$ -dimensional vector space over $K$ , and every $b \in B$ has an open neighborhood $U$ with a homeomorphism

Φ_{U} : π^{- 1} (U) ⟶ U \times K^{n}

over $U$ whose restriction to each fiber is a linear isomorphism

E_{b} ≅ {b} \times K^{n} .

For smooth vector bundles, $B$ is a smooth manifold, $E$ is a smooth manifold, $π$ is smooth, and local product charts are smooth with fiberwise-linear transition maps ^{[Milnor-Stasheff §2]}.

A section of $E$ is a map $s : B \to E$ with $π \circ s = id_{B}$ . The set of continuous or smooth sections is denoted $Γ (E)$ .

Key theorem with proof [Intermediate+]

Theorem (sections form a vector space). Let $π : E \to B$ be a vector bundle over $K$ . The set $Γ (E)$ of continuous sections is a vector space over $K$ under pointwise addition and scalar multiplication.

Proof. If $s, t \in Γ (E)$ , then $s (b)$ and $t (b)$ lie in the same fiber $E_{b}$ for every $b \in B$ . Since $E_{b}$ is a vector space, define

(s + t) (b) = s (b) + t (b), (a s) (b) = a s (b) .

These maps still land in $E_{b}$ , so $π ((s + t) (b)) = b$ and $π ((a s) (b)) = b$ . Thus $s + t$ and $a s$ are sections.

Continuity is local. On a product chart $π^{- 1} (U) ≅ U \times K^{n}$ , sections are represented by maps $U \to K^{n}$ . Pointwise addition and scalar multiplication of such coordinate maps are continuous, so the resulting sections are continuous. The vector-space axioms hold fiber by fiber because each $E_{b}$ is a vector space 01.01.03. $□$

Bridge. The construction here builds toward 03.05.04 (connection on a vector bundle), where the same data is upgraded, and the symmetry side is taken up in 03.06.04 (pontryagin and chern classes). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none — the project-level vector-bundle API needed for this unit is not yet present in Mathlib.

[object Promise]

The missing pieces include local linear product charts, transition maps into general linear groups, smooth sections, frame bundles, and functorial bundle operations.

Advanced results [Master]

A rank- $n$ vector bundle is equivalently a principal $GL_{n} (K)$ -bundle together with the standard associated-bundle construction. Given the frame bundle $Fr (E)$ , one recovers

E ≅ Fr (E) \times_{GL_{n} (K)} K^{n} .

This equivalence is the bridge between vector-bundle characteristic classes and principal-bundle Chern-Weil theory 03.05.01 ^{[Husemoller §3]}.

Vector bundles admit natural operations: direct sum, tensor product, dual, exterior powers, pullback, and Hom bundles. Characteristic classes are designed to behave predictably under these operations, especially direct sum and pullback 03.06.04.

Over a compact Hausdorff space, complex vector bundles form a commutative monoid under direct sum. The Grothendieck group completion is $K^{0} (B)$ , the starting point of topological K-theory 03.08.07 ^{[Atiyah K-Theory §1.1]}.

Synthesis. This construction generalises the pattern fixed in 01.01.03 (vector space), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

Transition functions. Choose local vector-bundle charts $Φ_{i} : π^{- 1} (U_{i}) \to U_{i} \times K^{n}$ . On overlaps, the comparison $Φ_{i} Φ_{j}^{- 1}$ fixes the base coordinate and is fiberwise linear. It has the form

(x, v) \mapsto (x, g_{ij} (x) v)

with $g_{ij} : U_{i} \cap U_{j} \to GL_{n} (K)$ . Associativity of chart comparison gives $g_{ik} = g_{ij} g_{j k}$ on triple overlaps.

Frame bundle. For each $b \in B$ , let $Fr (E)_{b}$ be the set of linear isomorphisms $K^{n} \to E_{b}$ . Right composition by $GL_{n} (K)$ gives a free transitive action. A local vector-bundle chart identifies $Fr (E) ∣_{U}$ with $U \times GL_{n} (K)$ , proving that $Fr (E) \to B$ is a principal bundle.

Associated-bundle recovery. Map $[φ, v] \in Fr (E) \times_{GL_{n} (K)} K^{n}$ to $φ (v) \in E$ . The quotient relation identifies $(φ A, v)$ with $(φ, A v)$ , and both map to the same vector. Fiberwise, every vector in $E_{b}$ is $φ (v)$ for some frame $φ$ , and freeness gives uniqueness modulo the quotient relation.

Connections [Master]

Vector space 01.01.03 — every fiber is a vector space and all local charts are fiberwise linear.
Principal bundle 03.05.01 — frame bundles are principal bundles, and vector bundles are associated bundles.
Pontryagin and Chern classes 03.06.04 — characteristic classes are invariants of real and complex vector bundles.
Bott periodicity 03.08.07 — topological K-theory is built from vector bundles.
Dirac operator 03.09.08 — spinor bundles and twisting bundles are vector bundles carrying differential operators.

Historical & philosophical context [Master]

Vector bundles grew from the study of tangent spaces, normal data, and transition functions in topology and differential geometry. Steenrod and later Husemoller organized bundles through local product data and structure groups; Milnor and Stasheff made vector bundles the central objects behind characteristic classes ^{[Milnor-Stasheff §2]}.

Atiyah's formulation of K-theory placed vector bundles into a stable algebraic setting: direct sum becomes addition, tensor product becomes multiplication, and formal differences become cohomological invariants of spaces ^{[Atiyah K-Theory §1.1]}.

Bibliography [Master]

Milnor, J. & Stasheff, J., Characteristic Classes, Princeton University Press, 1974. §2.
Husemoller, D., Fibre Bundles, Springer, 3rd ed., 1994. §3.
Atiyah, M. F., K-Theory, W. A. Benjamin, 1967. §1.1.
Steenrod, N., The Topology of Fibre Bundles, Princeton University Press, 1951.

Wave 2 Phase 2.1 unit #3. Produced as the vector-bundle prerequisite for characteristic classes, K-theory, and Dirac-type constructions.