dual vector bundle

(see also *Chern-Weil theory*, parameterized homotopy theory)

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

Given a vector bundle, its *dual* is the vector bundle obtained by passing fiber-wise to the dual vector space.

- Given a smooth manifold $X$ with tangent bundle $T X$, the dual vector bundle is the cotangent bundle $T^\ast X$.

Let $X$ be a topological space and let $E_i \overset{p_i}{\to} X$ be two topological vector bundles over $X$, of finite rank of a vector bundle. Then a homomorphism of vector bundles

$f \;\colon\; E_1 \rightarrow E_2$

is equivalently a section of the tensor product of vector bundles of $E_2$ with the dual vector bundle of $E_1$.

$Hom_{Vect(X)}(E_1, E_2)
\;\simeq\;
\Gamma_X( E_1^\ast \otimes_X E_2 )
\,.$

Moreover, this section is a trivializing section (this example) precisely if the corresponding morphism is an isomorphism.