tensor product of vector bundles

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

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

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see also **algebraic topology**

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Given two vector bundles $V_1 \to X$ and $V_2 \to X$ over the same base space $X$, their *tensor product of vector bundles* $V_1 \otimes V_2 \to X$ is the vector bundle over $X$ whose fiber over any point is the tensor product of vector spaces (i.e. the tensor product of modules) of the respective fibers of $V_1$ and $V_2$.

The tensor product of vector bundles makes the category Vect(X) into a symmetric monoidal category, in fact a distributive monoidal category.

**(tensor product of topological vector bundles)**

Let $X$ be a topological space, and let $E_1 \to X$ and $E_2 \to X$ be two topological vector bundles over $X$.

Let $\{U_i \subset X\}_{i \in I}$ be an open cover with respect to which both vector bundles locally trivialize (this always exists: pick a local trivialization of either bundle and form the joint refinement of the respective open covers by intersection of their patches). Let

$\left\{
(g_1)_{i j}
\colon U_i \cap U_j \to GL(n_1)
\right\}
\phantom{AAA}
\text{and}
\phantom{AAA}
\left\{
(g_2)_{i j}
\colon
U_i \cap U_j \longrightarrow GL(n_2)
\right\}$

be the transition functions of these two bundles with respect to this cover.

For $i, j \in I$ write

$\array{
(g_1)_{i j} \otimes (g_2)_{i j}
&\colon&
U_i \cap U_j
&\longrightarrow&
GL(n_1 \cdot n_2)
}$

be the pointwise tensor product of vector spaces of these transition functions

Then the *tensor product bundle* $E_1 \otimes E_2$ is the one glued from this tensor product of the transition functions (by this construction):

$E_1 \otimes E_2
\;\coloneqq\;
\left(
\left( \underset{i}{\sqcup} U_i \right)
\times
\left(
\mathbb{R}^{n_1 \cdot n_2}
\right)
\right)/ \left( \left\{ (g_1)_{i j} \otimes (g_2)_{i j} \right\}_{i,j \in I} \right)
\,.$

Let $X$ be a topological space and let $E_i \overset{p_i}{\to} X$ be a 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 $E_2$ 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.

Last revised on June 5, 2018 at 16:55:48. See the history of this page for a list of all contributions to it.