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tensor product of vector bundles

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

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see also algebraic topology

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Idea

Given two vector bundles V 1XV_1 \to X and V 2XV_2 \to X over the same base space XX, their tensor product of vector bundles V 1V 2XV_1 \otimes V_2 \to X is the vector bundle over XX 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 1V_1 and V 2V_2.

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

Definition

Definition

(tensor product of topological vector bundles)

Let XX be a topological space, and let E 1XE_1 \to X and E 2XE_2 \to X be two topological vector bundles over XX.

Let {U iX} iI\{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

{(g 1) ij:U iU jGL(n 1)}AAAandAAA{(g 2) ij:U iU jGL(n 2)} \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,jIi, j \in I write

(g i) ij(g 2) ij : U iU j GL(n 1n 2) \array{ (g_i)_{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 1E 2E_1 \otimes E_2 is the one glued from this tensor product of the transition functions (by this construction):

E 1E 2((iU i)×( n 1n 2))/({(g 1) ij(g 2) ij} i,jI). 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) \,.

Examples

Proposition

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

f:E 1E 2 f \;\colon\; E_1 \rightarrow E_2

is equivalently a section of the tensor product of E 2E_2 with the dual vector bundle of E 1E_1:

Hom Vect(X)(E 1,E 2)Γ X(E 1 * XE 2). 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.

Revised on July 4, 2017 11:45:25 by Urs Schreiber (195.37.209.183)