nLab external tensor product of vector bundles

Idea

Given two vector bundles $E_1 \to X_1$ and $E_2 \to X_2$, then their external tensor product $E_1 \boxtimes E_2 \to X_1 \times X_2$ is the tensor product of vector bundles on the product space $X_1 \times X_2$ of the two pullback bundles to this space, along the canonical projection maps $pr_i \colon X_1 \times X_2 \to X_i$.

More abstracty, this is the external tensor product in the indexed monoidal category of vector bundles indexed over suitable spaces.

Definition

Let $X_1$ and $X_2$ be topological spaces and let $E_1 \to X_1$ and $E_2 \to X_2$ be topological vector bundles.

The product topological space $X_1 \times X_2$ comes with two continuous projection functions

$$X_1 \overset{pr_1}{\longleftarrow} X_1 \times X_2 \overset{pr_2}{\longrightarrow} X_2 \,.$$

This gives rise to the pullback bundles $pr_1^\ast E_1 \to X_1 \times X_2$ and $pr_2^\ast E_2 \to X_1 \times X_2$.

The external tensor product $E_1 \boxtimes E_2$ is the tensor product of vector bundles of these pullback bundles:

$$E_1 \boxtimes E_2 \coloneqq (pr_1^\ast E_1) \otimes_{X_1 \times X_2} (pr_2^\ast E_2)$$

which is again naturally a vector bundle over th product space

$$E_1 \boxtimes E_2 \longrightarrow X_1 \times X_2 \,.$$

Properties

Proposition

(external product theorem in topological K-theory)

For $X$ a compact Hausdorff space then the external tensor product of vector bundles over $X$ and over the 2-sphere $S^2$ is an isomorphism of topological K-theory rings:

$$K(X) \otimes K(S^2) \overset{\simeq}{\longrightarrow} K(X \times S^2) \,.$$

Related entries

• direct sum of vector bundles, inner product on vector bundles

• fundamental product theorem in topological K-theory