nLab
external tensor product of vector bundles

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

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

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see also algebraic topology

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Idea

Given two vector bundles E 1X 1E_1 \to X_1 and E 2X 2E_2 \to X_2, then their external tensor product E 1E 2X 1×X 2E_1 \boxtimes E_2 \to X_1 \times X_2 is the tensor product of vector bundles on the product space X 1×X 2X_1 \times X_2 of the two pullback bundles to this space, along the canonical projection maps pr i:X 1×X 2X ipr_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 1X_1 and X 2X_2 be topological spaces and let E 1X 1E_1 \to X_1 and E 2X 2E_2 \to X_2 be topological vector bundles.

The product topological space X 1×X 2X_1 \times X_2 comes with two continuous projection functions

X 1pr 1X 1×X 2pr 2X 2. 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 *E 1X 1×X 2pr_1^\ast E_1 \to X_1 \times X_2 and pr 2 *E 2X 1×X 2pr_2^\ast E_2 \to X_1 \times X_2.

The external tensor product E 1E 2E_1 \boxtimes E_2 is the tensor product of vector bundles of these pullback bundles:

E 1E 2(pr 1 *E 1) X(pr 2 *E 2) E_1 \boxtimes E_2 \coloneqq (pr_1^\ast E_1) \otimes_X (pr_2^\ast E_2)

which is again naturally a vector bundle over th product space

E 1E 2X 1×X 2. E_1 \boxtimes E_2 \longrightarrow X_1 \times X_2 \,.

Properties

Proposition

(external product theorem in topological K-theory)

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

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

Revised on May 28, 2017 09:22:23 by Urs Schreiber (92.218.150.85)