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 abstractly, 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

• external direct sum of vector bundles

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

• fundamental product theorem in topological K-theory

References

The notion of external tensor product of vector bundles originates in discussion of topological K-theory:

• Michael Atiyah, §2.6 in: K-theory, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin (1967) [pdf, pdf]

• Raoul Bott, p. 19 of: Lectures on $K(X)$, Benjamin (1969) [pdf, pdf]

• Robert Switzer, Rem. 13.51 in Algebraic Topology – Homotopy and Homology, Grundlehren 212 Springer (1975) [doi:10.1007/978-3-642-61923-6_12]

• Max Karoubi, §4.9 in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer (1978) [pdf, doi:10.1007/978-3-540-79890-3]

• Klaus Wirthmüller, p. 40 of: Vector bundles and K-theory (2012) [pdf]

It is also briefly mentioned in a context of differential geometry in:

• Werner Greub, Stephen Halperin, Ray Vanstone, p. 84 of: Connections, Curvature, and Cohomology Volume 1: De Rham Cohomology of Manifolds and Vector Bundles, Academic Press (1973) [ISBN:978-0-12-302701-6]

Discussion in the context of the K-theory classification of topological phases of matter:

• Bruno Mera, The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator, Phys. Rev. B 102 155150 (2020) [arXiv:2009.02268, doi:10.1103/PhysRevB.102.155150]

Discussion in the variant of categories of quasicoherent sheaves in (derived) algebraic geometry:

• Alexei Bondal, Michel Van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Mosc. Math. J. 3 1 (2003) 1-36 [arXiv:math/0204218]

• David Ben-Zvi, John Francis, David Nadler, Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry, J. Amer. Math. Soc. 23 4 (2010) 909-966 [arXiv:0805.0157]