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Given two vector bundles and , then their external tensor product is the tensor product of vector bundles on the product space of the two pullback bundles to this space, along the canonical projection maps .
More abstracty, this is the external tensor product in the indexed monoidal category of vector bundles indexed over suitable spaces.
Let and be topological spaces and let and be topological vector bundles.
The product topological space comes with two continuous projection functions
This gives rise to the pullback bundles and .
The external tensor product is the tensor product of vector bundles of these pullback bundles:
which is again naturally a vector bundle over th product space
(external product theorem in topological K-theory)
For a compact Hausdorff space then the external tensor product of vector bundles over and over the 2-sphere is an isomorphism of topological K-theory rings:
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 , 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:
Discussion in the context of the K-theory classification of topological phases of matter:
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]
Last revised on May 23, 2023 at 14:52:08. See the history of this page for a list of all contributions to it.