vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
By default, by a real vector bundle over a plain topological space one means a vector bundle of real vector spaces associated to an $O(n)$-principal bundle for $O(n)$ the orthogonal group.
This is in contrast notably to complex vector bundles or quaternionic vector bundles.
In the context of KR-theory (“Real K-theory”) and following Atiyah 1966 p. 368 one considers a more general notion (which subsumes the plain notion above, see Exp. below):
(Atiyah Real vector bundles)
A Real vector bundle (capital “R” for disambiguation) over a Real space – namely over
a $\mathbb{Z}_2$-equivariant space
hence a topological space equipped with a continuous involution
is (according to Atiyah 1966 p. 368):
a complex vector bundle $p \colon E \to X$ over the underlying topological space $X$
whose total space $E$ is equipped with a continuous involution $(\text{-})^\ast \,\colon\, E \to E$
such that:
the bundle projection $p$ is a $\mathbb{Z}_2$-equivariant map, hence such that the following diagram commutes
the involution on $E$ is fiber-wise anti-linear, ie. for $x \in X$ the following diagram commutes:
A homomorphism of such Real vector bundles is a homomorphism of the underlying complex vector bundles which respects all the involutions.
(real vector bundles as Real vector bundles)
In the special case that the involution on the base space $X$ is trivial, $\sigma = id_X$, a Real vector bundle in the sense of Def. is a complex vector bundle over $X$ equipped continuously with a complex anti-linear involution on each fiber $(\text{-})^\ast \,\colon\, E_x \to E_x$.
The resulting $\pm 1$-eigenspace-decomposition realizes each fiber as the complexification
of a $\mathbb{R}$-vector space and this decomposition is preserved by homomorphisms of Real vector bundles over such $X$.
Therefore Real vector bundles over a Real space $X$ whose involution is trivial are equivalent to plain real vector bundles over $X$ as above.
The complex numbers $\mathbb{C}$ equipped with their involution by complex conjugation under their usual multiplication operation form a monoid object internal to the category of $\mathbb{R}$-vector spaces equipped with $\mathbb{R}$-linear involution, the “Real complex numbers”
The Real vector bundles of Def. are equivalently the vector bundles internal to the topos $\mathbb{Z}_2 Set$ with the “ground monoid” taken to be the Real complex numbers (1).
Real vector bundles in the plain sense are the default notion of vector bundles, and as such are discussed in essentially every reference on the topic, see the list there.
The notion of “Real vector bundle” over a Real space in the sense of KR-theory was introduced in:
Michael Atiyah, p. 368 in: K-theory and reality, The Quarterly Journal of Mathematics. Oxford. Second Series 17 1 (1966) 367-386 [doi:10.1093/qmath/17.1.367, pdf, ISSN:0033-5606]
Allan L. Edelson, Real Vector Bundles and Spaces with Free Involutions, Transactions of the American Mathematical Society 157 (1971) 179-188 [doi:10.2307/1995841, jstor:1995841]
Allan L. Edelson, Real Line Bundles on Spheres, Proceedings of the American Mathematical Society 27 3 (1971) 579-583 [doi:10.2307/2036501, jstor:2036501]
(on Real line bundles over spheres with involution action)
Last revised on September 6, 2023 at 17:52:07. See the history of this page for a list of all contributions to it.