Contents

bundles

# Contents

## Definition

### Plain notion

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 KR-theory

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):

###### Definition

(Atiyah Real vector bundles)
A Real vector bundle (capital “R” for disambiguation) over a Real space – namely over

• $\mathbb{Z}_2 \curvearrowright X \;\in\; \mathbb{Z}_2 Top \;=\; Func\big( \mathbf{B}\mathbb{Z}_2 ,\, Top \big)$

hence a topological space equipped with a continuous involution

$\sigma \,\colon\, X \longrightarrow X ,\;\;\;\;\; \sigma \circ \sigma = id_X$

is (according to Atiyah 1966 p. 368):

1. a complex vector bundle $p \colon E \to X$ over the underlying topological space $X$

2. whose total space $E$ is equipped with a continuous involution $(\text{-})^\ast \,\colon\, E \to E$

such that:

1. the bundle projection $p$ is a $\mathbb{Z}_2$-equivariant map, hence such that the following diagram commutes

$\array{ E &\overset{(\text{-})^\ast}{\longrightarrow}& E \\ \mathllap{{}^p}\big\downarrow && \big\downarrow\mathrlap{{}^p} \\ X &\underset{\sigma}{\longrightarrow}& X }$
2. the involution on $E$ is fiber-wise anti-linear, ie. for $x \in X$ the following diagram commutes:

$\array{ \mathbb{C} \times E_x &\overset{(\text{-})\cdot(\text{-})}{\longrightarrow}& E_x \\ \mathllap{ {}^{ \overline{(\text{-})} \times (\text{-})^\ast } } \big\downarrow && \big\downarrow\mathrlap{{}^{ (\text{-})^\ast }} \\ \mathbb{C} \times E_{\sigma(x)} &\underset{(\text{-})\cdot(\text{-})}{\longrightarrow}& E_{\sigma(x)} }$

A homomorphism of such Real vector bundles is a homomorphism of the underlying complex vector bundles which respects all the involutions.

###### Remark

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 numbers

(1)$\mathbb{Z}_2 \curvearrowright \mathbb{C} \;\;\; \in \;\; Mon\big( Func(\mathbf{B}\mathbb{Z}_2, Mod_{\mathbb{R}}) \big) \,.$

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 numbers (1).

###### Example

(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

$E_x \;\simeq\; V_x \oplus \mathrm{i} V_x \;\simeq\; V_x \otimes_{\mathbb{R}} \mathbb{C}$

of a $\mathbb{R}$-vector space and this decomposition is preserved by homomorphisms of Real vector bundles over such $X$, which are thus given by complexifications of morphisms of real vector bundles.

Therefore the full subcategory of Real vector bundles over a Real space $X$ whose involution is trivial is equivalent to that of plain real vector bundles over $X$ as above. (cf. Atiyah 1966, p. 369)

In fact, more is true: the full subcategory of Real vector bundles on all objects whose base carries the trivial $\mathbb{Z}/2$-action is equivalent to the category VectBund of real vector bundles over varying base spaces

(2)$\array{ \underset{X \in Top}{\textstyle{\int}} \mathbb{R}Vect_X &\xrightarrow{\; \sim \;}& \underset{ (X,id) \in \mathbb{Z}/2 Top }{\textstyle{\int}} RealVect_X \,. }$

###### Remark

Similarly to (3), there is a subcategory-inclusion of complex vector bundles over varying base spaces into Real vector bundles over varying bases spaces with free action

(3)$\array{ \underset{X \in Top}{\textstyle{\int}} \mathbb{C}Vect_X &\xrightarrow{\; \sim \;}& \underset{ (\mathbb{Z}/2 \times X) \in \mathbb{Z}/2 Top }{\textstyle{\int}} RealVect_{\mathbb{Z}/2 \times X} \\ E && E & \overline{E} \\ \big\downarrow &\mapsto& \big\downarrow & \big\downarrow \\ X && \{0\} \times X & \{1\} \times X \, }$

where $\overline{E}$ denotes the vector bundle with fiber-wise anti-linear structure. However, as is this is not a full functor: On the right, if a bundle morphism is nontrivial on the base labels $0 \leftrightarrow 1$, the over the remaining base space it corresponds to a fiberwise anti-linear map.

To make the above into an equivalence of categories one can equip the Real vector bundles moreover with involutions which square to -1 – ie. complex structures internal to Real vector bundles! – and require the maps to respect these further involutions. Then the functor which takes a “Real vector bundle with complex structure” over a free $\mathbb{Z}/2$-space to the $+\mathrm{i}$-eigenvalue bundle (of the internal complex structure), constitutes an equivalence with complex vector bundles.

In this sense the term “Real vector bundle” is quite appropriate.

## References

### Plain notion

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.

### In KR-theory

The notion of “Real vector bundle” over a Real space in the sense of KR-theory was introduced in:

Last revised on November 9, 2023 at 14:07:35. See the history of this page for a list of all contributions to it.