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.

###### 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 $X$.

Therefore Real vector bundles over a Real space $X$ whose involution is trivial are equivalent to plain real vector bundles over $X$ as above.

(cf. Atiyah 1966, p. 369)

###### 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 complex 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 complex numbers (1).

### 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: