Structures on vector bundles

Question

I am reading the book K theory by Atiyah. In page number 32, he defines some additional structure on a vector bundle $V$. I have understood the definitions there. But there is a statement that says

up to homotopy, self-conjugate, orthogonal, sympletic are essentially equivalent to self-dual, real, quaternion.

Now I am unable to realize the notion of equivalence between vector bundles with additional structures $(V,T)$! Also where does the homotopy come from?

After that it says

results from preceeding sections extend immediately to real and quaternion vector bundles,

I don't see how this happens and moreover I don't understand what would be the precise formulations of the previous results when we have additional structures such as those.

Thanks in advance.

Answer 1

What Atiyah means is the following. Recall the general fact that fiber bundles on a space $X$ with fiber $F$ and structure group $G$ are classified by homotopy classes of maps $X \to BG$, where $BG$ is the classifying space of $G$. An important property of the construction $G \mapsto BG$ is that it is homotopy invariant: that is, if $H \to G$ is a map of topological groups inducing a homotopy equivalence, then the induced map $BH \to BG$ is also a homotopy equivalence. There are various important examples of such maps showing that various bundle classifications therefore coincide.

The general structure of these examples is that if $G$ is a connected Lie group then it has a maximal compact subgroup $K$, and furthermore the inclusion $K \to G$ is a homotopy equivalence, hence so is the map $BK \to BG$, and so the classification of fiber bundles with structure group $G$ is equivalent to the classification of fiber bundles with structure group $K$. Here are the important examples:

• When $G = \text{GL}_n(\mathbb{R})$ (so that, taking $F = \mathbb{R}^n$, we are classifying real vector bundles), the maximal compact is $K = \text{O}(n)$. This tells us that the classification of real vector bundles is the same as the classification of orthogonal vector bundles. Morally speaking this is because the space of Riemannian metrics on a real vector bundle is contractible.
• When $G = \text{GL}_n(\mathbb{C})$ (so that, taking $F = \mathbb{C}^n$, we are classifying complex vector bundles), the maximal compact is $K = \text{U}(n)$. This tells us that the classification of complex vector bundles is the same as the classification of unitary vector bundles. Morally speaking this is because the space of Hermitian metrics on a complex vector bundle is contractible.
• When $G = \text{GL}_n(\mathbb{H})$ (so that, taking $F = \mathbb{H}^n$, we are classifying quaternionic vector bundles), the maximal compact is $K = \text{Sp}(n)$. This tells us that the classification of complex vector bundles is the same as the classification of $\text{Sp}(n)$-bundles. Morally speaking this is because the space of (not sure what the correct term is here) metrics on a quaternionic vector bundle is contractible.

(I hesitate to say "symplectic vector bundles" above because that term is ambiguous. It could also refer to the use of the structure groups $G = \text{Sp}_{2n}(\mathbb{C})$ or $G = \text{Sp}_{2n}(\mathbb{R})$. It turns out that $\text{Sp}(n)$ is also the maximal compact of the former group, but the maximal compact of the latter is $\text{U}(n)$.)