Splitting principle
I forget what I was saying here;
I had a geometric point.
"a geometric complement to Peter May's paper [cite: @ Hopf]"
The splitting principle says
Basically identify cohomology of $\GL$ ($\gl$?)
with invariant cohomology of the torus.
That is,
$T \to \GL$
so $H^*(\GL) \to H^*(T)$.
This latter map is an injection,
and the cohomology of the torus is easy.
More than an injection, actual identify with the invariants.
Schur-Weyl duality?
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In application to vector bundles,
you say: ``Given a vector bundle $E \to X$, pull it back to
the associated flag bundle, over which it is trivial''
[this also feels like the orientation cover]
This self-reference is formally a bit similar to how a tangent bundle has
an almost complex structure, and a cotangent bundle has a symplectic
structure: you look at the tangent bundle *of* the tangent bundle etc.
$T(TM)$ and $T(T^*M)$