# Pointed bundles as short exact sequences

Let $\pi: E \to B$ be a pointed continuous surjection, and let $F = \pi^{-1}(b_0)$ be the fiber over the basepoint (base fiber for short). Then $$* \to F \to E \stackrel{\pi}{\to} B \to *$$ is a short exact sequence, zero morphisms being defined as usual in categories with zero objects.

We immediately notice that left-splitting bundles are those with the bundle space retracting onto the base fiber, and right-splitting bundles are those that have a global section (this is especially nice when $\pi$ is a Cartan principal bundle).

UPD: as shown here: www.math.toronto.edu/selick/mat1345/notes.pdf, for locally trivial bundles left-splitting implies triviality, and thus for locally trivial principal bundles splitting of the either kind implies triviality, which I think is interesting.

Has this connection been pursued before? It most likely was, but it's never stated explicitly in what papers I've seen on the web. Are there any nice results?

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Alexei, don't you think this is [algebraic-topology] rather than [general-topology]? –  Asaf Karagila Oct 2 '11 at 6:35
@Asaf, I don't know what to think anymore after one of my questions on bundles got 'nerfed' from [algebraic-topology] to [general-topology] :) –  Alexei Averchenko Oct 2 '11 at 6:37