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Given a connected topological space $E$, under which conditions is it possible to find a subspace $B$ such that $E$ can be regarded as a (rank $n$) vector bundle over $B$?

Is it possible to find the conditions and the $B$'s if one moves to the more rigid differentiable, holomorphic or algebraic setting?

What if we restrict to the case: dim $E$ = 2, dim $B$ = 1? When is a surface the total space of a line bundle?

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    $\begingroup$ I doubt you will find a satisfactory answer in the literature in much generality, but if $E$ is a simply-connected manifold that has $B$ (a manifold) as a deformation-retract, then it is a vector bundle under some reasonable conditions, by the minimal handle theorem -- closely related to the h-cobordism theorem. A theorem of this sort appears in Kosinski's "Differential Topology" textbook. $\endgroup$
    – Ryan Budney
    Commented Aug 22, 2020 at 5:23
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    $\begingroup$ Fibre-bundles are a little easier to identify. Under "reasonable circumstances" assuming manifolds everywhere, all you need is some compactness and the map being a submersion. This is just a careful application of the implicit function theorem. $\endgroup$
    – Ryan Budney
    Commented Aug 22, 2020 at 6:35
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    $\begingroup$ There also is the Soul Theorem in Riemannian geometry $\endgroup$
    – Sebastian Goette
    Commented Aug 31, 2020 at 19:21

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The total space of a vector bundle is homotopy-equivalent to the base. Hence, for example, the only connected surfaces which are total spaces of real line bundles are the plane, the annulus and the Möbius strip.

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In smooth manifolds, Grabowski and Rotkiewicz - Higher vector bundles and multi-graded symplectic manifolds has a condition for when a monoid action $(\mathbb{R}^+, \cdot, 1)$ on a manifold $E$ induces a vector bundle structure where $E$ is the total space. I have a similar result in a recent paper (Vector bundles and differential bundles in the category of smooth manifolds), so that a morphism $\lambda:E \to TE$ induces a vector bundle where $E$ is the total space whenever $\lambda$ satisfies some coherences and a certain pullback diagram (these are called differential bundles in a tangent category). The total space is obtained by splitting the idempotent $p \circ \lambda:E \to E$, where $p$ is the tangent projection (it's a consequence of the coherences on $\lambda$ that $p\circ \lambda$ is an idempotent).

I don't know of any similar results that hold for general topological vector bundles, but I would be interested in seeing them!

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