How to recognize a vector bundle?

MathOverflow Asked by user163840 on December 30, 2020

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?

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 $$pcirc lambda$$ 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!