How to recognize a vector 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!

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.