How to recognize a vector bundle?

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?

MathOverflow Asked by user163840 on December 30, 2020

2 Answers

2 Answers

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!

Answered by Ben MacAdam on December 30, 2020

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.

Answered by Gael Meigniez on December 30, 2020

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