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This is a question that has appeared in various forms on MathOverflow, see here and here, for example. But as opposed to these more algebraic questions, I am interested in the purely topological version, and I would like to compile a list of techniques to approach this kind of problem.

For my own interests I will be working in the smooth category, though I'm sure the techniques generalize to a broader category of topological spaces. I will use the convention that an open manifold is a manifold that is not compact and has no boundary, and a closed manifold is a compact manifold without boundary.

Let $M$ be a manifold and $N$ a submanifold. Let $E \to N$ be a topological vector bundle. When can $E$ extend to a vector bundle $F \to M$ where $F|_N = E$? I am particularly interested in two specific cases, where $M$ is closed and $N$ is open (both as a subspace and as a manifold), or when $M$ is closed and $N$ is a codimension $1$ closed submanifold.

The primary tool for problems like this lies in obstruction theory of course. This theory has always seemed somewhat confusing to me, and I think not only I but many others would benefit from a breakdown of techniques and results from obstruction theory applicable to the above problem. Good references would also be welcome.

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    $\begingroup$ It seems to me like the fundamental theorem of obstruction theory, say Theorem 7.1 of Kirk and Davis, is very literally the exact situation of your question when you remember that vector bundles are classified by maps to $BO(n)$. I think there is not much more to say. We know a fair few homotopy groups of $BO(n)$, especially when $n=\infty$ so it often turns out that one can actually work with the obstructions in practice. $\endgroup$
    – Connor Malin
    Commented Jan 18 at 1:41
  • $\begingroup$ @ConnorMalin Thank you for the reference! And that's good news to hear about how there's not more than needed than this theorem. I sometimes hear the words "complete obstruction" or something similar in reference to an obstruction being necessary and sufficient for its intended purpose. I was lead to believe this means having this obstruction prevents the object in question from existing, and failing to have this obstruction always allows the object to exist. Is the correct definition? And can one obtain complete obstructions for the problem in my question with what you mentioned? $\endgroup$
    – Paul Cusson
    Commented Jan 18 at 1:48
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    $\begingroup$ Yes, the theorem provides complete obstructions relevant to extending maps up to a homotopy which stays constant on a subspace. You might find it helpful to think about this theorem in the context of mapping into the Eilenberg-MacLane space $K(A,n)$ which is perhaps a more familiar topic. $\endgroup$
    – Connor Malin
    Commented Jan 18 at 1:55

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