Let $X_1$ and $X_2$ be two $D^2$ bundles over a (not necessarily orientable) closed connected surface $\Sigma$. I looking for necessary and/or sufficient conditions that show that $\partial X_1 = \partial X_2$. As I guess, maybe we need the Euler classes to be equal up to sign?
2026-03-29 20:04:31.1774814671
Boundary of disk bundles over surfaces
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To begin with, you want both disk bundles to be orientable or nonorientable simultaneously (a bundle is orientable iff it is trivial over the 1-skeleton of the base). For orientable bundles with orientable base ("totally orientable"), the $\pm$ Euler number is the complete invariant; otherwise the Euler number is not even well-defined. However, one can define the "virtual Euler number", obtained by taking a finite totally orientable cover $p: X'\to X$ and setting $e(X):= e(X')/deg(p)$. I think, this is a complete invariant. Check "The geometries of 3-manifolds", Bulletin of the London Mathematical Society (1983), by Peter Scott.