A line bundle in an n-dimensional vector bundle E over an n-manifold M is given by a cross-section in the corresponding projective bundle \({\mathbb{P}}E\). A boundary value problem (\({\mathbb{P}}E,f)\) is a pair such that f is a cross-section of \({\mathbb{P}}E| \partial M\), a solution F of this boundary value problem is an extension of f to a cross-section F of \({\mathbb{P}}E| M\setminus \Delta\) where \(\Delta\) \(\subset M\setminus \partial M\) is called the defect set. This of course can be viewed at as an extension problem for line bundles. The essential defect sets are points and (n-2)-dimensional complexes. The local behavior at the former is described by a Hopf index.
In the most important case, the behavior along an (n-2)-dimensional defect submanifold X is described by a line bundle over \(M\odot X\) \((= M\) blown up along X), with possible point defects and subject to a natural homotopy condition relating it to a double covering of M branched along X. Allowing defects even over the boundary \(\partial M\), it is shown that a general ”normed” boundary value problem with a given branch manifold \(X\subset M\) has a ”normed” solution. (This is only a crude formulation of the elaborate results stated in §7.) For \(\partial M=\emptyset\), a defect formula computes the Euler number of E in terms of the Hopf indices of the point defects and the Euler number of the defect line bundle.
A final result shows that a boundary value problem with given branch manifold in \(\partial M\) can be solved if dim \(M\leq 4\). For dim M\(=5\), there is a counterexample. The introductory discussion explains how these problems are motivated by questions arising in physics.