Fiber bundles with degenerations and their applications to computing fundamental groups
Let \(p : E \to B\) be a surjective morphism of connected smooth manifolds, with generic fiber \(F\). The author derives a short exact sequence relating the fundamental groups of \(F\), \(B\), and \(E\) under the assumptions that (i) \(p\) is a locally trivial bundle projection except over a union of codimension two submanifolds of \(B\), and (ii) \(p\) has a local section at each point of \(B\). In case \(p\) has a sufficiently generic global section the sequence looks like the tail of the long exact homotopy sequence, with a correction term determined by the monodromies over loops in \(B\) linking the singular locus of \(p\). These results are applied to the fundamental groups of complements of hypersurfaces, using generic projection, and to fundamental groups of hyperplane complements, using iterated linear projections with singular fibers. In particular, the author computes presentations of the generalized pure braid groups of type \(B_ n\) and \(D_ n\).
surjective morphism of connected smooth manifolds
presentations of the generalized pure braid groups
scientific article; zbMATH DE number 486901