LGRT 1681
One way to study the topology of a 4-manifold M is by looking at surjections it admits onto S^2. Such maps almost always have singularities, but they can be homotoped so that the singularities are of two types: Lefschetz and indefinite fold. Lefschetz singularities have garnered tremendous acclaim, but today we pay a homage to their underdog cousin, indefinite folds. We’ll start with the necessary definitions, then consider a surjection M\to S^2 that has indefinite folds. We’ll see what the folds tell us about the topology of M. By the end, we’ll be able to list all 4-manifolds that admit surjections onto S^2 whose singular set is a connected component of indefinite folds.