Lefschetz fibration

A Lefschetz fibration is a proper map from an oriented smooth manifold $X^{2n}$ to an oriented surface, with isolated critical points modeled in orientation-compatible complex coordinates byAway from the critical values it is a proper submersion, hence a smooth fiber bundle by the Ehresmann fibration theorem. Properness is needed for this conclusion; the local critical-point model alone cannot ensure that fibers do not escape to infinity.^[ref]

In real dimension four, a regular fiber is a compact surface. Near a critical value, a circle in that surface shrinks to a node. Blowing up the base points of a Lefschetz pencil supplies a central family of examples: each exceptional sphere becomes a section of the resolved fibration.