Search

We prove that a closed smooth 4-manifold has shadow-complexity zero if and only if it is a kind of four-dimensional graph manifold, which decomposes into some particular blocks along embedded copies of S2 × S1, plus some complex projective spaces. We deduce a classification of all smooth 4-manifolds with finite fundamental group and shadow-complexity zero. [ABSTRACT FROM PUBLISHER]

We show that only finitely many links in a closed 3-manifold can have homeomorphic complements, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are only finitely many closed smooth 4-manifolds constructed from some Kirby diagram with bounded number of crossings, discs, and strands, or from some Turaev special shadow with bounded number of vertices. (These are the 4-dimensional analogues of Heegaard diagrams and special spines for 3-manifolds.) We therefore get two filtrations on the set of all closed smooth orientable 4-manifolds with finite sets. The two filtrations are equivalent after linear rescalings, and their cardinality grows at least as nc · n. [ABSTRACT FROM AUTHOR]