Corks, exotic 4-manifolds and genus functions

TL;DR

This paper demonstrates the existence of infinitely many exotic smooth structures on certain 4-manifolds, explores their genus functions, and introduces a genus function type as a diffeomorphism invariant, with applications to exotic knotting.

Contribution

It proves that handlebodies without 3- and 4-handles can have infinitely many exotic structures with equivalent genus functions, and introduces a new genus function type as a diffeomorphism invariant.

Findings

Existence of infinitely many exotic smooth structures on certain 4-manifolds.

Genus functions of these manifolds are pairwise equivalent.

Algebraic inequivalences of genus functions are stable under connected and boundary sums.

Abstract

We prove that every 4-dimensional oriented handlebody without 3- and 4-handles can be modified to admit infinitely many exotic smooth structures, and moreover prove that their genus functions are pairwise equivalent. We furthermore show that for any 4-manifold admitting an embedding into a symplectic 4-manifold with weakly convex boundary, its genus function is algebraically realized as those of infinitely many pairwise exotic 4-manifolds. In addition, we prove that algebraic inequivalences of genus functions are stable under connected sums and boundary sums with a certain type of 4-manifolds having arbitrarily large second Betti numbers. Besides, we introduce a notion of genus function type for diffeomorphism invariants, and show that any such invariant shares properties similar to all the preceding results and yields lower bounds for the values of genus functions. As an application of our exotic 4-manifolds, we also prove that for any (possibly non-orientable) 4-manifold, every submanifold of codimension at most one satisfying a mild condition can be modified to admit infinitely many exotically knotted copies.