Nonexistence of twists and surgeries generating exotic 4-manifolds

TL;DR

The paper proves that certain exotic smooth structures on 4-manifolds cannot be generated by twisting or surgery on fixed submanifolds, showing limitations of these methods in producing all smooth structures.

Contribution

It demonstrates that no single fixed submanifold or surgery can generate all smooth structures on certain 4-manifolds, revealing fundamental limitations of twisting and surgery techniques.

Findings

Existence of 4-manifolds not generated by twisting small submanifolds.

No universal submanifold can produce all smooth structures via twisting.

Similar non-generability results hold for surgeries.

Abstract

It is well known that for any exotic pair of simply connected closed oriented 4-manifolds, one is obtained from the other by twisting a compact contractible submanifold via an involution on the boundary. By contrast, here we show that for each positive integer $n$, there exists a simply connected closed oriented 4-manifold $X$ such that, for any compact (not necessarily connected) codimension zero submanifold $W$ with $b_1(\partial W)<n$, the set of all smooth structures on $X$ cannot be generated from $X$ by twisting $W$ and varying the gluing map. As a corollary, we show that there exists no `universal' compact 4-manifold $W$ such that, for any simply connected closed 4-manifold $X$, the set of all smooth structures on $X$ is generated from a 4-manifold by twisting a fixed embedded copy of $W$ and varying the gluing map. Moreover, we give similar results for surgeries.