A cut-and-paste mechanism to introduce fundamental group and construct new four-manifolds

TL;DR

This paper presents a straightforward cut-and-paste technique for constructing four-manifolds that modifies the fundamental group while maintaining key topological invariants, aiding in discovering exotic smooth structures.

Contribution

It introduces a new, simple cut-and-paste method to alter fundamental groups and construct four-manifolds, streamlining the search for exotic smooth structures.

Findings

Constructed new four-manifolds with altered fundamental groups

Unveiled exotic irreducible smooth structures on closed four-manifolds

Facilitated the search for fixed-point free involutions

Abstract

We introduce a simple cut-and-paste mechanism to construct both orientable and nonorientable four-manifolds from a given initial one. This mechanism alters the fundamental group while preserving other essential topological invariants. It avoids codimension two cut-and-paste fundamental group computations and fast tracks the search for fixed-point free involutions. The mechanism proves useful to unveil novel exotic irreducible smooth structures on closed four-manifolds with finite cyclic fundamental group, which include $\Q$-homology real projective four-spaces.