Pair of pants decomposition of 4-manifolds

TL;DR

This paper explores the decomposition of smooth closed 4-manifolds into pairs of pants, extending concepts from tropical geometry and complex hypersurfaces, and constructs examples including manifolds with arbitrary finitely presented groups as their fundamental groups.

Contribution

It introduces a method to decompose 4-manifolds into pairs of pants and demonstrates that any finitely presented group can be realized as the fundamental group of such a decomposed 4-manifold.

Findings

Every finitely presented group is the fundamental group of a decomposed 4-manifold.

Constructs numerous examples of 4-manifolds decomposing into pairs of pants.

Extends the concept of pair of pants decomposition from surfaces to 4-manifolds.

Abstract

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $\mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open regular neighborhood of $n+2$ generic hyperplanes from $\mathbb{CP}^n$. As is well-known, every compact surface of genus $g\geqslant 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension 4. Which smooth closed 4-manifolds decompose into pairs of pants? We address this problem here and construct many examples: we prove in particular that every finitely presented group is the fundamental group of a 4-manifold that decomposes into pairs of pants.