Pair of pants decomposition of $4$–manifolds

Abstract

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in ${ℂ\mathbb{P}}^{n+1}$ 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 complex hyperplanes from ${ℂ\mathbb{P}}^n$.

As is well-known, every compact surface of genus $g\ge 2$ 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.