Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in decomposes into pairs of pants: a pair of pants is a real compact –manifold with cornered boundary obtained by removing an open regular neighborhood of generic complex hyperplanes from .
As is well-known, every compact surface of genus decomposes into pairs of pants, and it is now natural to investigate this construction in dimension . Which smooth closed –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 –manifold that decomposes into pairs of pants.
"Pair of pants decomposition of $4$–manifolds." Algebr. Geom. Topol. 17 (3) 1407 - 1444, 2017. https://doi.org/10.2140/agt.2017.17.1407