Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in CPn+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 CPn. As is well-known, every compact surface of genus g â\u89¥ 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.
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