#### Volume 17, issue 3 (2017)

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Pair of pants decomposition of $4$–manifolds

### Marco Golla and Bruno Martelli

Algebraic & Geometric Topology 17 (2017) 1407–1444
##### Abstract

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

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.

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##### Keywords
4-manifolds, pair of pants
##### Mathematical Subject Classification 2010
Primary: 57M99, 57N13
##### Publication
Received: 30 June 2015
Revised: 18 May 2016
Accepted: 11 July 2016
Published: 17 July 2017
##### Authors
 Marco Golla Department of Mathematics Uppsala University Box 480 SE-751 06 Uppsala Sweden Bruno Martelli Mathematics Department “Tonelli” Università di Pisa Largo Pontecorvo 5 I-56127 Pisa Italy