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

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
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.

##### 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