## Algebraic & Geometric Topology

### Pair of pants decomposition of $4$–manifolds

#### Abstract

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 $g ≥ 2$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 17, Number 3 (2017), 1407-1444.

Dates
Received: 30 June 2015
Revised: 18 May 2016
Accepted: 11 July 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841402

Digital Object Identifier
doi:10.2140/agt.2017.17.1407

Mathematical Reviews number (MathSciNet)
MR3677932

Zentralblatt MATH identifier
1376.57020

Keywords
4-manifolds pair of pants

#### Citation

Golla, Marco; Martelli, Bruno. Pair of pants decomposition of $4$–manifolds. Algebr. Geom. Topol. 17 (2017), no. 3, 1407--1444. doi:10.2140/agt.2017.17.1407. https://projecteuclid.org/euclid.agt/1510841402

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