## Algebraic & Geometric Topology

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

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

Marco Golla and Bruno Martelli

#### Abstract

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in ${\u2102\mathbb{P}}^{n+1}$ 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 ${\u2102\mathbb{P}}^{n}$.

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.

#### 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

**Subjects**

Primary: 57M99: None of the above, but in this section 57N13: Topology of $E^4$ , $4$-manifolds [See also 14Jxx, 32Jxx]

**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