Pair of pants decomposition of 4–manifolds

Golla, Marco; Martelli, Bruno

doi:10.2140/agt.2017.17.1407

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Pair of pants decomposition of $4$–manifolds

Marco Golla and Bruno Martelli

Algebraic & Geometric Topology 17 (2017) 1407–1444

DOI: 10.2140/agt.2017.17.1407

Abstract

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in ${ℂ ℙ}^{n + 1}$ decomposes into pairs of pants: a pair of pants is a real compact $2 n$ –manifold with cornered boundary obtained by removing an open regular neighborhood of $n + 2$ generic complex hyperplanes from ${ℂ ℙ}^{n}$ .

As is well-known, every compact surface of genus $g \geq 2$ 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

References

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

Volume 17, issue 3 (2017)