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Pair of pants decomposition of $4$–manifolds

Marco Golla and Bruno Martelli

Algebraic & Geometric Topology 17 (2017) 1407–1444

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in 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 n.

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.

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