Simple crystallizations of 4-manifolds

Biplab Basak 1  and Jonathan Spreer 2
  • 1 Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
  • 2 School of Mathematics and Physics, The University of Queensland, Brisbane QLD 4072, Australia


Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination with known results this yields minimal crystallizations of all simply connected PL 4-manifolds of “standard” type, that is, all connected sums of ℂℙ2, S2 × S2, and the K3 surface. In particular, we obtain minimal crystallizations of a pair of homeomorphic but non-PL-homeomorphic 4-manifolds. In addition, we give an elementary proof that the minimal 8-vertex crystallization of ℂℙ2 is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of ℂℙ2 by the minimum of four edge contractions.

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Advances in Geometry is a mathematical journal which publishes original research articles of excellent quality in the area of geometry. Geometry is a field of long-standing tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity, and geometric ideas and the geometric language permeate all of mathematics.