Open Access
2011 Combinatorial Properties of the $K^3$ Surface: Simplicial Blowups and Slicings
Jonathan Spreer, Wolfgang Kühnel
Experiment. Math. 20(2): 201-216 (2011).


The 4-dimensional abstract Kummer variety $K^4$ with 16 nodes leads to the $K^3$ surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with a minimal 16-vertex triangulation of $K^4$, we resolve its 16 isolated singularities—step by step—by simplicial blowups. As a result we obtain a 17-vertex triangulation of the standard PL $K^3$ surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from real projective 3-space onto the 2-sphere with the minimum number of vertices. Moreover, we study simplicial Morse functions and the changes of their levels between the critical points. In this way we obtain slicings through the $K^3$ surface of various topological types.


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Jonathan Spreer. Wolfgang Kühnel. "Combinatorial Properties of the $K^3$ Surface: Simplicial Blowups and Slicings." Experiment. Math. 20 (2) 201 - 216, 2011.


Published: 2011
First available in Project Euclid: 6 October 2011

zbMATH: 1279.57018
MathSciNet: MR2821391

Primary: 57Q15
Secondary: 14E15 , 14J28 , 52B70 , 57Q25

Keywords: $K^3$ surface , Combinatorial manifold , combinatorial pseudomanifold , intersection form , Kummer variety , resolution of singularities , simplicial Hopf map

Rights: Copyright © 2011 A K Peters, Ltd.

Vol.20 • No. 2 • 2011
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