Experimental Mathematics

Combinatorial Properties of the $K^3$ Surface: Simplicial Blowups and Slicings

Jonathan Spreer and Wolfgang Kühnel

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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.

Article information

Experiment. Math., Volume 20, Issue 2 (2011), 201-216.

First available in Project Euclid: 6 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57Q15: Triangulating manifolds
Secondary: 14J28: $K3$ surfaces and Enriques surfaces 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 57Q25: Comparison of PL-structures: classification, Hauptvermutung 52B70: Polyhedral manifolds

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


Spreer, Jonathan; Kühnel, Wolfgang. Combinatorial Properties of the $K^3$ Surface: Simplicial Blowups and Slicings. Experiment. Math. 20 (2011), no. 2, 201--216. https://projecteuclid.org/euclid.em/1317924411

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