## Experimental Mathematics

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

#### Abstract

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

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

Dates
First available in Project Euclid: 6 October 2011

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