Combinatorial Properties of theK3 Surface: Simplicial Blowups and Slicings

Abstract

The 4-dimensional abstract Kummer variety K 4 with 16 nodes leads to the K3 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 K3 surface. A key step is the construction of a triangulated version of the mapping cylinder of the Hopf map from the 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 K3 surface of various topological types.Problem 2. Find two concrete combinatorial triangulations of a 4-manifold such that the underlying PL manifolds are homeomorphic but not PL homeomorphic. It is known that some compact topological 4-manifolds admit exotic PL structures. Furthermore any combinatorial triangulation induces a unique PL structure and thus a unique smooth structure.Concerning Problem 1 there are pairs of non-orientable and orientable surfaces with the same minimum number of vertices, e.g., the two surfaces with χ = −10 admit triangulations with the f -vector (12, 66, 44) but no smaller triangulations. Moreover, the existence of pairs of non-homeomorphic lens

Heuristics for Sphere Recognition

“…To ensure the discretization has been carried out correctly, we want to confirm the manifold property. In practice, this test effectively detects the majority of construction errors [1,33,30]. 3.…”

Simple crystallizations of 4-manifolds

“…Now, the procedure to construct simple contracted pseudotriangulations essentially performs bistellar moves and edge contractions at random where moves reducing the complexity of the triangulation (3-moves, 4-moves and edge contractions) are performed with a higher probability. Using this strategy we were able to obtain 40 651 simple contracted pseudotriangulations from the combinatorial manifold PL-homeomorphic to the K3 surface a d due to Kühnel and the second author [50] and 19 129 simple contracted pseudotriangulations PL-homeomorphic to the minimum 16-vertex combinatorial manifold homeomorphic to the K3 surface due to Casella and Kühnel [16]. We believe that the number of simple contracted pseudotriangulations of the K3 surface is orders of magnitude larger than the numbers provided above.…”

“…We believe that the number of simple contracted pseudotriangulations of the K3 surface is orders of magnitude larger than the numbers provided above. Note that both versions of the K3 surface are conjectured to be PL-homeomorphic [50]. This conjecture could be settled by finding a simple crystallization which occurs in both the list of simple contracted pseudotriangulations.…”

Cataloguing PL 4-Manifolds by Gem-Complexity

“…Moreover, we point out that the program performing automatic recognition of PL-homeomorphic 4-manifolds may be a useful tool to approach open problems related to different triangulations of the same TOP 4-manifold, which are conjectured to represent the same PL 4-manifold, too. The first candidates are the two known 16-vertices and 17-vertices triangulations of the K3-surface: see [27] and [45], together with the attempts to settle the conjecture described in [10], [11] and [9].…”

“…In particular, an application of Γ4-class to the case of the 16-vertices (resp. 17-vertices) triangulation of the K3-surface (obtained in [27] and [45] respectively) is in progress. The idea is similar to the one described in [9], [10] and [11], but the elementary moves involved in the automatic procedures are different (blob and flips, together with dipole eliminations and ρ-pair switching, instead of edge-contraction and bistellar moves).…”