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 $\mathbb{CP}^2$, $S^2 \times S^2$, 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 $\mathbb{CP}^2$ is unique and its associated pseudotriangulation is related to the 9-vertex combinatorial triangulation of $\mathbb{CP}^2$ by the minimum of four edge contractions.Comment: 23 pages, 7 figures. Minor update, replacement of Figure 7. To appear in Advances in Geometr

Optimal Morse matchings reveal essential structures of cell complexes that lead to powerful tools to study discrete geometrical objects, in particular, discrete 3-manifolds. However, such matchings are known to be NP-hard to compute on 3-manifolds through a reduction to the erasability problem. Here, we refine the study of the complexity of problems related to discrete Morse theory in terms of parameterized complexity. On the one hand, we prove that the erasability problem is W [ P ]-complete on the natural parameter. On the other hand, we propose an algorithm for computing optimal Morse matchings on triangulations of 3-manifolds, which is fixed-parameter tractable in the treewidth of the bipartite graph representing the adjacency of the 1- and 2-simplices. This algorithm also shows fixed-parameter tractability for problems such as erasability and maximum alternating cycle-free matching. We further show that these results are also true when the treewidth of the dual graph of the triangulated 3-manifold is bounded. Finally, we discuss the topological significance of the chosen parameters and investigate the respective treewidths of simplicial and generalized triangulations of 3-manifolds.

There are many fundamental algorithmic problems on triangulated 3-manifolds whose complexities are unknown. Here we study the problem of finding a taut angle structure on a 3-manifold triangulation, whose existence has implications for both the geometry and combinatorics of the triangulation. We prove that detecting taut angle structures is NP-complete, but also fixed-parameter tractable in the treewidth of the face pairing graph of the triangulation. These results have deeper implications: the core techniques can serve as a launching point for approaching key decision problems such as unknot recognition and prime decomposition of 3-manifolds.

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

In this sequel to earlier papers by three of the authors, we obtain a new bound on the complexity of a closed 3-manifold, as well as a characterisation of manifolds realising our complexity bounds. As an application, we obtain the first infinite families of minimal triangulations of Seifert fibred spaces modelled on Thurston's geometry SL 2 (R). IntroductionGiven a closed, irreducible 3-manifold, its complexity is the minimum number of tetrahedra in a (pseudo-simplicial) triangulation of the manifold. This number agrees with the complexity defined by Matveev [12] unless the manifold is S 3 , RP 3 or L(3, 1). We denote the complexity of M by c(M). The only known infinite families of closed 3-manifolds for which an exact value of the complexity is known are spherical space forms [7,8,9].Asymptotic bounds for an infinite family of double branched coverings of certain alternating closed 3-braids are given by Ni and Wu [14] using the results of [9]. New lower bounds on the complexity of closed 3-manifolds were recently given with different methods by Cha [2, 3].In this paper, we continue to be interested in determining the exact complexity and minimal triangulations of infinite classes of closed 3-manifolds. In order to describe our results, we need the following definition.Let M be a closed, orientable, irreducible, connected 3-manifold, and let S be an embedded closed surface dual to a given ϕ ∈ H 1 (M; Z 2 ). An analogue of Thurston's norm [16] is defined in [9] as follows. If S is connected, let χ − (S) = max{0, −χ(S)}, and otherwise letwhere the sum is taken over all connected components of S. Note that S i is not necessarily orientable. Define:|| ϕ || = min{χ − (S) | S dual to ϕ}.The surface S dual to ϕ ∈ H 1 (M; Z 2 ) is said to be Z 2 -taut if no component of S is a sphere and χ(S) = −|| ϕ ||. As in [16], one observes that (after possibly deleting compressible tori) every component of a Z 2 -taut surface is non-separating and geometrically incompressible. This follows from the fact that the Klein bottle does not embed in RP 3 .

A triangulation of a closed 2-manifold is tight with respect to a field of characteristic two if and only if it is neighbourly; and it is tight with respect to a field of odd characteristic if and only if it is neighbourly and orientable. No such characterization of tightness was previously known for higher dimensional manifolds. In this paper, we prove that a triangulation of a closed 3-manifold is tight with respect to a field of odd characteristic if and only if it is neighbourly, orientable and stacked. In consequence, the Kühnel-Lutz conjecture is valid in dimension three for fields of odd characteristic.Next let F be a field of characteristic two. It is known that, in this case, any neighbourly and stacked triangulation of a closed 3-manifold is F-tight. For closed, triangulated 3-manifolds with at most 71 vertices or with first Betti number at most 188, we show that the converse is true. But the possibility of the existence of an F-tight, non-stacked triangulation on a larger number of vertices remains open. We prove the following upper bound theorem on such triangulations. If an F-tight triangulation of a closed 3-manifold has n vertices and first Betti number β 1 , then (n − 4)(617n − 3861) ≤ 15444β 1 . Equality holds here if and only if all the vertex links of the triangulation are connected sums of boundary complexes of icosahedra.MSC 2010 : 57Q15, 57R05.

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. A focus is given to dimension 3 where slicings are normal surfaces. In the case of 2-neighborly 3-manifolds and quadrangulated slicings, a lower bound on the number of quadrilaterals of normal surfaces depending on the genus g is presented. It is shown to be sharp for infinitely many values of g. Furthermore we classify slicings of combinatorial 3-manifolds with a maximum number of edges in the slicing.Comment: 18 pages, 9 figure

Tight triangulations are exotic, but highly regular objects in combinatorial topology. A triangulation is tight if all its piecewise linear embeddings into a Euclidean space are as convex as allowed by the topology of the underlying manifold. Tight triangulations are conjectured to be strongly minimal, and proven to be so for dimensions ≤ 3. However, in spite of substantial theoretical results about such triangulations, there are precious few examples. In fact, apart from dimension two, we do not know if there are infinitely many of them in any given dimension.In this paper, we present a computer-friendly combinatorial scheme to obtain tight triangulations, and present new examples in dimensions three, four and five. Furthermore, we describe a family of tight triangulated d-manifolds, with 2 d−1 ⌊d/2⌋!⌊(d−1)/2⌋! isomorphically distinct members for each dimension d ≥ 2. While we still do not know if there are infinitely many tight triangulations in a fixed dimension d > 2, this result shows that there are abundantly many.MSC 2010 : 57Q15, 57R05.