Combinatorial properties of the K3 surface: Simplicial blowups and slicings

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.Comment: 31 pages, 3 figure

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The purpose of this note is to establish a connection between the notion of (n−2)-tightness in the sense of N.H. Kuiper and T.F. Banchoff and the total absolute curvature of compact submanifolds-with-boundary of even dimension in Euclidean space. The argument used is a certain geometric inequality similar to that of S.S. Chern and R.K. Lashof where equality characterizes (n−2)-tightness

Hopf triangulations of spheres and equilibrium triangulations of projective spaces

Following work by the first author and Banchoff, we investigate triangulations of real and complex projective spaces of real and complex dimension $k$ that are adapted to the decomposition into "zones of influence" around the points $[1,0,\ldots,0],$ $\ldots,$ $[0,\ldots,0,1]$ in homogeneous coordinates. The boundary of such a "zone of influence" must admit a simplicial version of the Hopf decomposition of a sphere into "solid tori" of various dimensions. We present such {\em Hopf triangulations} of $S^{2k-1}$ for $k \leq 4$, and give candidate triangulations for arbitrary $k$. In the complex case, a crucial role of this construction is the central $k$-torus as the intersection of all "zones of influence". Candidate triangulations of the $k$-torus with $2^{k+1}-1$, $k\geq 1$, vertices -- possibly the minimum numbers -- are well known. They admit an involution acting like complex conjugation and an automorphism of order $k+1$ realising the cyclic shift of coordinate directions in $\mathbb{C}P^k$. For $k=2$, this can be extended to what we call a {\em perfect equilibrium triangulation} of $\mathbb{C}P^2$, previously described in the literature. We prove that this is no longer possible for $k=3$, and no perfect equilibrium triangulation of $\mathbb{C}P^3$ exists. In the real case, the central torus is replaced by its fixed-point set under complex conjugation: the vertices of a $k$-dimensional cube. We revisit known equilibrium triangulations of $\mathbb{R}P^k$ for $k\leq 2$, and describe new equilibrium triangulations of $\mathbb{R}P^3$ and $\mathbb{R}P^4$. Finally, we discuss the most symmetric and vertex-minimal triangulation of $\mathbb{R}P^4$ and present a tight polyhedral embedding of $\mathbb{R}P^3$ into 6-space. No such embedding was known before.Comment: 31 pages, 8 figures, 5 pages of appendi

Liouville's theorem in conformal geometry

AbstractLiouville's theorem states that all conformal transformations of En and Sn (n⩾3) are restrictions of Möbius transformations. As a generalization, we determine all conformal mappings of semi-Riemannian manifolds preserving pointwise the Ricci tensor. It turns out that, up to isometries, they are essentially of the same type as in the classical case but they can exist for metrics different from the Euclidean metric and spherical metric

simpcomp -- A GAP toolbox for simplicial complexes

simpcomp is an extension (a so called package) to GAP, the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abstract) simplicial complexes, provides functions to construct new complexes from existing ones and an extensive library of triangulations of manifolds.Comment: 4 page

PL Morse theory in low dimensions

We discuss a PL analogue of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level, it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular we show that in low dimensions $d \leq 4$ a homologically regular point on a PL $d$-manifold is always strongly regular. Examples show that this fails to hold in higher dimensions $d \geq 5$. One of our constructions involves an 8-vertex embedding of the dunce hat into a polytopal 4-sphere with 8 vertices such that a regular neighborhood is Mazur's contractible 4-manifold.Comment: 24 pages, 3 figure

A generic impact-scoring system applied to alien mammals in Europe

We present a generic scoring system that compares the impact of alien species among members of large taxonomic groups. This scoring can be used to identify the most harmful alien species so that conservation measures to ameliorate their negative effects can be prioritized. For all alien mammals in Europe, we assessed impact reports as completely as possible. Impact was classified as either environmental or economic. We subdivided each of these categories into five subcategories (environmental: impact through competition, predation, hybridization, transmission of disease, and herbivory; economic: impact on agriculture, livestock, forestry, human health, and infrastructure). We assigned all impact reports to one of these 10 categories. All categories had impact scores that ranged from zero (minimal) to five (maximal possible impact at a location). We summed all impact scores for a species to calculate "potential impact" scores. We obtained "actual impact" scores by multiplying potential impact scores by the percentage of area occupied by the respective species in Europe. Finally, we correlated species' ecological traits with the derived impact scores. Alien mammals from the orders Rodentia, Artiodactyla, and Carnivora caused the highest impact. In particular, the brown rat (Rattus norvegicus), muskrat (Ondathra zibethicus), and sika deer (Cervus nippon) had the highest overall scores. Species with a high potential environmental impact also had a strong potential economic impact. Potential impact also correlated with the distribution of a species in Europe. Ecological flexibility (measured as number of different habitats a species occupies) was strongly related to impact. The scoring system was robust to uncertainty in knowledge of impact and could be adjusted with weight scores to account for specific value systems of particular stakeholder groups (e.g., agronomists or environmentalists). Finally, the scoring system is easily applicable and adaptable to other taxonomic groups

Scheinbar konfliktfrei aneinander vorbei. Eine Retrospektive auf die Generationsbeziehungen in den achtziger Jahren in der DDR

Die seit den siebziger Jahren immer offensichtlicher werdenden gesellschaftlichen Krisenprozesse in der DDR haben auch entscheidende Veränderungen in den Lebenszusammenhängen der Heranwachsenden und im Verhältnis zu vorangegangenen Generationen eingeleitet. Infolge der zunehmenden Durchstaatlichung ihrer Lebensvähältnisse wurde für die Jugendlichen immere mehr die Erfahrung sozialer Schließungen in ihren Bildungs- und Berufschancen, begleitet von einem Legitimationsverfall schulischer Lernangebote und politischer Partizipationsmöglichkeiten, bestimmend. Andererseits fand gegenläufig dazu ein Wandel in den soziokulturell und medial vermittelten Lernprozessen statt, der als »nischenhafte« Modernisierung des Alltagskulturellen aufgefaßt werden kann. Dieser Prozeß hat ebenso eine gewisse Erosion in den traditionellen Autoritätsverhältnissen und E,fahrungsbezügen zur Erwachsenengeneration begünstigt

Hamiltonian submanifolds of regular polytopes

We investigate polyhedral $2k$-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex {\it $k$-Hamiltonian} if it contains the full $k$-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called {\it super-neighborly triangulations}) we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the $d$-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of 7 copies of $S^2 \times S^2$. By this example all regular cases of $n$ vertices with $n < 20$ or, equivalently, all cases of regular $d$-polytopes with $d\leq 9$ are now decided.Comment: 26 pages, 4 figure