Combinatorial and algebraic properties of balanced simplicial complexes

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Title: Combinatorial and algebraic properties of balanced simplicial complexes
Authors: Venturello, Lorenzo
Thesis advisor: Prof. Dr. Martina Juhnke-Kubitzke
Thesis referee: Prof. Dr. Julian Pfeifle
Abstract: Simplicial complexes are mathematical objects whose importance stretches from topology to commutative algebra and combinatorics. In this thesis we focus on the family of balanced simplicial complexes. A d-dimensional simplicial complex is balanced if its 1-skeleton can be properly (d+1)-colored, as in the classical graph theoretic sense. Equivalently, a d-dimensional complex is balanced iff it admits a non-degenerate simplicial projection to the d-simplex. We present results on these complexes from a number of different points of view. After two introductory chapters, we exhibit in chapter 3 an infinite family of balanced counterexamples to Stanley's partitionability conjecture. These complexes, which are in addition constructible, answer a question of Duval et al. in the negative. Next we shift to combinatorial topology, and study cross-flips, i.e., local moves on balanced manifolds introduced by Izmestiev, Klee and Novik, which preserve both the coloring and the topological type. In chapter 4 we provide an explicit description and enumeration of an interesting subset of these moves and use it to prove a Pachner-type theorem. Indeed, we show that any two balanced combinatorial manifolds with boundary which are PL-homeomorphic can be transformed one into the other by a sequence of shellings and inverse shellings which preserve both the coloring and the topological type at each step. This solves a problem proposed by Izmestiev, Klee and Novik. Chapter 5 is devoted to the study of certain algebraic invariants of simplicial complexes in the balanced case. Here upper bounds for the graded Betti numbers of the Stanley-Reisner ring of balanced simplicial complexes are investigated in several level of generalities, and we show that they are sharper than in the general case. First, we employ Hochster formula to obtain inequalities for the case of arbitrary balanced complexes. Next, we focus on the balanced Cohen-Macaulay case and we obtain two upper bounds via two different strategies. Using similar ideas we also bound the Betti numbers in the linear strand of balanced normal d-pseudomanifolds, for d>2. Finally, we explicitly compute graded Betti numbers of the class of stacked cross-polytopal spheres, and conjecture that they provide a sharp upper bound for those of all balanced pseudomanifolds with the same dimension and number of vertices. In the last chapter, we implement cross-flips on balanced surfaces and 3-manifolds, and use this computer program to search for balanced manifolds on few vertices, possibly vertex-minimal. Reducing the barycentric subdivision of vertex minimal triangulations, we find a long list of balanced triangulations of interesting spaces on few vertices. Among those stand out a balanced vertex-minimal triangulation of the dunce hat (11-vertices) and of the 2- and 3-dimensional real projective space (9 and 16 vertices respectively). Using obstructions from knot theory and a careful choice of flips we find a balanced non-shellable 3-sphere and a balanced shellable non-vertex-decomposable 3-sphere on 28 and 22 vertices respectively. These are the smallest instances known in the literature.
Subject Keywords: Balanced simplicial complexes; Combinatorial manifolds; Graded Betti numbers
Issue Date: 19-Nov-2019
License name: Attribution 3.0 Germany
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Type of publication: Dissertation oder Habilitation [doctoralThesis]
Appears in Collections:FB06 - E-Dissertationen

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