New Applications of Topological Methods in Discrete Geometry
Albert Haase · Universitätsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver · 2017
In this dissertation we present new applications of topological methods to problems in discrete geometry. The topological proof strategy we use goes back to Lovász' 1978 proof of the Kneser conjecture: reduce the question of whether a geometric statement is true to the nonexistence of an equivariant map between a test space and a configuration space, where both spaces have the same non-trivial group acting on them. The introduction (Chapter 1) is followed by two chapters in which the topological proof strategy is applied to the Grünbaum--Hadwiger--Ramos hyperplane mass partition problem, which is due to Grünbaum, Hadwiger, and Ramos: Given positive integers j,k the problem asks for the smallest dimension d such that any choice of j convex bodies in R^d or, more generally, any choice of j absolutely continuous finite Borel measures on R^d can be cut into 2^k equal pieces by k hyperplanes. In Chapter 2 we give a critical review of the progress that has been made on the Grünbaum --Hadwiger--Ramos hyperplane mass partition problem and point out mistakes and gaps in the recent articles. This shows that the problem is still wide open. The main new result of Chapter 2 is a correct solution of the problem in the case of two hyperplanes and 2^t+1 measures. It is obtained by a degree calculation of a restriction of the test map. In Chapter 3 we use a different approach based on relative equivariant obstruction theory to verify the solutions of the problem in the cases of two hyperplanes and 2^t-1 respectively 2^t +1 measures and obtain a correct solution of the problem in the case of two hyperplanes and 2^t measures. We also obtain solutions in the cases of three hyperlanes and two respectively four measures. In Chapter 4 we study the problem, to what extent the well-known variant of the topological proof strategy based on the connectivity of the configuration space and a theorem by Dold can be used to answer the question, when a matroid (viewed as a simplicial complex) can be mapped to R^d such that the images of no k pairwise disjoint faces intersect. An answer to this question would give rise to a Tverberg-type theorem for matroids. Our main result is a counterexample to a conjecture by Bárány, Kalai, and Meshulam concerning the connectivity of one of the two possible configuration spaces. Furthermore, we establish the connectivity of the other possible configuration space. Finally, we prove a tight Tverberg-type theorem for the family of matroids arising as counterexamples. Together, our results imply that the topological proof strategy based on the connectivity of the configuration space and Dold's thoerem does not lead to an optimal Tverberg-type result in the case of matroids.