Equivariant topology methods in discrete geometry
Benjamin Matschke · Universitätsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver · 2011
The first chapter of this thesis is on the colored Tverberg problem, which is joint work with Pavle Blagojević and Günter Ziegler [BMZ09], [BMZ11a], [BMZ11b]. First we present a new and tight colored version of Tverberg's theorem that implies the Bárány-Larman conjecture for primes minus one and asymptotically in general. This in turn improves the bounds in the second selection lemma, which is used in computational complexity for example to bound the number of halving sets of an n-set in R^d . Then we generalize our theorem to a transversal version, a colored version of the Tverberg-Vrećica conjecture, which is a unifying theorem in the sense that it implies the ham sandwich theorem and the center transversal theorem. Finally we generalize our theorem to maps into manifolds. Two results of independent interest are a new parameterized Borsuk-Ulam-type theorem for equivariant vector bundles and the calculation of the Fadell-Husseini index of joins of chessboard complexes. The second chapter is on inscribing squares and rectangles into closed curves in the plane. The results are disjoint from the ones in [Mat08, Chap. III], and they will appear in [Mat09] and [Mat11]. We present two new classes of Jordan curves that fulfill Toeplitz' still unsolved square peg problem from 1911, that is, these curves inscribe squares. One of them strictly contains all previously known classes; the other one is the first known open set of such curves. Then we disprove Cantarella's conjecture on the parity of inscribed squares for immersed plane curves and give the right answer, also for inscribed rectangles. We give another class of Jordan curves that inscribes rectangles of aspect ratio sqrt(3), which is the first known partial result for an aspect ratios other than 1. The appendix summarizes two papers on polytopes. The first one is joint work with Francisco Santos and Christophe Weibel [MSW11] on 5-spindles with large width, which are a building block for new counter-examples of the Hirsch conjecture. The second is joint work with Julian Pfeifle and Vincent Pilaud [MPP11] on productsimplicial-neighborly polytopes, where we construct polytopes that interpolate between being neighborly and cubically neighborly.