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Báo cáo toán học: "Tetrahedra on deformed spheres and integral group cohomology"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Tetrahedra on deformed spheres and integral group cohomology. | Tetrahedra on deformed spheres and integral group cohomology Pavle V. M. Blagojevic Mathematicki Institut Knez Michailova 35 1 11001 Beograd Serbia pavleb@mi.sanu.ac.rs Gunter M. Ziegler Inst. Mathematics MA 6-2 TU Berlin D-10623 Berlin Germany ziegler@math.tu-berlin.de Submitted Aug 28 2008 Accepted Jun 4 2009 Published Jun 10 2009 Mathematics Subject Classification 55S91 55M20 52C99 Dedicated to Anders Bjorner on the occasion of his 60th birthday Abstract We show that for every injective continuous map f S2 R3 there are four distinct points in the image of f such that the convex hull is a tetrahedron with the property that two opposite edges have the same length and the other four edges are also of equal length. This result represents a partial result for the topological Borsuk problem for R3. Our proof of the geometrical claim via Fadell-Husseini index theory provides an instance where arguments based on group cohomology with integer coefficients yield results that cannot be accessed using only field coefficients. 1 Introduction The motivation for the study of the existence of particular types of tetrahedra on deformed 2-spheres is twofold. The topological Borsuk problem as formulated by Soibelman in 1977 6 estimate the minimal Borsuk partition number for the unit ball in R for general metrics along with the square peg problem 5 first posed by Toeplitz 1911 does every Jordan curve contain the vertices of a square inspire the search for possible polytopes with nice metric properties whose vertices lie on the continuous images of spheres. Beyond their intrinsic interest these problems can be used as testing grounds for tools from equivariant topology e.g. for comparing the strength of Fadell-Husseini index theory with ring resp. field coefficients. Supported by the grant 144018 of the Serbian Ministry of Science and Technological development Supported by the German Research Foundation DFG THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R16 1 The following .