Hyperplane mass partitions via relative equivariant obstruction theory

Albert Haase ; Pavle V.M. Blagojević ; Florian Frick ; Günter M. Ziegler

Documenta mathematica, Tome 21 (2016), pp. 735-771

Résumé

The Grünbaum–Hadwiger–Ramos hyperplane mass partition problem was introduced by Grünbaum (1960) in a special case and in general form by Ramos (1996). It asks for the «admissible» triples (d,j,k) such that for any j masses in Rd there are k hyperplanes that cut each of the masses into 2k equal parts. Ramos' conjecture is that the Avis–Ramos necessary lower bound condition dk≥j(2k−1) is also sufficient. We develop a «join scheme» for this problem, such that non-existence of an Sk±​-equivariant map between spheres (Sd)∗k→S(Wk​⊕Uk⊕j​) that extends a test map on the subspace of (Sd)∗k where the hyperoctahedral group Sk±​ acts non-freely, implies that (d,j,k) is admissible. For the sphere (Sd)∗k we obtain a very efficient regular cell decomposition, whose cells get a combinatorial interpretation with respect to measures on a modified moment curve. This allows us to apply relative equivariant obstruction theory successfully, even in the case when the difference of dimensions of the spheres (Sd)∗k and S(Wk​⊕Uk⊕j​) is greater than one. The evaluation of obstruction classes leads to counting problems for concatenated Gray codes. Thus we give a rigorous, unified treatment of the previously announced cases of the Grünbaum–Hadwiger–Ramos problem, as well as a number of new cases for Ramos' conjecture.

Détail

DOI : 10.4171/dm/544

Classification : 51N20, 52A35, 55N25, 55R20
Mots-clés : hyperplane mass partition problem, equi­variant topological combinatorics, equivariant obstruction theory