2004/02/25 by Pavle V. M. Blagojević, Pavle V. M. Blagojevic, Blagojevic, Pavle V. M. · 1 citation
Computer Science · Mathematics · #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #math.AT #math.CO #msc:52C15 #msc:55S91
paper · pdf · doi:10.48550/arxiv.math/0402400
The exposition is radicaly improved. The proof is more general. The obstruction cocycle got the clear geometric interpretation
arxiv created 2004/07/27 · arxiv updated 2009/12/01
\noindent The simultaneous partition problems are classical problems of the combinatorial geometry which have the natural flavor of the equivariant topology. The k-fan partition problems have attracted a lot of attention \citeAki2000, \citeBaMa2001, \citeBaMa2002 and forced some hard concrete combinatorial calculations in the equivariant cohomology \cite% Bl-Vr-Ziv. These problems can be reduced, by a beautiful scheme of \cite% BaMa2001, to a \textquotedblright typical\textquotedblright question of the existence of a \mathbbD2n equivariant map f:V2(ℝ% 3)→ Wn-∪ A(α), where V2(ℝ% 3) is the space of all orthonormal 2-frames in ℝ3 and % Wn-∪ A(α) is the complement of the appropriate arrangement. We introduce the target extension scheme which allow us to use the equivariant obstruction theory as a tool for proving that: for every two proper measures on the sphere S2, and any α=(a,a+b,b)∈ ℝ>03, there exists an α-partition of theses measures by a 3-fan. \noindent The significance of these results, among other, is that, beside negative results \citeBl-Vr-Ziv, the equivariant obstruction theory can pull off some positive results, which were not attained by other means.