Mathematics – Combinatorics
Scientific paper
2004-02-25
Mathematics
Combinatorics
The exposition is radicaly improved. The proof is more general. The obstruction cocycle got the clear geometric interpretation
Scientific paper
\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 \cite{Aki2000}, \cite{BaMa2001}, \cite{BaMa2002} 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 $\mathbb{D}_{2n}$ equivariant map $f:V_{2}(\mathbb{R}% ^{3})\to W_{n}-\cup \mathcal{A}(\alpha)$, where $V_{2}(\mathbb{R}% ^{3})$ is the space of all orthonormal 2-frames in $\mathbb{R}^{3}$ and $% W_{n}-\cup \mathcal{A}(\alpha)$ is the complement of the appropriate arrangement. We introduce the \textit{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 $S^{2}$, and any $\alpha =(a,a+b,b)\in \mathbb{R}_{>0}^{3}$, there exists an $\alpha $-partition of theses measures by a 3-fan. \noindent The significance of these results, among other, is that, beside negative results \cite{Bl-Vr-Ziv}, the equivariant obstruction theory can pull off some positive results, which were not attained by other means.
No associations
LandOfFree
Topology of partition of measures by fans and the second obstruction does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Topology of partition of measures by fans and the second obstruction, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Topology of partition of measures by fans and the second obstruction will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-484941