Topology of partition of measures by fans and the second obstruction

TL;DR

This paper explores the topology of measure partitions by fans, introducing a new scheme to prove the existence of certain partitions using equivariant obstruction theory, thus advancing understanding in combinatorial geometry.

Contribution

It introduces the target extension scheme, enabling the use of equivariant obstruction theory to establish positive partition results that previous methods could not achieve.

Findings

Existence of -partitions for any two measures on the sphere

Introduction of the target extension scheme for equivariant topology

Positive results in measure partition problems using obstruction theory

Abstract

\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.