Topology of partition of measures by fans and the second obstruction

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 [1], [2], [17], [11] and forced some hard concrete combinatorial calculations in the equivariant cohomology [3]. These problems can be reduced, by a beautiful scheme of [1], to a ”typical” question of the existence of D2n equivariant map f : V2(R ) → Wn − ∪A(α), where V2(R ) is the space of all orthonormal 2-frames in R and Wn − ∪A(α) is the complement of the appropriate arrangement. The existence of D2n equivariant maps and the failure of equivariant methods in cases of 4-fans, 2-measures [3] and 2-fans, 3-measures [17] focus our attention to the 3-fan, 2-measures partition problem. With the help of the equivariant obstruction theory, even though the first obstruction to the existence of D2n equivariant map f : V2(R ) → Wn − ∪A(α) is zero [17], we prove that for α = (a, a + b, b), the second obstruction is not zero and so that there exists an 3-fan partition of every two spherical measures on S. To prove the central result, that the family of all second obstructions that live in H D2n(S , π2(Wn − ∪A(α))) are different from zero for all choices of 2skeleton extensions, we introduce the target extension scheme.