2,405 research outputs found
The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes
We compute the complete Fadell-Husseini index of the 8 element dihedral group
D_8 acting on S^d \times S^d, both for F_2 and for integer coefficients. This
establishes the complete goup cohomology lower bounds for the two hyperplane
case of Gr"unbaum's 1960 mass partition problem: For which d and j can any j
arbitrary measures be cut into four equal parts each by two suitably-chosen
hyperplanes in R^d? In both cases, we find that the ideal bounds are not
stronger than previously established bounds based on one of the maximal abelian
subgroups of D_8.Comment: new version revised according to referee's comments, 44 pages, many
diagrams; a shorter version of this will appear in Topology and its
Applications (ATA 2010 proceedings
Tverberg plus constraints
Many of the strengthenings and extensions of the topological Tverberg theorem
can be derived with surprising ease directly from the original theorem: For
this we introduce a proof technique that combines a concept of "Tverberg
unavoidable subcomplexes" with the observation that Tverberg points that
equalize the distance from such a subcomplex can be obtained from maps to an
extended target space.
Thus we obtain simple proofs for many variants of the topological Tverberg
theorem, such as the colored Tverberg theorem of Zivaljevic and Vrecica (1992).
We also get a new strengthened version of the generalized van Kampen-Flores
theorem by Sarkaria (1991) and Volovikov (1996), an affine version of their
"j-wise disjoint" Tverberg theorem, and a topological version of Soberon's
(2013) result on Tverberg points with equal barycentric coordinates.Comment: 15 pages; revised version, accepted for publication in Bulletin
London Math. Societ
Some more amplituhedra are contractible
The amplituhedra arise as images of the totally nonnegative Grassmannians by
projections that are induced by linear maps. They were introduced in Physics by
Arkani-Hamed \& Trnka (Journal of High Energy Physics, 2014) as model spaces
that should provide a better understanding of the scattering amplitudes of
quantum field theories. The topology of the amplituhedra has been known only in
a few special cases, where they turned out to be homeomorphic to balls. The
amplituhedra are special cases of Grassmann polytopes introduced by Lam
(Current Developments in Mathematics 2014, Int.\ Press). In this paper we show
that that some further amplituhedra are homeomorphic to balls, and that some
more Grassmann polytopes and amplituhedra are contractible.Comment: 7 pages, to appear in Selecta Mathematic
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