Authors: Blagojević, Pavle 
Ziegler, Günter
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: The ideal-valued index for a dihedral group action, and mass partition by two hyperplanes
Journal: Topology and its Applications
Volume: 158
Issue: 12
First page: 1326
Last page: 1351
Issue Date: 1-Aug-2011
Rank: M23
ISSN: 0166-8641
DOI: 10.1016/j.topol.2011.05.008
We compute the complete Fadell-Husseini index of the dihedral group D8=(Z{double-struck}2)2⋊Z{double-struck}2 acting on Sd×Sd for F{double-struck}2 and for Z{double-struck} coefficients, that is, the kernels of the maps in equivariant cohomology. HD8*(pt,F{double-struck}2)→HD8*(Sd×Sd,F{double-struck}2) and. HD8*(pt,Z{double-struck})→HD8*(Sd×Sd,Z{double-struck}). This establishes the complete cohomological lower bounds, with F{double-struck}2 and with Z{double-struck} coefficients, for the two-hyperplane case of Grünbaum'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{double-struck}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 D8.
Keywords: Bockstein spectral sequence | Dihedral group | Equivariant cohomology | Fadell-Husseini index | Mass partitions | Serre spectral sequence
Publisher: Elsevier
Project: European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC, Grant agreement No. 247029-SDModels
Advanced Techniques of Cryptology, Image Processing and Computational Topology for Information Security 

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