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 | Abstract: | 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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