Authors: Baralić, Đorđe 
Blagojević, Pavle 
Karasev, Roman
Vučić, Aleksandar
Title: Index of Grassmann manifolds and orthogonal shadows
Journal: Forum Mathematicum
Volume: 30
Issue: 6
First page: 1539
Last page: 1572
Issue Date: 1-Nov-2018
Rank: M22
ISSN: 0933-7741
DOI: 10.1515/forum-2018-0058
In this paper, we study the Z/2 action on the real Grassmann manifolds Gn (R2n) and ∼Gn (R2n) given by taking the (appropriately oriented) orthogonal complement.We completely evaluate the relatedZ/2 Fadell-Husseini index utilizing a novel computation of the Stiefel-Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For n = 2a (2b + 1), k = 2a+1-1, a convex body C in R 2n, and k real-valued functions α1, . . . , αk continuous on convex bodies in R2n with respect to the Hausdorff metric, there exists a subspace V ⊆ R 2n such that projections of C to V and its orthogonal complement V have the same value with respect to each function αi, that is, αi(pV(C)) = αi(pV (C)) for all 1 ≤ i ≤ k.
Keywords: Cohomology of Grassmann manifolds | existence of equivariant maps | Fadell-Husseini ideal-valued index
Publisher: de Gruyter
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 
Advanced Techniques of Cryptology, Image Processing and Computational Topology for Information Security 
Mathematical Sciences Research Institute (MSRI) 
Federal professorship program, grant 1.456.2016/1.4
Russian Foundation for Basic Research, grant 18-01-00036

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