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Computational topology of equivariant maps from spheres to complements of arrangements

Authors: Pavle V. M. Blagojevic, Sinisa T. Vrecica and Rade T. Zivaljevic
Journal: Trans. Amer. Math. Soc. 361 (2009), 1007-1038
MSC (2000): Primary 52A37, 55S35; Secondary 55M35
Published electronically: August 19, 2008
MathSciNet review: 2452832
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Abstract: The problem of the existence of an equivariant map is a classical topological problem ubiquitous in topology and its applications. Many problems in discrete geometry and combinatorics have been reduced to such a question and many of them resolved by the use of equivariant obstruction theory. A variety of concrete techniques for evaluating equivariant obstruction classes are introduced, discussed and illustrated by explicit calculations. The emphasis is on $ D_{2n}$-equivariant maps from spheres to complements of arrangements, motivated by the problem of finding a $ 4$-fan partition of $ 2$-spherical measures, where $ D_{2n}$ is the dihedral group. One of the technical highlights is the determination of the $ D_{2n}$-module structure of the homology of the complement of the appropriate subspace arrangement, based on the geometric interpretation for the generators of the homology groups of arrangements.

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Additional Information

Pavle V. M. Blagojevic
Affiliation: Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia

Sinisa T. Vrecica
Affiliation: Mathematical Faculty, University of Belgrade, Belgrade, Serbia

Rade T. Zivaljevic
Affiliation: Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade, Serbia

Keywords: Partition of measures, $k$-fans, equivariant obstruction theory
Received by editor(s): June 10, 2005
Received by editor(s) in revised form: April 3, 2006, and May 7, 2007
Published electronically: August 19, 2008
Additional Notes: This research was supported by grants 144018 and 144026 of the Serbian Ministry of Science, Technology and Ecology.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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