Colorful versions of the Lebesgue, KKM, and Hex theorem

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• 作者：Đorđe Baralić p>djbaralic@mi.sanu.ac.rs" class="auth_mail" title="E-mail the corresponding authorp> ; Rade Živaljević p>rade@mi.sanu.ac.rs" class="auth_mail" title="E-mail the corresponding authorp>
• 关键词：Lebesgue theorem ; KKM theorem ; Simple polytopes ; Quasitoric manifolds ; Cup-length
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：146
• 期：Complete
• 页码：295-311
• 全文大小：464 K

文摘

Following and developing ideas of R. Karasev (2014) pan id="bbr0100">[10]pan>, we extend the Lebesgue theorem (on covers of cubes) and the Knaster–Kuratowski–Mazurkiewicz theorem (on covers of simplices) to different classes of convex polytopes (colored in the sense of M. Joswig). We also show that the n-dimensional Hex theorem admits a generalization where the n-dimensional cube is replaced by a n-colorable simple polytope. The use of specially designed quasitoric manifolds, with easily computable cohomology rings and the cohomological cup-length, offers a great flexibility and versatility in applying the general method.