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Sperner's Lemma, the Brouwer Fixed-Point Theorem, and Cohomology
| Content Provider | Semantic Scholar |
|---|---|
| Author | Ivanov, Nikolai V. |
| Copyright Year | 2009 |
| Abstract | The proof of the Brouwer fixed-point Theorem based on Sperner's Lemma is often presented as an elementary combinatorial alternative to advanced proofs based on algebraic topology. The goal of this note is to show that: (i) the combinatorial proof of Sperner's Lemma can be considered as a cochain-level version, written in the combinatorial language, of a standard cohomological argument; (ii) the standard deduction of the Brouwer Theorem from Sperner's Lemma is similar to the usual deduction of the Brouwer theorem from the No-Retraction Theorem and is closely related to the notion of a simplicial approximation. In order to make these connections transparent, we included the above mentioned standard arguments, so the note is self-contained modulo the basic notions of (simplicial) cohomology theory. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/0906.5193v1.pdf |
| Alternate Webpage(s) | http://arxiv.org/pdf/0906.5193v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
