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A combinatorial analog of a theorem of F.J. Dyson. (English) Zbl 1196.54073
F. J. Dyson’s theorem [Ann. Math. (2) 54, 534–536 (1951; Zbl 0045.02901)] states that for any real-valued function on $$S^2$$ there are two orthogonal diameters whose endpoints are mapped to the same point. The authors give a combinatorial proof of this theorem exploiting Tucker labellings: Let $$T$$ be a triangulation of $$S^2$$ which is invariant under the antipodal map $$A$$. A Tucker labelling $$\ell$$ assigns the values $$\pm1$$ to the vertices of $$T$$ such that $$\ell(-v)=\ell(v)$$. The authors show that for the linear extension $$L$$ of $$\ell$$ there exists a polygonal simple closed path that is invariant under $$A$$ and is mapped to zero under $$L$$. They then show that this result is “equivalent” to Dyson’s theorem.

##### MSC:
 54H25 Fixed-point and coincidence theorems (topological aspects)
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##### References:
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