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A combinatorial Lefschetz fixed-point formula. (English) Zbl 0771.55008
The author’s summary: “Let \(K\) be any (finite) simplicial complex, and \(K'\) a subdivision of \(K\). Let \(\varphi: K'\to K\) be a simplicial map, and, for all \(j\geq 0\), let \(\varphi_ j\) denote the algebraical number of \(j\)-simplices \({\mathcal G}\) of \(K'\) such that \({\mathcal G}\subset\varphi({\mathcal G})\). From Hopf’s alternating trace formula it follows that \(\varphi_ 0-\varphi_ 1+\varphi_ 2-\dots=L(\varphi)\), the Lefschetz number of the simplicial map \(\varphi: X\to X\). Here \(X\) denotes the space of \(| K|\) (or \(| K'|)\). A purely combinatorial proof of the case \(K=a\) closed simplex (now \(L(\varphi)=1\)) is given, thus solving a problem posed by Ky Fan in 1978”.

55U10 Simplicial sets and complexes in algebraic topology
55M25 Degree, winding number
05A99 Enumerative combinatorics
Full Text: DOI
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