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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”.

##### MSC:
 55U10 Simplicial sets and complexes in algebraic topology 55M25 Degree, winding number 05A99 Enumerative combinatorics
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##### References:
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