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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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[1] Brown, A.B; Cairns, S.S, Strengthening of Sperner’s lemma applied to homology, (), 113-114 · Zbl 0097.38702
[2] Cohen, D.I.A, On the sperner lemma, J. combin. theory, 2, 585-587, (1967) · Zbl 0163.18104
[3] Fan, K, Simplicial maps from an orientable n-pseudomanifold into sm with the octahedral triangulation, J. combin. theory, 2, 588-602, (1967) · Zbl 0149.41302
[4] Fan, K, Some properties of convex sets related to fixed point theorems, Math. ann., 266, 519-537, (1984) · Zbl 0515.47029
[5] Hopf, H, A new proof of the Lefschetz formula on invariant points, (), 149-153 · JFM 54.0610.01
[6] Lefschetz, S, Intersections and transformations of complexes and manifolds, Trans. amer. math. soc., 28, 1-49, (1926) · JFM 52.0572.02
[7] Maunder, C.R.F, Algebraic topology, (1970), Cambridge Univ. Press London · Zbl 0205.27302
[8] Sperner, E, Neuer beweis für die invarianz der dimensionszahl und des gebietes, Abh. math. sem. univ. Hamburg, 6, 265-272, (1928) · JFM 54.0614.01
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