A COMBINATORIAL LEFSCHETZ FIXED-POINT FORMULA

Let K be any (finite) simplicial complex, and K′ a subdivision of K. Let φ{symbol}: K′ → K be a simplicial map, and, for all j ≥ 0, let φ{symbol}j denote the algebraical number of j-simplices G of K′ such that G ⊃ φ{symbol}(G). From Hopf's alternating trace formula it follows that φ{symbol}0 - φ{symbol}1 + φ{symbol}2 - ... = L(φ{symbol}), the Lefschetz number of the simplicial map φ{symbol}: X → X. Here X denotes the space of |K| (or |K′|). A purely combinatorial proof of the case K = a closed simplex (now L(φ{symbol}) = 1) is given, thus solving a problem posed by Ky Fan in 1978. © 1992.