Euler characteristic formulas for simplicial maps (Q2710562)

The authors study a certain local Euler characteristic of a finite simplicial complex which assigns to each vertex its contribution to the usual global Euler characteristic. Unlike the global Euler characteristic, the local one may change under subdivision. Several formulae are obtained. These are applied, in particular, to smooth maps from a smooth closed surface into a smooth \(3\)-manifold. Here the authors obtain generalizations of results of \textit{T. F. Banchoff} [Proc. Am. Math. Soc. 46, 407-413 (1974; Zbl 0309.57017)], \textit{A. Szücs} [Bull. Lond. Math. Soc. 18, 60-66 (1986; Zbl 0563.57015)], \textit{S. Izumiya} and \textit{W. L. Marar} [J. Geom. 52, No. 1-2, 108-119 (1995; Zbl 0840.57010)], \textit{K. Houston} [Topology Appl. 91, No. 3, 197-219 (1999; Zbl 0943.57012)], and others.