Euler-Poincaré characteristic--a case of topological self-convincing

Milosav M. Marjanović

In this paper we establish a topological property of geometric objects (lines, surfaces and solids) called Euler-Poincaré characteristic. Since the paper is intended for a large profile of mathematics teachers, our approach is entirely intuitive and majority of readers can omit two addenda whose understanding requires a solid knowledge of topology. E-P characteristic is an integer which we calculate here decomposing lines into fibers being finite sets of points, surfaces into fibers being lines and solids into fibers being surfaces. When an object is subjected to a ``plastic'' deformation its shape and size changes as well as the alternating sums resulting from the method of calculation, but the value of these sums stays unchanged. This fact serves to convince the reader that E-P characteristic is a stable topological property. The same fact gives this approach an advantage over usually practiced ones which require a triangulation of geometric objects.