Various Disguises of Euler Characteristic

I want to share a paper I wrote about 3,5 years ago about Euler Characteristic. Euler Characteristic is a number to help us distinguish one manifolds from another. In this paper, I don’t write one or two.. but four different equivalent definitions of Euler Characteristic. Well probably not really equivalent but if we restrict our case to orientable compact differentiable manifolds, then they are indeed equivalent. One definition isĀ  combinatorial while another one may be more geometrical in nature, or algebraic or even involves integral calculus in manifolds. This is the first time I see the interplay of so many branches of Mathematics at once. Also, on after each definition I give example (with pictures :)) on which case the definition can be used to compute Euler Characteristic easily. Download the paper here