Ivancevic_Applied-Diff-Geom

1076 Applied Differential Geometry: A Modern Introductionpieces. Therefore, the Euler characteristic is a topological invariant, i.e.,any two geometric figures that are homeomorphic to each other have thesame Euler characteristic.More specifically, a standard way to analyze a geometric Figure is tofragment it into other more familiar objects and then to examine how thesepieces fit together. Take for example a surface M in the Euclidean 3D space.Slice M into pieces that are curved triangles (this is called a triangulation ofthe surface). Then count the number F of faces of the triangles, the numberE of edges, and the number V of vertices on the tesselated surface. Now,no matter how we triangulate a compact surface Σ, its Euler characteristic,χ(Σ) = F − E + V , will always equal a constant which is characteristic ofthe surface and which is invariant under diffeomorphisms φ : Σ → Σ ′ .At higher dimensions this can be again defined by using higher dimensionalgeneralizations of triangles (simplexes) and by defining the Eulercharacteristic χ(M) of the nD manifold M to be the alternating sum:{number of points} − {number of 2-simplices} +{number of 3-simplices} − {number of 4-simplices} + ...n∑i.e., χ(M) = (−1) k (number of faces of dimension k).k=0and then define the Euler characteristic of a manifold as the Euler characteristicof any simplicial complex homeomorphic to it. With this definition,circles and squares have Euler characteristic 0 and solid balls have Eulercharacteristic 1.The Euler characteristic χ of a manifold is closely related to its genusg as χ = 2 − 2g. 18Recall that a more standard topological definition of χ(M) isχ(M) =n∑(−1) k b k (M), (6.121)k=0where b k are the kth Betti numbers of M.18 Recall that the genus of a topological space such as a surface is a topologicallyinvariant property defined as the largest number of nonintersecting simple closed curvesthat can be drawn on the surface without separating it, i.e., an integer representing themaximum number of cuts that can be made through it without rendering it disconnected.This is roughly equivalent to the number of holes in it, or handles on it. For instance: apoint, line, and a sphere all have genus 0; a torus has genus 1, as does a coffee cup as asolid object (solid torus), a Möbius strip, and the symbol 0; the symbols 8 and B havegenus 2; etc.