Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 1490D manifold is a point. A 1D compact and connected manifold can eitherbe a line element or a circle, and it is intuitively clear (and can easily beproven) that these two spaces are topologically different. In 2D, there isalready an infinite number of different topologies: a 2D compact and connectedsurface can have an arbitrary number of handles and boundaries,and can either be orientable or non–orientable (see figure 3.2). Again, it isintuitively quite clear that two surfaces are not homeomorphic if they differin one of these respects. On the other hand, it can be proven that any twosurfaces for which these data are the same can be continuously mapped toone another, and hence this gives a complete classification of the possibletopologies of such surfaces.Fig. 3.2 Three examples of 2D manifolds: (a) The sphere S 2 is an orientable manifoldwithout handles or boundaries. (b) An orientable manifold with one boundary and onehandle. (c) The Möbius strip is an unorientable manifold with one boundary and nohandles.A quantity such as the number of boundaries of a surface is called atopological invariant. A topological invariant is a number, or more generallyany type of structure, which one can associate to a topological space, andwhich does not change under continuous mappings. Topological invariantscan be used to distinguish between topological spaces: if two surfaces havea different number of boundaries, they can certainly not be topologicallyequivalent. On the other hand, the knowledge of a topological invariantis in general not enough to decide whether two spaces are homeomorphic:a torus and a sphere have the same number of boundaries (zero), but areclearly not homeomorphic. Only when one has some complete set of topologicalinvariants, such as the number of handles and boundaries in the 2Dcase, is it possible to determine whether or not two topological spaces arehomeomorphic. In more than 2D, many topological invariants are known,but for no dimension larger than two has a complete set of topological invariantsbeen found. In 3D, it is generally believed that a finite number ofcountable invariants would suffice for compact manifolds, but this is not rig-