Delta complexes, Betti numbers and torsion | Algebraic Topology 35 | NJ Wildberger скачать в хорошем качестве

Delta complexes, Betti numbers and torsion | Algebraic Topology 35 | NJ Wildberger

We continue our investigation of homology by computing the homology groups of a torus. For this we use the framework of delta-complexes, a somewhat general and flexible approach to simplicial complexes that allows us to use just two triangles in the standard square planar representation of a torus with opposite edges identified. While this means that all three corners of each triangle is actually identified with one point, if we go through the algebraic formalities of computing the homology groups, we get the same answer as with a more complicated geometrical simplical subdivision. So we actually make the computation, discuss some generalities on subgroups of Z plus Z, and then move to the computation of the homology of the projective plane, for which a new phenomenon appears called torsion: a homology group which is actually a finite commutative group, not a free abelian group. In fact H_1 (P) is just Z/2Z or Z_2, the group with two elements. This corresponds to a cycle which is not a boundary, but which has the property that twice that cycle is a boundary! Then we introduce the Betti numbers b_0, b_1, b_2 etc. which are the ranks of the homology groups H_0, H_1, H_2 etc, and mention the celebrated result that the alternating sum of Betti numbers gives us the Euler characteristic of the space. This is the last lecture in this series for a while! ************************ Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at   / njwildberger   Your support would be much appreciated. Here are the Insights into Mathematics Playlists:    • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист   Here are the Wild Egg Maths Playlists (some available only to Members!)    • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист      • Плейлист   ************************