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Add Computing homology groups | Algebraic Topology | NJ Wildberger
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- Computing homology groups | Algebraic Topology | NJ Wildberger
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- Video
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- 2012-10-12
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- The definition of the homology groups H_n(X) of a space X, say a simplicial complex, is quite abstract: we consider the complex of abelian groups generated by vertices, edges, 2-dim faces etc, then define boundary maps between them, then take the quotient of kernels mod boundaries at each stage, or dimension. To make this more understandable, we give in this lecture an in-depth look at some examples. Here we start with the simplest ones: the circle and the disk. For each space it is necessary to look at each dimension separately. The 0-th homology group H_0(X) measures the connectivity of the space X, for a connected space it is the infinite cyclic group Z of the integers. The first homology group H_1 measures the number of independent non-trivial loops in the space (roughly). The second homology group H_2 measures the number of independent non-trivial 2-dim holes in the space, and so on.
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- https://www.youtube.com/watch?v=YNBi4Ix3cY0
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Add Homology
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- Homology
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- In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, to other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes.
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- https://en.wikipedia.org/?curid=142432
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