Add resource "Homology Theory — A Primer" Accepted
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Add The Fundamental Group — A Primer
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 The Fundamental Group — A Primer
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 Web
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 20130113
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 Our eventual goal is to get comfortable with the notion of the “homology group” of a topological space. That is, to each topological space we will associate a group (really, a family of groups) in such a way that whenever two topological spaces are homeomorphic their associated groups will be isomorphic. In other words, we will be able to distinguish between two spaces by computing their associated groups (if even one of the groups is different). In general, there may be many many ways to associate a group with an object (for instance, it could be a kind of symmetry group or a group action). But what we want to do, and what will motivate both this post and the post on homology, is figure out a reasonable way to count holes in a space. Of course, the difficult part of this is determining what it means mathematically to have a “hole” in a space.
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 https://jeremykun.com/2013/01/12/thefundamentalgroupaprimer/
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Add Homology Theory — A Primer
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 Homology Theory — A Primer
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 Web
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 20130404
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 This series on topology has been long and hard, but we’re are quickly approaching the topics where we can actually write programs. For this and the next post on homology, the most important background we will need is a solid foundation in linear algebra, specifically in rowreducing matrices (and the interpretation of rowreduction as a change of basis of a linear operator). Last time we engaged in a whirlwind tour of the fundamental group and homotopy theory. And we mean “whirlwind” as it sounds; it was all over the place in terms of organization. The most important fact that one should take away from that discussion is the idea that we can compute, algebraically, some qualitative features about a topological space related to “ndimensional holes.”
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 https://jeremykun.com/2013/04/03/homologytheoryaprimer/
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Add Computing Homology
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 Computing Homology
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 Web
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 20130410
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 In our last post in this series on topology, we defined the homology group. Specifically, we built up a topological space as a simplicial complex (a mess of triangles glued together), we defined an algebraic way to represent collections of simplices called chains as vectors in a vector space, we defined the boundary homomorphism as a linear map on chains, and finally defined the homology groups as the quotient vector spaces. In this post we will be quite a bit more explicit. Because the chain groups are vector spaces and the boundary mappings are linear maps, they can be represented as matrices whose dimensions depend on our simplicial complex structure. Better yet, if we have explicit representations of our chains by way of a basis, then we can use rowreduction techniques to write the matrix in a standard form.
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 https://jeremykun.com/2013/04/10/computinghomology/
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Add Jeremy Kun created The Fundamental Group — A Primer
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Add Jeremy Kun created Homology Theory — A Primer
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Add Jeremy Kun created Computing Homology
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