Computing Homology


Resource | v1 | created by janarez |
Type Web
Created 2013-04-10
Identifier unavailable

Description

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 row-reduction techniques to write the matrix in a standard form.

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created by Jeremy Kun

My name is Jeremy Kun. I’m currently an engineer at Google. I earned a PhD in mathematics from the Un...

follows Homology Theory — A Primer

This series on topology has been long and hard, but we’re are quickly approaching the topics where we...


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janarez
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Last part of the homology series. Contains code and is the easiest to comprehend. Does not make sense to read without the previous post.