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Tag Archives: homology
Exact sequence (Quotient space)
Exact sequence (Quotient space) If is a space and is a nonempty closed subspace that is a deformation retract of some neighborhood in , then there is an exact sequence where is the inclusion and is the quotient map . … Continue reading
Reduced Homology
Define the reduced homology groups to be the homology groups of the augmented chain complex where . We require to be nonempty, to avoid having a nontrivial homology group in dimension 1. Relation between and Since , vanishes on and … Continue reading
Persistent Homology Algorithm
Algorithm for Fields In this section we describe an algorithm for computing persistent homology over a field. We use the small filtration as an example and compute over , although the algorithm works for any field. A filtered simplicial complex … Continue reading
Singular Homology
A singular simplex in a space is a map . Let be the free abelian group with basis the set of singular simplices in . Elements of , called singular chains, are finite formal sums for and . A boundary … Continue reading
Universal Property of Quotient Groups (Hungerford)
If is a homomorphism and is a normal subgroup of contained in the kernel of , then “factors through” the quotient uniquely. This can be used to prove the following proposition: A chain map between chain complexes and induces homomorphisms … Continue reading
Some Homology Definitions
Chain Complex A sequence of homomorphisms of abelian groups with for each . th Homology Group is the free abelian group with basis the open simplices of . chains Elements of , called chains, can be written as finite formal … Continue reading
How to calculate Homology Groups (Klein Bottle)
This post will be a guide on how to calculate Homology Groups, focusing on the example of the Klein Bottle. Homology groups can be quite difficult to grasp (it took me quite a while to understand it). Hope this post … Continue reading
Video on Computing Homology Groups
This video follows after the previous video on Simplicial Complexes. If you are looking for quality Math textbooks to study from (including Linear Algebra and Calculus, the two most popular Math courses), check out my page on Recommended Math Books … Continue reading