# 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

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## 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