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A simplicial map f: S → T determines a homomorphism of homology groups H k (S) → H k (T) for each integer k. This is the homomorphism associated to a chain map from the chain complex of S to the chain complex of T. Explicitly, this chain map is given on k-chains by
A chain homotopy offers a way to relate two chain maps that induce the same map on homology groups, even though the maps may be different. Given two chain complexes A and B, and two chain maps f, g : A → B, a chain homotopy is a sequence of homomorphisms h n : A n → B n+1 such that hd A + d B h = f − g. The maps may be written out in a ...
In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology.In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain complexes take their terms in an abelian category, so that we can talk about cohomology, then the cone of a map f being acyclic means that the map is a quasi ...
Diffusion maps exploit the relationship between heat diffusion and random walk Markov chain.The basic observation is that if we take a random walk on the data, walking to a nearby data-point is more likely than walking to another that is far away.
The n-dimensional sphere S n admits a CW structure with two cells, one 0-cell and one n-cell.Here the n-cell is attached by the constant mapping from to 0-cell. Since the generators of the cellular chain groups (,) can be identified with the k-cells of S n, we have that (,) = for =,, and is otherwise trivial.
Let = be a -graded algebra, with product , equipped with a map : of degree (homologically graded) or degree + (cohomologically graded). We say that (,,) is a differential graded algebra if is a differential, giving the structure of a chain complex or cochain complex (depending on the degree), and satisfies a graded Leibniz rule.
In topological data analysis, a persistence barcode, sometimes shortened to barcode, is an algebraic invariant associated with a filtered chain complex or a persistence module that characterizes the stability of topological features throughout a growing family of spaces. [1]
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