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Extending persistence using Poincaré and Lefschetz duality. (English) Zbl 1189.55002
Let $$K$$ be a simplicial complex and $$f$$ an injective function on its vertices, so that the vertices $$\{u_1,\dots,u_n\}$$ are ordered: $$f(u_1)<\cdots<f(u_n)$$. Define $$K_i$$ to be the subcomplex spanned by $$\{u_1,\dots,u_i\}$$. For fixed $$p$$, inclusion gives a sequence of homology groups $$H_p(K_i)$$ with coefficients in the field of order two. A class in $$H_p(K_i)$$ is said to be born at $$K_i$$ if it is not in the image of $$H_p(K_{i-1})\rightarrow H_p(K_i)$$, and the class dies at $$K_j$$ if its image in $$H_p(K_j)$$ is in the image of $$H_p(K_{i-1})\rightarrow H_p(K_j)$$, but its image in $$H_p(K_{j-1})$$ is not in the image of $$H_p(K_{i-1})\rightarrow H_p(K_{j-1})$$. A persistence diagram consists of the multiset of points $$(f(u_i),f(u_j))$$ in the extended plane for which there is a class that is born at $$K_i$$ and dies at $$K_j$$; note that the second coordinate of such a point may be infinite, since a class does not necessarily die.
To define the notion of extended persistence, the authors let $$L_{n-i}$$ be the subcomplex of $$K$$ spanned by $$\{u_{i+1},\dots,u_n\}$$ and define $$K_{n+i}\doteq(K,L_i)$$, thereby extending the above filtration $$\{K_i\}_{1\leq i\leq n}$$ to allow for indices $$1\leq i\leq 2n$$. All classes in extended persistence necessarily die. The authors discuss using extended persistence to choose an optimal homology basis, an algorithm to compute extended persistence, and the stability of extended persistence diagrams under different choices of the function $$f$$. Also discussed is the case when $$K$$ is a simplicial manifold of dimension $$d$$: Poincaré and Lefschetz duality imply a pairing between $$p$$-th persistence and $$(d-p)$$-th persistence, and in particular, their diagrams are reflections of each other.
The authors target an audience that is familiar with the homology of a simplicial complex and with some knowledge of computational topology; knowledge of persistent homology and Morse function theory is helpful but not necessary. The article is well-written, but the overall exposition could have been more focused.

##### MSC:
 55N99 Homology and cohomology theories in algebraic topology 68W30 Symbolic computation and algebraic computation 57Q99 PL-topology
##### Keywords:
persistence; computational topology
HEMO
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##### References:
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