By L. Auslander, R. Tolimieri

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The dimension of each topological feature is coded in the color of the corresponding diagram point: white for 0, black for 1. — From Chazal et al. [72]. A natural question to ask is whether all R-indexed ﬁltrations of a topological space X are sublevel-sets ﬁltrations of functions f : X → R. This is not exactly the case, as the following simple example shows. 5. Take for X the discrete space {a, b, c}, and let X be the following ﬁltration of X indexed over R: ⎧ ⎪ if i ≤ 0 ⎨ {a} {a, b} if i ∈ (0, 1) Xi = ⎪ ⎩ {a, b, c} if i ≥ 1.

This provides us with an algorithm for computing persistence: build the matrix of the boundary operator of Kσ , then compute its Smith normal form over k[t] using Gaussian elimination on the rows and columns, then read oﬀ the interval decomposition. The algorithm is the same as for non-persistent homology, except the ring of coeﬃcients is k[t] instead of Z. This approach for computing persistence was ﬁrst suggested by Zomorodian and Carlsson [243], who also observed that it is in fact suﬃcient to reduce the boundary matrix to a column-echelon form over the ﬁeld k itself, which has two beneﬁts: ﬁrst, Gaussian elimination only has to be applied on the columns of the matrix; second, and more importantly, there is no more need to compute greatest common divisors between polynomials.

This diagram is called a level-sets zigzag of f , denoted Z generically. There are as many such diagrams as there are ﬁnite increasing sequences of real values. However, in some cases there is a class of such zigzags that is ‘canonical’. For instance, when X is a compact manifold and f is a Morse function, there is a ﬁnite set of values c1 < · · · < cn such that the preimage of each open interval S = (−∞, c1 ), (c1 , c2 ), · · · , (cn−1 , cn ), (cn , +∞) is homeomorphic to a product of the form Y × S, where Y has ﬁnite-dimensional homology and f is the projection onto the factor S.