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Alexander Duality for Functions: the Persistent Behavior of Land and Water and Shore

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Alexander Duality for Functions: the Persistent Behavior of Land and Water and Shore Alexander Duality for Functions: the Persistent Behavior of Land and Water and Shore Herbert Edelsbrunner∗ Michael Kerber IST Austria IST Austria Klosterneuburg, Austria Klosterneuburg, Austria [email protected] [email protected] ABSTRACT information contained in the sequence of sublevel sets. If we This note contributes to the point calculus of persistent substitute reduced for standard homology groups, we get a homology by extending Alexander duality from spaces to slightly modified reduced persistence diagram. real-valued functions. Given a perfect Morse function f : As between homology groups, we can observe relation- Sn+1 → [0, 1] and a decomposition Sn+1 = U ∪ V into two ships between persistence diagrams. A prime example is (n + 1)-manifolds with common boundary M, we prove el- Poincar´eduality, which says that the p-th and the (n − p)- ementary relationships between the persistence diagrams of th homology groups of an orientable n-manifold are isomor- f restricted to U, to V, and to M. phic. More precisely, this is true if homology is defined for field coefficients, which is what we assume throughout this paper. The extension to functions says that the diagram Categories and Subject Descriptors is symmetric with respect to reflection across the vertical F.2.2 [Analysis of Algorithms and Problem Complex- axis; see [3]. Here, we change the homological dimension ity]: Nonnumerical Algorithms and Problems—Geometrical of a dot from p to n − p whenever we reflect it across the problems and computations, Computations on discrete struc- axis. This paper contributes new relationships by extending tures Alexander duality from spaces to functions. To state our re- sults, we assume a perfect Morse function, f : Sn+1 → [0, 1], Keywords which for the sphere has no critical points other than a min- imum and a maximum, and a decomposition of the (n + 1)- Algebraic topology, homology, Alexander duality, Mayer- dimensional sphere into two (n + 1)-manifolds with bound- Vietoris sequences, persistent homology, point calculus. ary, Sn+1 = U ∪ V, whose intersection is the common n- manifold boundary M = U ∩ V. Our first result says that 1. INTRODUCTION the reduced persistence diagrams of f restricted to U and to Persistent homology is a recent extension of the classi- V are reflections of each other. We call this the Land and cal theory of homology; see e.g. [6]. Given a real-valued Water Theorem. Our second result relates land with shore. function on a topological space, it measures the importance Ignoring some modifications, it says that the persistence dia- of a homology class by monitoring when the class appears gram of f restricted to M is the disjoint union of the diagram and when it disappears in the increasing sequence of sub- of f restricted to U and of its reflection. The modifications level sets. A technical requirement is that the function be become unnecessary if we assume that the minimum and tame, which means it has only finitely many homological maximum of f both belong to a common component of V. critical values, and each sublevel set has finite rank homol- We call this the Euclidean Shore Theorem. ogy groups. Pairing up the births and deaths, and drawing In the example that justifies the title of this paper, and each pair of values as a dot (a point in the plane), we get a the names of our theorems, we let U be the planet Earth, not multiset which we refer to as the persistence diagram of the including the water and the air. To a coarse approximation, function. It is a combinatorial summary of the homological U is homeomorphic to a 3-ball, sitting inside the Universe, S ∗ which we model as a 3-sphere, . The function we consider . and Departments of Computer Science and of Math- is the negative gravitational potential of the Earth, which is ematics, Duke University, Durham, North Carolina; De- defined on the entire Universe. The sea is then a sublevel set launay Lab of Discrete and Computational Geometry, V S Yaroslavl’ State University, Russia; Geomagic, Research Tri- of this function restricted to , which is the closure of − angle Park, North Carolina. U. With these definitions, our results relate the persistence diagram of the gravitational potential restricted to the Earth with the shape of the sea as its water level rises. Also, the Euclidean Shore Theorem applies, unmodified, expanding Permission to make digital or hard copies of all or part of this work for the relationship to include the sea floor, which is swept out personal or classroom use is granted without fee provided that copies are by the shoreline as the water level rises. not made or distributed for profit or commercial advantage and that copies Besides developing the mathematical theory of persistent bear this notice and the full citation on the first page. To copy otherwise, to homology, there are pragmatic reasons for our interest in republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. the extension of Alexander duality to functions. Persistence SCG’12, June 17–20, 2012, Chapel Hill, North Carolina, USA. has fast algorithms, so that the bulk of the work is often Copyright 2012 ACM 978-1-4503-1299-8/12/06 ...$10.00. in the construction of the space and the function for which equal to 0, otherwise. Furthermore, we use relative homol- we compute persistence. A point in case is the analysis of ogy, which is defined for pairs of spaces, Y ⊆ X. Taking a the biological process of cell segregation started in [7]. Mod- pair relaxes the requirement of a chain to be called a cy- eling the process as a subset of space-time, the function of cle, namely whenever its boundary is contained in Y, which interest is time which, after compactifying space-time to S4, includes the case when the boundary is empty. We write has no critical points other than a minimum and a maxi- Hp(X, Y) for the p-th relative homology group of the pair, and 4 mum. The subset U of S is a union of cells times time, βp(X, Y) = rank Hp(X, Y) for the p-th relative Betti number. whose boundary is a 3-manifold. We can represent U by a As before, we will suppress the dimension from the notation X Y X Y 1-parameter family of alpha complexes, whose disjoint union by introducing H( , )= Lp∈Z Hp( , ). has the same homotopy type; see e.g. [6, Chapter III]. How- As examples, consider the (n+1)-dimensional sphere, S = ever, the boundary of that disjoint union is not necessarily Sn+1, and a closed hemisphere, H ⊆ S, which is a ball of a 3-manifold. Using our Euclidean Shore Theorem, we can dimension n + 1. In standard homology, we have β0(S) = compute the persistence diagram of the function on the 3- β0(H)= βn+1(S) = 1 while all other Betti numbers are zero. manifold without ever constructing the 3-manifold. In reduced homology, we have β˜n+1(S) = 1 while all other Outline. Sections 2 and 3 introduce background on homol- reduced Betti numbers are zero. In particular, β˜p(H) = 0 ogy and persistent homology. Sections 4 and 5 present our for all p. In relative homology, we have βn+1(S, H) = 1 two results. Section 6 concludes the paper. while all other relative Betti numbers are zero. In particular, β0(S, H) = 0, which may be confusing at first but makes 2. HOMOLOGY sense because every point on the sphere can be connected by a path to a point in the hemisphere and is thus a 0- Starting with a brief introduction of classical homology boundary. groups, we present the relevant background on Alexander duality and Mayer-Vietoris sequences. More comprehensive Alexander duality. Recall that a perfect Morse function is discussions of these topics can be found in textbooks of al- one whose number of critical points equals the sum of Betti gebraic topology, such as [8, 10]. numbers of the space. For a sphere, this number is 2: a min- imum and a maximum. Throughout the remainder of this Background. The p-dimensional homology of a topological section, we assume a perfect Morse function f : S → [0, 1], space, X, is a mathematical language to define, count, and whose values at the minimum and the maximum are 0 and reason about the p-dimensional connectivity of X. There 1. We also assume two (n + 1)-manifolds with boundary, U are different but essentially equivalent theories depending and V, whose union is S, and whose intersection is the com- on the choices one makes in the representation of the space, mon boundary, M. For technical reasons, we require that the selection of cycles, and the meaning of addition. For our U, V, and M are embedded in the sphere in a way that is purpose, the most elementary of these theories will suffice: compatible with taking the simplicial homology and with a simplicial complex, K, that triangulates X, formal sums applying the Mayer-Vietoris sequence and Alexander dual- of p-simplices with zero boundary as p-cycles, and adding ity. Specifically, we assume that U and V are triangulable with coefficients in a field, F. Most algorithms on homology closed cofibrations. In other words, there is a deformation assume this model, in particular the ones developed within retraction from an open neighborhood of U to U, and simi- persistent homology. In this model, we call a formal sum larly for V. Furthermore, we assume that the restriction of of p-simplices a p-chain, a p-cycle if its boundary is empty, f to M is tame, which implies that its restrictions to U and and a p-boundary if it is the boundary of a (p + 1)-chain.
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