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. V are also tame. For each t ∈ R, we write S = f −1[0,t] The p-boundaries form a subgroup of the p-cycles, which t for the sublevel set of f, and U = U ∩ S , V = V ∩ S , form a subgroup of the p-chains: B ⊆ Z ⊆ C . The p-th t t t t p p p M = M ∩ S for the sublevel sets of the restrictions of f to homology group is the quotient of the p-cycles over the p- t t U, V, M. Similarly, we write St = f −1[t, 1] for the super- boundaries: H = Z /B . Its elements are sets of p-cycles p p p level set, and Ut = U ∩ St, Vt = V ∩ St, Mt = M ∩ St for that differ from each other by p-boundaries. More fully, we the superlevel sets of the restrictions. Since we will apply denote the p-th homology group by H (K, F), or by H (X, F) p p Mayer-Vietoris and Alexander duality to sub- and superlevel to emphasize that the group is independent of the simplicial sets, we make the same technical assumptions as formulated complex we choose to triangulate the space. However, we above for U, V, and M. For later reference, we call a pair will fix an arbitrary field F and drop it from the notation. For of (n + 1)-manifolds with boundary that satisfy those condi- field coefficients, H (X) is necessarily a vector space, which p tions a complementary decomposition of S, noting that the is fully described by its rank, βp(X) = rank Hp(X) such that X two (n + 1)-manifolds are indeed complementary except for H (X) ≃ Fβp( ), called the p-th Betti number of X. Finally, p the shared boundary. we will often drop the dimension from the notation and write The basic version of Alexander duality is a statement H(X)= L H (X) for the direct sum. p∈Z p about a complementary decomposition of the sphere; see Besides standard homology, we will frequently use reduced [10, page 424]. More specifically, it states that H (U) and homology groups, H˜ (X), which are isomorphic to the non- q p H (V) are isomorphic, where q = n−p, except for p = 0 and reduced groups except possibly for dimensions p = 0, −1. p q = 0 when the 0-dimensional group has an extra generator. To explain the difference, we note that β (X) counts the 0 This implies components of X. In contrast, β˜0(X) = rank H˜ 0(X) counts V U the gaps between components or, equivalently, the edges β0( ) = βn( ) + 1, (1) that are needed to merge the components into one. Hence, β (V) = β (U), (2) ˜ X X X p q β0( ) = β0( ) − 1, except when is empty, in which case V U βn( ) = β0( ) − 1, (3) β˜0(X) = β0(X) = 0. To distinguish this case from a sin- gle component, we have β˜−1(X) equal to 1 if X = ∅, and for 1 ≤ p ≤ n − 1. We will also need the version that deals with a complementary decomposition of the (n + 1)- the sublevel sets of M and U. More specifically, we get dimensional ball, Bn+1; see [10, page 426]. Let t ∈ (0, 1) M U U denote a regular value of f|M. Note first that St is homeo- βp( ) = βp( )+ βq( ), (13) Bn+1 U t morphic to . By excision, the homology groups of t β (Mt) = β (Ut)+ βq(U, U ), (14) U −1 U p p relative to t ∩ f (t) are isomorphic to those of relative t t t t β (M, M ) = β (U, U )+ βq(Ut), (15) to U . Alexander duality states that Hq(U, U ) and Hp(Vt) p p are isomorphic, where q + p = n, as before, except for p = 0, for all 0 ≤ p ≤ n. Here, we combine (6), (7), (8) with (1), when V has an extra component. This implies t (2), (3) to get (13). Similarly, we combine (9), (10) with (4), t β (Vt) = β (U, U ) + 1, (4) (5) to get (14). Finally, we exploit the symmetry between 0 n U V V U Ut and and combine (11), (12) with (4), (5) to get (15). βp( t) = βq( , ), (5) t Note that (14) and (15) imply βp(Mt) = βq(M, M ), which for 1 ≤ p ≤ n. is a consequence of Lefschetz duality; see [10]. Mayer-Vietoris. We can connect the homology groups of An example. We illustrate the above relationships with an U and V with those of M and S using the Mayer-Vietoris se- example. Let M be a 2-dimensional torus in S3. Accordingly, quence of the decomposition. This sequence is exact, mean- U and V are the two solid tori that decompose the 3-sphere ing the image of every map equals the kernel of the next and intersect in M. This is sketched in Figure 1, where we map. Counting the trivial homology groups, the sequence is assume that U is the part of space surrounded by M, while infinitely long, with three terms per dimension: V is the space surrounding M. The only non-zero Betti numbers of the solid torus are β0(U)= β1(U) = 1. We get . . . → Hp+1(S) → Hp(M) → Hp(U) ⊕ Hp(V) → Hp(S) → ..., β (M)= β0(U)+ β2(U) = 1, where the maps between homology groups of the same di- 0 M U U mension are induced by the inclusions. The only non-trivial β1( )= β1( )+ β1( ) = 2, S M U U homology groups of are in dimensions 0 and n + 1, with β2( )= β2( )+ β0( ) = 1 ranks β0(S)= βn+1(S) = 1. It follows that for 1 ≤ p ≤ n−1, the groups defined by M, U, V are surrounded by trivial from (13). These are the correct Betti numbers of the 2- groups. This implies that the groups of M and the direct dimensional torus. Next, choose t so that the sublevel set of U sums of the groups of U and V are isomorphic. For p = 0 is half the solid torus. Its only non-zero Betti number is and p = n, the non-trivial homology groups of S prevent this isomorphism, and we get Ut M U V β0( ) = β0( )+ β0( ) − 1, (6) M U V βp( ) = βp( )+ βp( ), (7) M U V βn( ) = βn( )+ βn( ) + 1, (8) for 1 ≤ p ≤ n − 1. We also have a Mayer-Vietoris sequence for the spaces St, Ut, Vt and Mt: f −1(t) . . .→ Hp+1(St) → Hp(Mt) → Hp(Ut) ⊕ Hp(Vt) → Hp(St) → . . .
For 0 ≤ t < 1, St is a point or a closed (n + 1)-dimensional ball, and its only non-trivial homology group is of dimension Ut 0, with rank β0(St) = 1. It follows that for 1 ≤ p ≤ n, the groups defined by Mt, Ut, Vt are surrounded by trivial M Figure 1: The level set defined by t splits the solid groups, which implies that the groups of t and the direct torus into two halves. sums of the groups of Ut and Vt are isomorphic. We get U β (Mt) = β (Ut)+ β0(Vt) − 1 (9) β0( t) = 1, and the only non-zero relative Betti number of 0 0 U Ut M U V the pair is β1( , ) = 1. We thus get βp( t) = βp( t)+ βp( t), (10) M U U Ut for 1 ≤ p ≤ n. We may also consider the relative homology β0( t)= β0( t)+ β2( , ) = 1, M U U Ut groups, again connected by a Mayer-Vietoris sequence: β1( t)= β1( t)+ β1( , ) = 1, t t M U U Ut . . . → Hp+1(S, S ) → Hp(M, M ) β2( t)= β2( t)+ β0( , ) = 0 U Ut V Vt S St → Hp( , ) ⊕ Hp( , ) → Hp( , ) → . . . from (14). These are the correct Betti numbers of the sub- M For 0 < t ≤ 1, the only non-trivial homology group of the level set of . Finally, we get t pair (S, S ) is in dimension n + 1, which implies M Mt U U Ut β0( , )= β2( t)+ β0( , ) = 0, t t t t t β (M, M ) = β (U, U )+ βp(V, V ), (11) M M U U U p p β1( , )= β1( t)+ β1( , ) = 1, M Mt U Ut V Vt M Mt U U Ut βn( , ) = βn( , )+ βn( , ) + 1, (12) β2( , )= β0( t)+ β2( , ) = 1 for 0 ≤ p ≤ n − 1. from (15). These are the correct relative Betti numbers of Separating manifold. Combining Alexander duality with the pair (M, Mt). Note again that the vector of Betti num- Mayer-Vietoris, we get relations between the homology of bers of Mt is the reverse of that of (M, Mt). in which the sublevel set grows from ∅ to X, and then again −tps 2−nd pass1−st pass during the second pass in which X minus the superlevel set shrinks back from X to ∅. When we collect the dots, we Birth separate the passes by drawing each coordinate axis twice. The result is a multiset of dots, which we refer to as the 0 0 1 1 Death persistence diagram of the function, and denote as Dgm(g), or Dgmp(g) if we restrict ourselves to the dots representing p-dimensional classes; see Figure 2, where we label each dot 1−st pass 2−nd passHor Vcl with the dimension of the classes it represents. We further −1 Ord 2 0Rel 3 distinguish four regions within the diagram: the ordinary, the horizontal, the vertical, and the relative subdiagrams, denoted as Ord(g), Hor(g), Vcl(g), and Rel(g). For example, Figure 2: The persistence diagram of the function a dot belongs to the ordinary subdiagram if its birth and restricted to the solid torus (dots drawn as circles), death both happen during the first pass; see again Figure 2. and to the complementary solid torus (dots drawn as squares), as sketched on the left. The white dots Reduced persistence diagrams. It is sometimes conve- belong to the reduced diagram, the black dots be- nient to use reduced instead of standard homology groups long to the standard diagram, and the shaded dots in the filtration. There are small differences caused by the belong to both. (−1)-dimensional class, which exists if the space is empty. As a first step, we introduce reduced versions of the rela- tive homology groups, which are isomorphic to the standard 3. PERSISTENCE relative groups, for all p. To define them, let ω be a new (dummy) vertex, write ω Xt for the cone of ω over Xt, and Starting with a brief introduction of persistent homology, t t let Xω = X ∪ ω X be the result of gluing the cone along we review its combinatorial expression as a point calculus; Xt to X. Then the reduced relative homology group of the see [6] and [1, 2] for more comprehensive discussions. ˜ t ˜ t t pair is Hp(X, X )= Hp(Xω). For example, if X = ∅, then ω Background. We take the step from homology to persis- forms a separate component, so that the reduced 0-th Betti tent homology by replacing a space with the sequence of number is equal to the number of components of X, just as t (closed and open) sublevel sets of a function on the space. β0(X, X ). The reduced persistence diagram is now defined To explain this, let X be compact and g : X → [0, 1] tame. as before, but for the filtration of reduced homology groups. X −1 As before, we write t = g [0,t] for the sublevel set and To describe this, we suppress the dimension and write X˜i for t −1 X = g [t, 1] for the superlevel set defined by t. Since g is ˜ X ˜ X ˜ Xt2m−i H( ti ), if 0 ≤ i
1−st pass 2−nd pass Lt(g) Rt(g)
(t,t) (t,t) Figure 3: The cascade combines the coordinates of the four black dots with 0 and 1 to form the five Figure 4: The persistence diagram of the function white dots. g = f|M, as defined in Figure 1; the circle dots also belong to the diagram of f|U, but the square dots do not. The rectangle on the left, Lt(g), represents the Cascades. We let ℓ+1 be the number of extreme dots and, homology of Mt, while the rectangle on the right, for a reason that will become clear shortly, denote their co- t Rt(g), represents the homology of (M, M ); compare ordinates with different indices as (uk,wk+1), for 0 ≤ k ≤ ℓ, with the Betti numbers computed for the example ordering them such that u0 < u1 < ... X0 → X1 → . . . → Xm persistence modules: ×× × ˜ p ˜ p ˜ p U0 → . . . → Um → . . . → U2m Ym ← Ym−1 ← . . . ← Y0. ↓ ↓ ↓ q q q j j ˜ ˜ ˜ V2m ← . . . ← Vm ← . . . ← V0, Write gi : Xi → Xj and hi : Yi → Yj for the maps upstairs and downstairs. We call the pairings compatible with these ′ ′ ′ where p + q = n. At the end of this section, we will prove maps if ξ, η = ξ , η for every ξ ∈ Xi and η ∈ Ym−j , ′ ′ − that there exists a sequences of compatible pairings between j m i p q where ξ = g (ξ ) and η = h − (η). The reason for consid- ˜ ˜ i m j Ui and V2m−i, for 0 ≤ i ≤ 2m. Using these pairings, we get ering compatible pairings is the following result. our first result, which we formulate using the superscript ‘T ’ for the operation that reflects a dot across the vertical axis Contravariant PE Theorem. Let X0 to Xm and Y0 of a persistence diagram and, at the same time, changes its to Ym be two filtrations contravariantly connected by non- degenerate and compatible pairings. Then there is a bijec- dimension from p to q = n − p. tion between the groups upstairs and downstairs such that a Land and Water Theorem. Let U and V be a comple- n+1 class is born at Xi and dies entering Xj iff the corresponding mentary decomposition of S = S , and let f : S → [0, 1] be class is born at Ym−j+1 and dies entering Ym−i+1. a perfect Morse function whose restriction to the n-manifold T ∗ M = U ∩ V is tame. Then Dgm˜ (f|V)= Dgm˜ (f|U) . Proof. Let Xi be the dual vector space of Xi, that is, the space of homomorphisms from Xi to F, and similarly let Proof. We apply the Contravariant PE Theorem to the ∗ ˜ p ˜ q ˜ p Ym−j be the dual vector space of Ym−j . We consider the persistence modules Ui and Vi : If a class α is born at Ui ˜ p linear maps and dies entering Uj then its corresponding class is born at ∗ ∗ V˜ q and dies entering V˜ q . If1 ≤ i < j ≤ m, then φ : Xi → Ym−j and ψ : Ym−j → Xi 2m−j+1 2m−i+1 the class α is represented by (si,sj ) in the p-th ordinary sub- defined by mapping x to φ(x), which sends y ∈ Ym−j to U j diagram of f| . The corresponding class is represented by gi (x),y , and by mapping y to ψ(y), which sends x ∈ Xi to (sj ,si) in the q-th relative subdiagram of f|V. The reflection m−i x, hm−j (y) . Since the pairings are compatible, we have maps the first and second coordinate-axes of the ordinary j m−i subdiagram to the second and first coordinate-axes of the φ(x)(y)= gi (x),y = x, hm−j (y) = ψ(y)(x). relative subdiagram. It follows that it maps (si,sj ) in the Therefore, φ and ψ are adjoint, which means that the as- former to (sj ,si) in the latter subdiagram. Other cases are ˜ ˜ sociated matrices are transposes of each other. It follows similar, and we conclude that Dgmp(f|U) and Dgmq(f|V) are ′ j that φ and ψ have the same rank. Note that φ = φ ◦ gi , reflections of each other. Writing this more succinctly gives ′ ∗ ′ ′ ′ where φ : Xj → Ym−j is defined by mapping x to φ (x ), the claimed relationship between the reduced persistence di- ′ which sends y ∈ Ym−j to x ,y . Since the pairing is non- agrams of f restricted to the two complementary subsets of ′ j degenerate, φ is an isomorphism. Hence, rank gi = rank φ, the sphere. Alexander pairing. We fill the gap in the proof of the α be a reduced p-cycle in U, β a reduced q-cycle in V, and ′ Land and Water Theorem by establishing compatible pair- β a (q + 1)-chain whose boundary is β. We define α, β A ings between the groups upstairs and downstairs. For the by counting the intersections between α and β′. Note that ′ sake of simplicity, we restrict ourselves to the case F = Z2. β is the preimage of β under the previously defined map- t While the argument for general fields is similar, it requires ping, ϕ : H˜ q+1(Ut,∂Ut) → H˜ q (V, V ), for any regular value t. oriented simplices, which is a technical formalism we prefer Therefore, the pairing in the middle is compatible with the to avoid. We begin with the pairing implicit in Lefschetz pairings in left half, again exploiting the compatibility of the duality for manifolds with possibly non-empty boundary: Lefschetz pairing. Alternatively, we can define the pairing by taking a (p + 1)-chain α′ with boundary α and count- , : H˜ (U ) × H˜ (U ,∂U ) → Z , ′ L p t q+1 t t 2 ing its intersections with β. In this case, α is the preimage ˜ V V ˜ U Ut which is defined by mapping two classes to the parity of the of α under the map ϕ : Hp+1( t,∂ t) → Hp( , ), and number of intersections between representing cycles. We compatibility with the pairings in the right half follows. In- get such a pairing for every regular value t, and these pair- deed, both definitions are equivalent, as already observed by Z ings are compatible with the horizontal maps induced by Lefschetz [9]. We give a simple proof for the case F = 2: inclusion of the sublevel sets; see [3]. Next, we reduce the Bridge Lemma. Let α and β be non-intersecting reduced dimension of the second factor using a mapping, ϕ, which cycles on the (n + 1)-sphere whose dimensions add up to we compose from four simpler mappings: n. Let α′ and β′ be chains whose boundaries are α and β, ′ ′ ϕ 1 t respectively. Then α , β L = α, β L. H˜ q+1(Ut,∂Ut) → H˜ q+1(S, S ∪ Vt) Proof. We can assume that α and β do not intersect. ϕ2 ˜ St V → Hq( ∪ t) The intersection of α′ and β′ is a 1-chain. Its boundary ϕ3 t t → H˜ q(S ∪ Vt, S ) consists of an even number of points, and is the disjoint ′ ′ ϕ4 union of intersections of α with β and of α with β . This H˜ V Vt → q( , ). implies that both types of intersections occur with the same The first mapping, ϕ1, is an isomorphism defined by exci- parity. sion. Next, ϕ2, is the connecting homomorphism of the ex- t act sequence of the pair (S, S ∪ Vt). Since the q-th reduced 5. SHORE S homology of is trivial, for all q = n + 1, ϕ2 is an isomor- This section presents our second result, which extends phism for all 0 ≤ q < n. It is a surjection for q = n. The the combination of Alexander duality and Mayer-Vietoris third mapping, ϕ3, is induced by inclusion. It occurs in the sequences from spaces to functions. exact sequence of the pair (St ∪ V , St). For 0 < t ≤ 1, the t Mayer-Vietoris sequence of filtrations. Assuming that reduced homology groups of St are all trivial, implying that f|M is tame, we write s < t < s <... Proof of (20). We compare Dgm0(f|M) with the disjoint Euclidean case. Considering that (13) to (15) give ele- union of Dgm0(f|U) and Dgm0(f|V), noting that their dots gant relations between the Betti numbers of the shore and all belong to the ordinary and horizontal subdiagrams. Con- the land, it is perhaps surprising that we need cascades to sider first a dot (a,b) in the ordinary subdiagram Ord0(f|M). formulate a similar result for persistence diagrams. Indeed, It represents a component in the sublevel set that is born (13) to (15) suggest that the persistence diagrams of f|M at a minimum x ∈ M, with f(x) = a, and that dies at a be the disjoint union of the persistence diagram of f|U and of its reflection. The example in Figure 5 shows that this with shared boundary, forming a complementary decompo- simple relation does not hold in general. Indeed, we have sition of the (n + 1)-dimensional sphere with each other. the dots (a, c), (b,d), (c, a), (d,b) in the standard diagram The second extension relates the persistence diagram of the of f|M, while the standard diagram of f|U contains (a,d) and function restricted to the n-manifold boundary with the di- (c, b). This clearly violates the suggested relation. However, agram of the function restricted to one (n + 1)-manifold. there is a natural setting in which the relation does hold. A key tool in its proof is the persistence module of Mayer- To state this result, we let A be a compact (n + 1)-manifold Vietoris sequences (or the Mayer-Vietoris sequence of per- with boundary in Rn+1. To be compatible with the rest sistence modules). This suggests the study of more general of this paper, we assume A is triangulable and there exists modules of exact sequences. Besides the hope to develop a an open neighborhood and a deformation retraction of that general purpose device that pervades persistent homology in neighborhood to A. the same way exact sequences pervade homology, we moti- vate the study with a few concrete questions. Euclidean Shore Theorem. Let n be a positive inte- ger, let A ⊂ Rn+1 be a compact (n+1)-manifold with bound- • Can our results be extended to the more general setting X A B ary satisfying the technical conditions as described, and sup- of a decomposition = ∪ of a topological space X pose that e : Rn+1 → R has no homological critical values , as considered by Di Fabio and Landi [4, 5]. and its restriction to ∂A is tame. Then • As pointed out to us by Yuriy Mileyko, an alternative T approach to generalizing our results for functions can Dgm(e|∂A)= Dgm(e|A) ⊔ Dgm(e|A) . be based on the generalization of Alexander duality Proof. We can extend e to a perfect Morse function f : given in [8, Proposition 3.46]. Combining this state- Sn+1 → R and A to a subset U of S = Sn+1 such that the ment of duality with the Mayer-Vietoris sequence has persistence diagrams of e restricted to A and to ∂A are the the potential to generalize our two Shore Theorems. same as those of f restricted to U and M = ∂U. It thus • Recall that our Euclidean Shore Theorem relates the suffices to show that the persistence diagram of f|M is the persistence diagram of a function f on an (n + 1)- disjoint union of the diagram of f|U and of its reflection. For A Rn+1 0