Inducing a map on homology from a correspondence
Shaun Harker, Hiroshi Kokubu, Konstantin Mischaikow, Paweł Pilarczyk
aa r X i v : . [ m a t h . A T ] N ov INDUCING A MAP ON HOMOLOGYFROM A CORRESPONDENCE
SHAUN HARKER, HIROSHI KOKUBU, KONSTANTIN MISCHAIKOW,AND PAWE L PILARCZYK
Abstract.
We study the homomorphism induced in homology by a closedcorrespondence between topological spaces, using projections from the graphof the correspondence to its domain and codomain. We provide assumptionsunder which the homomorphism induced by an outer approximation of a con-tinuous map coincides with the homomorphism induced in homology by themap. In contrast to more classical results we do not require that the projectionto the domain have acyclic preimages. Moreover, we show that it is possible toretrieve correct homological information from a correspondence even if somedata is missing or perturbed. Finally, we describe an application to com-binatorial maps that are either outer approximations of continuous maps orreconstructions of such maps from a finite set of data points. Introduction
The focus of this paper is on effective methods for computing f ∗ : H ∗ ( X, A ) → H ∗ ( Y, B ), the homomorphism induced on homology by a continuous map f : ( X, A ) → ( Y, B ), in situations where the map f and/or the underlying spaces are only knownvia finite approximations. This problem arises naturally in the context of topolog-ical data analysis or the use of algebraic topological invariants to study nonlineardynamics. In general, in these settings, at best one has bounds for the action of f ,though estimates of questionable certainty are more likely. With this in mind, weuse correspondences to represent f (see Section 2 for definitions) and complexes torepresent the domain and codomain.To put the goals of this paper into perspective, consider the simpler case of f : X → Y . Let F ⊂ X × Y be a correspondence. Let p : F → X and q : F → Y bethe canonical projection maps. Assume A1: p ( F ) = X , A2: f ( x ) ∈ q ( p − ( x )), for all x ∈ X , and A3: p − ( x ) is acyclic for all x ∈ X .Recall the following classical result of Vietoris [16]. Theorem 1.1.
Let Z and X be compact metric spaces. Let p : Z → X be a surjec-tive continuous function. If p − ( x ) is acyclic for every x ∈ X then p ∗ : H ∗ ( Z ) → H ∗ ( X ) is an isomorphism. Date : November 18, 2014.2010
Mathematics Subject Classification.
Primary 55M99. Secondary 55-04. Key words andphrases: homology, homomorphism, continuous map, time series, combinatorial approximation,grid, multivalued map, correspondence, acyclicity.
In the context of this paper, an immediate consequence is that f ∗ = q ∗ ◦ ( p ∗ ) − .On a theoretical level the computation of f ∗ in terms of acyclic correspondences hasa long tradition dating back at least to the work of Eilenberg and Montgomery [8].To the best of our knowledge, the first explicit algorithms based on this idea arepresented in [14], where it is implemented in the context of cubical complexes. Analternative approach, based on discrete Morse theory and applicable to arbitrarycomplexes, is presented in [10]. Code based on these implementations can be foundat [7].From the perspective of applications, all three assumptions A1 – A3 are problem-atic. In the context of data analysis, the domain used to compute f ∗ is a complex X constructed from a finite data set and the true domain X of f is unknown. Thus,as is demonstrated in Example 5.9, there are situations in which the appropriateassumption is that X ⊂ X , in which case A1 fails.Again, in the context of data analysis one expects that the data representing f is corrupted by bounded noise, but in many situations one does not have aclear estimate on the size of the noise. Thus, as indicated in Example 5.7, thecorrespondence F constructed using the data may fail to satisfy A2 .Finally, even in settings where complete knowledge of the function f can beassumed, the assumption of acyclicity may be too strong. Consider a smoothfunction f : R n → R n and assume that we are interested in computing f ∗ restrictedto a compact subset X represented in terms of a cubical complex, where X is a setdefined in terms of the action induced by f . Problems of this type occur naturallyin nonlinear dynamics, where ( X, A ) might represent an index pair and f ∗ becomesa representative for the associated Conley index (see [1, 4, 5, 13] for more detailsand concrete examples).Using the smoothness, there are a variety of methods by which rigorous boundson the images of n -dimensional cubes can be computed [6]. It is natural to constructa correspondence as follows. For each n -dimensional cube, define the correspon-dence in terms of the set of cubes that cover the outer approximation of the image,based on the rigorous bounds (see Figure 1). To define the correspondence at theintersection of two n -dimensional cubes in such a way that the compactness as-sumption of Theorem 1.1 is satisfied, it is natural to declare the correspondence interms of the union of the correspondences over the n -dimensional cubes. However,since in general the union of acyclic sets need not be acyclic, at this point one canno longer assume that A3 is satisfied.Keeping these examples in mind, we provide an outline for this paper. After pro-viding necessary preliminary definitions in Section 2, we introduce in Section 3 theconcepts of homological completeness and homological consistency which allow us tocharacterize whether the correspondence contains sufficient information to recoverthe appropriate homological information about the domain and the image into therange, respectively. In particular, this leads to our main result Theorem 3.10 inwhich we guarantee the correct computation of f ∗ : H ∗ ( X, A ) → H ∗ ( Y, B ) based oncorrespondences that do not necessarily satisfy A1 – A3 . To extend the applicabilityof this result, in Section 4 we focuses on relating homological information betweendifferent correspondences.As is suggested above, our results have an application in the context of writingsoftware for computational homology. For example, to apply [10, 14] it is not onlynecessary to obtain an outer approximation of the true underlying function f but NDUCING A MAP ON HOMOLOGY FROM A CORRESPONDENCE 3 f Figure 1.
Suppose that an outer bound for the image (shadedin dark gray) of the 2-dimensional cube shown in the left is givenin terms of a rectangular set (shaded in medium gray) that is notaligned with the grid. If one covers it with the minimal collectionof grid elements that intersect the rectangular set (all shaded inbright gray) then the obtained set is acyclic, but not convex. Theminimal convex covering contains twice the number of cubes andthus is a much poorer approximation of the dynamics induced by f .also to ensure that the outer approximation is acyclic-valued. In order to guaran-tee this, the associated software relies on performing homology computation usinguniform grids and grid-aligned bounding rectangles. As suggested in Figure 1, thisresults in undesirable overestimates. The results of this paper relax the require-ment to homological completeness and consistency, which permits algorithms thatare more robust, allowing for a wider choice of complexes and tighter outer ap-proximations. To emphasize this, we conclude in Section 5 with examples thatspecifically highlight the relevance of homological completeness and consistency.2. Preliminaries.
Let X and Y be topological spaces. Let A ⊂ X and B ⊂ Y . A correspondence from ( X, A ) to (
Y, B ) is a pair (
F, F ′ ) of relations such that F ⊂ X × Y and F ′ ⊂ A × B , and F ′ ⊂ F . We identify each relation with its graph. In particular,we consider the following special correspondences: a map , where every x ∈ X corresponds to exactly one y ∈ Y through F , and every a ∈ A corresponds to exactlyone b ∈ B through F ′ ; and a multivalued map , where every x ∈ X correspondsto at least one y ∈ Y through F , and every a ∈ A corresponds to at least one b ∈ B through F ′ . If ( F, F ′ ) is a map from ( X, A ) to (
Y, B ) then F ′ is uniquelydetermined by F and A , and obviously F ( A ) ⊂ B , which justifies the usual notation F : ( X, A ) → ( Y, B ). We remark that requiring that F ( A ) ⊂ B in the case ofgeneral correspondences is too restrictive (see Section 5 and [14]); therefore, workingexplicitly with pairs of relations is necessary.A correspondence ( F, F ′ ) is closed if both F and F ′ are closed as subsets of X × Y .A correspondence ( G, G ′ ) is an enlargement of ( F, F ′ ) if F ⊂ G and F ′ ⊂ G ′ , andis denoted by ( F, F ′ ) ⊂ ( G, G ′ ). Note that an enlargement of a [multivalued] mapis a multivalued map. A map f : ( X, A ) → ( Y, B ) is a selector of a correspondence
S. HARKER, H. KOKUBU, K. MISCHAIKOW, AND P. PILARCZYK ( F, F ′ ) from ( X, A ) to (
Y, B ) if (
F, F ′ ) is an enlargement of f . Observe that if acorrespondence ( F, F ′ ) has a selector then ( F, F ′ ) is a multivalued map.A relation F ⊂ X × Y is upper semicontinuous if for every x ∈ X , and for everyneighborhood V of F ( x ), there exists a neighborhood U of x such that F ( U ) ⊂ V .A correspondence ( F, F ′ ) is upper semicontinuous if both F and F ′ are uppersemicontinuous. Recall the following classical result [3]: Theorem 2.1.
Let F ⊂ X × Y be a relation. If F ( x ) is closed for every x ∈ X ,the domain of F is closed, and the range Y of F is compact, then F is closed (asa subset of X × Y ) if and only if F is upper semicontinuous. We consider homology with coefficients in an arbitrary ring R , which will befixed throughout the paper. We leave the freedom to choose the most appropriatehomology theory to work with for the reader; however, in the examples we shalluse cubical homology [12].A set is acyclic with respect to R if its homology is isomorphic with the homologyof a single point, that is, H ∼ = R and H q ∼ = { } for all q = 0. A correspondence( F, F ′ ) is acyclic if the image of every point by F is an acyclic set, and also by F ′ whenever applicable. Definition 2.2.
Let (
F, F ′ ) be a closed correspondence from ( X, A ) to (
Y, B ). Let p : ( F, F ′ ) → ( X, A ) and q : ( F, F ′ ) → ( Y, B ) be the canonical projections. Let e p ∗ denote the isomorphism induced by p ∗ between the quotient space H ∗ ( F, F ′ ) / ker p ∗ and im p ∗ . Let e q ∗ denote the homomorphism induced by q ∗ between the quotientspace H ∗ ( F, F ′ ) / ker p ∗ and H ∗ ( Y, B ) /q ∗ (ker p ∗ ). The homomorphism ( F, F ′ ) ∗ in-duced in homology by ( F, F ′ ) is defined by( F, F ′ ) ∗ := e q ∗ ◦ e p ∗− : im p ∗ → H ∗ ( Y, B ) /q ∗ (ker p ∗ ) . Remark . If f : X → Y is a continuous map then p ∗ is an isomorphism, and thehomomorphism obtained by Definition 2.2 applied to f coincides with the homo-morphism induced in homology by the continuous map f in the usual sense.3. Characterizing homomorphisms induced in homology by acorrespondence
We begin by defining the notion of homological completeness and homologicalconsistency for correspondences, and by establishing their basic properties. Thenwe introduce the definition of the homomorphism induced in homology by a corre-spondence, and we prove our main result, Theorem 3.10.Our goal is to use a correspondence (
F, F ′ ) for which we can compute ( F, F ′ ) ∗ to determine the induced map on homology of a continuous function f . To do thisrequires and understanding of the information that is lost or preserved by e q ∗ and e p ∗− . We begin by considering p ∗ . Definition 3.1.
A closed correspondence (
F, F ′ ) from ( X, A ) to (
Y, B ) is homo-logically complete if the homomorphism p ∗ induced in homology by the naturalprojection p : ( F, F ′ ) → ( X, A ) is an epimorphism.
Lemma 3.2. If ( F, F ′ ) ⊂ ( G, G ′ ) are closed correspondences from ( X, A ) to ( Y, B ) ,and p F : ( F, F ′ ) → ( X, A ) and p G : ( G, G ′ ) → ( X, A ) are the natural projections,then im p F ∗ ⊂ im p G ∗ . NDUCING A MAP ON HOMOLOGY FROM A CORRESPONDENCE 5
Y XB A ( F, F ′ ) f pq Figure 2.
Notation used in the paper: (
F, F ′ ) is a correspondencefrom ( X, A ) to (
Y, B ), and f : ( X, A ) → ( Y, B ) is a continuousselector. The maps p : ( F, F ′ ) → ( X, A ) and q : ( F, F ′ ) → ( Y, B )are the canonical projections onto X and Y , respectively. Notethat given the correspondence ( F, F ′ ), in general p does not satisfythe acyclicity assumption of Theorem 1.1, but if ker p ∗ ⊂ ker q ∗ then f ∗ can be determined from the projections (see Theorem 3.10). Proof.
Since obviously p F = p G ◦ ι , where ι : ( F, F ′ ) → ( G, G ′ ) is the inclusion, thesame holds true after applying the homology functor: p F ∗ = p G ∗ ◦ ι ∗ . The conclusionfollows. (cid:3) Proposition 3.3.
Let ( F, F ′ ) ⊂ ( G, G ′ ) be closed correspondences from ( X, A ) to ( Y, B ) . If ( F, F ′ ) is homologically complete then so is ( G, G ′ ) .Proof. This result follows immediately from Lemma 3.2. (cid:3)
Remark . Since the projection from the graph of a continuous map onto the do-main of the map is a homeomorphism, a continuous map is homologically complete.By Remark 3.4 and Proposition 3.3, we immediately conclude the following:
Corollary 3.5.
If a closed correspondence ( F, F ′ ) has a continuous selector then ( F, F ′ ) is homologically complete. We now change our focus to q ∗ . Definition 3.6.
A closed correspondence (
F, F ′ ) from ( X, A ) to (
Y, B ) is homo-logically consistent if ker p ∗ ⊂ ker q ∗ , where p : ( F, F ′ ) → ( X, A ) and q : ( F, F ′ ) → ( Y, B ) are the natural projections.
Proposition 3.7.
Let ( F, F ′ ) ⊂ ( G, G ′ ) be closed correspondences from ( X, A ) to ( Y, B ) . If ( G, G ′ ) is homologically consistent then so is ( F, F ′ ) .Proof. Let p F , p G , q F , q G be the respective projections. Consider the inclusion ι : ( F, F ′ ) → ( G, G ′ ). Note that p F = p G ◦ ι and q F = q G ◦ ι . Take any c ∈ H ∗ ( F, F ′ ) S. HARKER, H. KOKUBU, K. MISCHAIKOW, AND P. PILARCZYK such that p F ∗ ( c ) = 0. Define c ′ := ι ∗ ( c ). Then p G ∗ ( c ′ ) = 0. Since ( G, G ′ ) is homo-logically consistent, q G ∗ ( c ′ ) = 0, and thus q F ∗ ( c ) = 0. Since the choice of c wasarbitrary, this reasoning shows that ( F, F ′ ) is homologically consistent. (cid:3) Remark . If p ∗ is an isomorphism then ( F, F ′ ) is obviously homologically consis-tent. In particular, if ( F, F ′ ) is a continuous map, or X and Y are compact metricspaces and ( F, F ′ ) is an acyclic upper semicontinuous multivalued map, then ( F, F ′ )is homologically consistent. However, it is not enough for a multivalued map to havea continuous selector to be homologically consistent (see Example 5.8). Corollary 3.9. If Y is compact and a closed correspondence ( F, F ′ ) from ( X, A ) to ( Y, B ) has an enlargement that is an acyclic upper semicontinuous multivaluedmap then ( F, F ′ ) is homologically consistent. The following result justifies the usefulness of Definition 2.2. See Figure 2 for asuggestive illustration of the set-up.
Theorem 3.10.
Let f : ( X, A ) → ( Y, B ) be a continuous selector of a closed corre-spondence ( F, F ′ ) from ( X, A ) to ( Y, B ) . If ( F, F ′ ) is homologically consistent then f ∗ = ( F, F ′ ) ∗ .Proof. By Corollary 3.5, the fact that (
F, F ′ ) has a continuous selector implies thatthe correspondence ( F, F ′ ) is homologically complete, and thus im p ∗ = H ∗ ( X, A ).Since (
F, F ′ ) is homologically consistent, ker p ∗ ⊂ ker q ∗ , and thus q ∗ (ker p ∗ ) = 0.Therefore, H ∗ ( Y ) /q ∗ (ker p ∗ ) ∼ = H ∗ ( Y ).Let us prove that f ∗ = e q ∗ ◦ ( e p ∗ ) − . Let j : ( f, f | A ) → ( F, F ′ ) denote the inclusion.Since ( f, f | A ) is homeomorphic with ( X, A ) by the natural projection p ◦ j , thishomeomorphism induces the isomorphism ( p ◦ j ) ∗ = p ∗ ◦ j ∗ in homology. Thefollowing diagram may help following the remaining part of the proof. H ∗ ( Y, B ) H ∗ ( f, f | A ) H ∗ ( F, F ′ ) H ∗ ( F, F ′ ) / ker p ∗ H ∗ ( X, A ) j ∗ ( q ◦ j ) ∗ ∼ =( p ◦ j ) ∗ πq ∗ p ∗ e q ∗ f p ∗ ∼ = f ∗ Let π : H ∗ ( F, F ′ ) → H ∗ ( F, F ′ ) / ker p ∗ denote the natural projection (which is anepimorphism). Then obviously p ∗ = e p ∗ ◦ π . As a consequence,( p ◦ j ) ∗ = p ∗ ◦ j ∗ = ( e p ∗ ◦ π ) ◦ j ∗ = e p ∗ ◦ ( π ◦ j ∗ ) , and thus ( e p ∗ ) − = ( π ◦ j ∗ ) ◦ (( p ◦ j ) ∗ ) − . The assumption that ker p ∗ ⊂ ker q ∗ implies that q ∗ = e q ∗ ◦ π (because ker π = ker p ∗ ).Therefore, e q ∗ ◦ ( e p ∗ ) − = e q ∗ ◦ ( π ◦ j ∗ ) ◦ (( p ◦ j ) ∗ ) − == (( e q ∗ ◦ π ) ◦ j ∗ ) ◦ (( p ◦ j ) ∗ ) − = ( q ∗ ◦ j ∗ ) ◦ (( p ◦ j ) ∗ ) − =( q ◦ j ) ∗ ◦ (( p ◦ j ) ∗ ) − , and this coincides with f ∗ (see Remark 2.3), which completes the proof. (cid:3) NDUCING A MAP ON HOMOLOGY FROM A CORRESPONDENCE 7 Extensions of a correspondence
Applications such as reconstructing the graph of a map from time series or usinga finite sample, possibly with some noise, to determine the graph of a map giverise to missing or perturbed data. Thus the observed correspondence (
F, F ′ ) maydiffer from the unknown true correspondence ( G, G ′ ). We show that it is possibleto retrieve some homological information about ( G, G ′ ) from ( F, F ′ ).4.1. Homological extension and homologically consistent enlargment.
Ahomological extension is meant to serve as a replacement of a correspondence thatextends the original one at the homological level. A homologically consistent en-largement is supposed to be a correction of a correspondence aimed at fixing theproblem of missing data. More specifically:
Definition 4.1.
Let (
F, F ′ ) and ( G, G ′ ) be closed correspondences from ( X, A ) to(
Y, B ). Let p F , p G , q F , and q G denote the respective projections. We say that( G, G ′ ) is a homological extension of ( F, F ′ ) if the following conditions hold:(1) im p F ∗ ⊂ im p G ∗ , which yields the natural inclusion homomorphism i : im p F ∗ → im p G ∗ ;(2) q G ∗ (ker p G ∗ ) ⊂ q F ∗ (ker p F ∗ ), which provides the natural projection homomor-phism j : H ∗ ( Y, B ) /q G ∗ (ker p G ∗ ) → H ∗ ( Y, B ) /q F ∗ (ker p F ∗ );(3) ( F, F ′ ) ∗ = j ◦ ( G, G ′ ) ∗ ◦ i .The name “extension” is motivated by the fact that the homomorphism j ◦ ( G, G ′ ) ∗ restricted to the domain of ( F, F ′ ) ∗ actually equals ( F, F ′ ) ∗ , so indeed( G, G ′ ) ∗ is an extension of ( F, F ′ ) ∗ up to an isomorphism. Moreover, the cor-respondence ( G, G ′ ) carries possibly more homological information than ( F, F ′ ).Indeed, ( F, F ′ ) ∗ can be recovered from ( G, G ′ ) ∗ by applying the maps i and j . Remark . The relation of being a homological extension defines a partial orderon closed correspondences from (
X, A ) to (
Y, B ).It is worth to mention that even if (
G, G ′ ) is a homological extension of ( F, F ′ )then it is not necessary that ( F, F ′ ) ⊂ ( G, G ′ ), nor that ( G, G ′ ) ⊂ ( F, F ′ ). However,a converse holds true to certain extent: Proposition 4.3.
A homologically consistent enlargement is a homological exten-sion.Proof.
Let (
G, G ′ ) be a homologically consistent enlargement of ( F, F ′ ). We showthat ( G, G ′ ) is a homological extension of ( F, F ′ ). For that purpose, we show thatconditions (1)–(3) of Definition 4.1 are satisfied. (1) By Lemma 3.2, im p F ∗ ⊂ im p G ∗ .(2) Since ( G, G ′ ) is homologically consistent, q G ∗ (ker p G ∗ ) = 0 ⊂ q F ∗ (ker p F ∗ ). (1)Suppose F ∗ ( x ) = y . Let ι : ( F, F ′ ) → ( G, G ′ ) be the inclusion map. We show( j ◦ ( G, G ′ ) ∗ ◦ i )( x ) = y . By definition of ( F, F ′ ) ∗ ( x ) = y , there exists c ∈ H ∗ ( F, F ′ )such that p F ∗ ( c ) = x and y = q F ∗ ( c ) + q F ∗ (ker p F ∗ ). Consider c ′ := ι ∗ ( c ) ∈ H ∗ ( G, G ′ ).By definition, ( G, G ′ ) ∗ ( p G ∗ ( c ′ )) = q G ∗ ( c ′ ) + q G ∗ (ker p G ∗ ) . S. HARKER, H. KOKUBU, K. MISCHAIKOW, AND P. PILARCZYK
Note that p F ∗ ( c ) = p G ∗ ( c ′ ) = x . Similarly, q F ∗ ( c ) = q G ∗ ( c ′ ). Hence( j ◦ ( G, G ′ ) ∗ ◦ i )( x ) = ( j ◦ ( G, G ′ ) ∗ )( x ) = j ( q G ∗ ( c ′ ) + q G ∗ (ker p G ∗ )) == j ( q F ∗ ( c ) + q G ∗ (ker p G ∗ )) = q F ∗ ( c ) + q G ∗ (ker p G ∗ ) + q F ∗ (ker q F ∗ ) . Since 0 = q G ∗ (ker p G ∗ ) ⊂ q F ∗ (ker q F ∗ ), we conclude that ( j ◦ ( G, G ′ ) ∗ ◦ i )( x ) = q F ∗ ( c ) + q F ∗ (ker q F ∗ ) = y , as desired. (cid:3) Remark . Proposition 4.3 shows that we can learn partial or total informationabout the homomorphism induced in homology by an unknown homologically con-sistent map even when some data is missing. In particular, if (
F, F ′ ) ∗ is nontrivialthen the same holds true for any of its homologically consistent enlargements. Remark . As an immediate consequence of Proposition 3.7, if there exists ahomologically consistent enlargement of a correspondence (
F, F ′ ) then ( F, F ′ ) ishomologically consistent. Note that by Proposition 3.3, homological completenesscarries over to enlargements; therefore, if ( F, F ′ ) is homologically complete then soare any of its homologically consistent enlargements.The concept of homologically consistent enlargements allows us to give the fol-lowing generalization of Theorem 3.10: Theorem 4.6.
Let f : ( X, A ) → ( Y, B ) be a continuous selector of a homologicallyconsistent enlargement ( G, G ′ ) of a closed correspondence ( F, F ′ ) from ( X, A ) to ( Y, B ) . If ( F, F ′ ) is homologically complete then f ∗ = ( F, F ′ ) ∗ .Proof. By Theorem 3.10, we have f ∗ = ( G, G ′ ) ∗ . Via Proposition 4.3, ( G, G ′ ) is ahomological extension of ( F, F ′ ). By homological completeness of ( F, F ′ ), the map i of Definition 4.1 is identity. By Proposition 3.7, ( F, F ′ ) is homologically consistent,and the map j of Definition 4.1 is identity as well. It follows from property (3) ofDefinition 4.1 that ( F, F ′ ) ∗ = ( G, G ′ ) ∗ = f ∗ . (cid:3) Existence of an acyclic enlargement.
We prove that if the images of aclosed correspondence are small enough then the correspondence has an acyclicenlargement, that is, an enlargement which is an acyclic multivalued map; see The-orem 4.13 below. The required size of the images is provided explicitly if we know aLebesgue number of a cover of Y that satisfies certain conditions. In particular, thisresult provides an explicit sufficient condition on how tight an approximation of anunknown continuous map must be in order to be sure that it provides meaningfulhomological information. Definition 4.7.
A collection U of closed subsets of a metric space Y is called a closed cover of Y if the interiors of the sets in U form an open cover of Y . Definition 4.8.
A closed cover U of a metric space Y is called a good closed cover of Y if the intersection of any finite collection of elements of U is either empty oracyclic. Remark . If U is a good closed cover of Y then any sub-cover V of Y (that is, aclosed cover V of Y such that V ⊂ U ) is a good closed cover of Y , too. In particular,if Y is compact and has a good closed cover U then it has a finite good closed cover.(The finite sub-cover of U that exists by the compactness of Y is a good choice.)A minor modification of the proof conducted in [2, pg. 42–43] for the case of anopen cover, allows us to claim the following. NDUCING A MAP ON HOMOLOGY FROM A CORRESPONDENCE 9
Proposition 4.10.
Every second countable Hausdorff manifold has a good closedcover.
Definition 4.11.
We say that a relation F ⊂ X × Y is inscribed into a closed cover U of Y if for every x ∈ X there exists U ∈ U such that F ( x ) ⊂ int U . We say thata correspondence ( F, F ′ ) is inscribed into a closed cover U of Y if F is inscribedinto U . (Note that then obviously F ′ ⊂ F is also inscribed into U .) Remark . If Y is a compact metric space then it follows from the Lebesgue’snumber lemma that if the diameters of F ( x ) are bounded by a Lebesgue’s numberof the open cover U ′ := { int U : U ∈ U} then F is inscribed into U . Theorem 4.13.
Let X and Y be compact metric spaces. Assume that Y has afinite good closed cover U . Let A ⊂ X be closed, and take B := S U ′ for some U ′ ⊂ U . Let ( F, F ′ ) be an upper semicontinuous multivalued map from ( X, A ) to ( Y, B ) inscribed into U . Then ( F, F ′ ) has an acyclic enlargement.Proof. Denote the elements of U by U , . . . , U r . For each i ∈ { , . . . , r } , define W i := { x ∈ X : F ( x ) ⊂ int U i } . Since F is inscribed into U , the collection { W , . . . , W r } is a cover of X . By semicontinuity, the sets W i are open. Define G ( x ) := T { U i ∈ U : x ∈ W i } and G ′ := G | A . Obviously, ( F, F ′ ) ⊂ ( G, G ′ ).Moreover, ( G, G ′ ) is acyclic, because U is a good closed cover. (cid:3) Remark . Let (
G, G ′ ) be an acyclic enlargement of ( F, F ′ ) in Theorem 4.13.Then by Vietoris-Begle Mapping Theorem 1.1, the projection p G : ( G, G ′ ) → ( X, A )induces an isomorphism in homology. Note that it follows that the correspondence(
G, G ′ ) is homologically consistent.5. Combinatorial maps and examples
When using computational methods for the determination of the homomorphisminduced in homology, one may represent a correspondence by means of a finitecombinatorial structure. We provide the essential definitions and then provide aseries of examples.5.1.
Preliminaries. A grid X for a compact set X is a finite collection of regularcompact subsets of X with disjoint interiors such that S X = S ξ ∈X ξ = X . The geometric realization of a set U ⊂ X is |U| := S U = S ξ ∈U ξ .Let X be a grid in X and let Y be a grid in Y . A combinatorial map is amultivalued map F : X −→→ Y , that is, a set-valued map F : X → Y , such that F ( ξ ) = ∅ for all ξ ∈ X . If A ⊂ X and
B ⊂ Y and F ( A ) := S {F ( ξ ) : ξ ∈ A} ⊂ B then we denote F : ( X , A ) −→→ ( Y , B ).The geometric realization of a multivalued map F : X −→→ Y is the relation F ⊂ X × Y defined as follows: ( x, y ) ∈ F if there exist ξ X ∈ X and ξ Y ∈ Y such that x ∈ ξ X , y ∈ ξ Y , and ξ Y ∈ F ( ξ X ). The geometric realization of a multivaluedmap F : ( X , A ) −→→ ( Y , B ) is the correspondence ( F, F ′ ) such that F is the geometricrealization of F : X −→→ Y and F ′ is the geometric realization of the multivalued map F ′ : A −→→ B , where F ′ ( ξ ) := F ( ξ ) for all ξ ∈ A . It is easy to see that the geometricrealization of a combinatorial map is an upper semicontinuous multivalued map. Definition 5.1.
A combinatorial map F : ( X , A ) −→→ ( Y , B ) is a representation of acontinuous map f : ( X, A ) → ( Y, B ), where X = |X | , A = |A| , Y = |Y| , B = |B| , if f ( ξ ) ⊂ |F ( ξ ) | for all ξ ∈ X . We remark that in many cases of practical interest, if an analytic description of f is given explicitly or implicitly then a representation of f is effectively computable.Furthermore, this representation can be chosen to approximate f sufficiently closelyto ensure that the assumptions of Theorem 4.13 are satisfied. A variety of efficienttools for this are provided by CAPD [6].To construct representations in the context of data we introduce the followingconcept. Definition 5.2.
Given a continuous map f : ( X, A ) → ( Y, B ), any finite subset S of the graph of f is called sampling data for f . Given grid representations( X , A ) and ( Y , B ) for ( X, A ) and (
Y, B ), respectively, define the combinatorial map F S : ( X , A ) −→→ ( Y , B ) induced by the samples S as follows: η ∈ F S ( ξ ) if and only if x ∈ | ξ | and f ( x ) ∈ | η | for some ( x, f ( x )) ∈ S .Assume that a combinatorial representation F : ( X , A ) −→→ ( Y , B ) of an unknowncontinuous map f is given and we are interested in the computation of f ∗ . Let( F, F ′ ) be the geometric realization of F . Then F is said to be homologically com-plete (respectively, consistent ) whenever ( F, F ′ ) is homologically complete (respec-tively, consistent). A combinatorial map G is a homologically consistent enlargementof F if the geometric realization of G is a homologically consistent enlargement ofthe geometric realization of F . Consider the projections p : ( F, F ′ ) → ( X, A ) and q : ( F, F ′ ) → ( Y, B ). If F is both homologically complete and homologically consis-tent, then we define(5.1) F ∗ := e q ∗ ◦ ( e p ∗ ) − , where e p ∗ and e q ∗ are the homomorphisms induced by p ∗ and q ∗ , respectively, on thequotient space H ∗ ( F, F ′ ) / ker p ∗ . Remark . The results in Section 4.1 transparently carry over to combinatorialmaps by applying them to their geometric realizations.
Remark . Observe that (5.1) is independent of the method chosen to compute p ∗ and q ∗ . A variety of different techniques to perform these computations can befound in [10, 14, 15].Returning to the setting of data, we have the following result. Proposition 5.5.
Let f : ( X, A ) → ( Y, B ) be a continuous map admitting a homog-ically consistent representation F : ( X , A ) −→→ ( Y , B ) . Let S be some sampling datafor f . Let F S be the combinatorial map induced by the samples S . Then F S ishomologically consistent. Moreover, if F S is homologically complete then f ∗ = F S ∗ .Proof. By Definition 5.2, F S ⊂ F . Then F S is homologically consistent by Propo-sition 3.7. The remaining result is an immediate consequence of Theorem 4.6. (cid:3) Remark . In fact, we can do somewhat better than Proposition 5.5 and tolerate noisy samples . A sufficient condition is that there exists a homologically consistentcorrespondence containing both f and the noisy samples. Given known bounds onthe noise, it may be possible to give a general result of this form using ideas alongthe lines of Section 4.2. We do not attempt this here, but please see the discussionin Example 5.7. NDUCING A MAP ON HOMOLOGY FROM A CORRESPONDENCE 11
Applications and Examples.
In what follows, we provide three specific ex-amples which illustrate the importance of homological consistency and homologicalcompleteness. We give an example satisfying homological completeness and ho-mological consistency, an example where homological consistency fails, and an ex-ample where homological completeness fails. To make the computations discussedin Examples 5.7 and 5.9 reproducible, the software used for these computations isavailable at [9].
Example . Our first example illustrates a successful application of Theorem 4.6.Consider a sampling from the a double winding map f (cos θ, sin θ ) = (cos 2 θ, sin 2 θ ) . Rather than using Proposition 5.5, we want to demonstrate the full strength ofTheorem 4.6. To this end we generated a combinatorial map F : X −→→ X by com-puting samples of f but adding a small level of random noise. The noise resultsin f not being a continuous selector of F . Nevertheless, as we demonstrate theconditions of Theorem 4.6 are satisfied and thus F can be used to obtain f ∗ .To generate our example we proceed as follows. We divide the plane region[ − , into 256 ×
256 grid elements and choose X to be those grid elements within .
01 distance (under the sup-norm) to the unit circle. We generate 3 ,
000 samples onthe unit circle of the plane, evaluate f on each sample point, add bounded noiseuniformly distributed in [ − . , . , and radially project the result to the unit circle.From these noisy samples of f we constructed a combinatorial map F . We checkcomputationally that (i) F does not contain f and that (ii) F is homologically com-plete. The fact that F admits a homologically consistent enlargement containing f follows from the results of Section 4.2, an appropriate choice of good cover, andthe observation that the noise was small. We omit the details, but remark that thisimplies that Theorem 4.6 is applicable.As a check we compute the homomorphism induced in homology by the closedcorrespondence ( F, F ′ ) corresponding to F and obtain H ∗ ( F, F ′ ) ∼ = ( Z , Z , , . . . ).Computing the induced map on homology of the projections p and q , we find that p ∗ ( g ) = q ∗ ( g ) = 0 for all but one 0-cycle basis element, where we had p ∗ ( g ) = q ∗ ( g )and all but one 1-cycle basis element, where we found q ∗ ( g ) = 2 p ∗ ( g ). Hence F ∗ = f ∗ , the induced map on homology of a double winding map, in accordancewith Theorem 4.6. Example . Our second example illustrates a very simple situation that showsthat if (
F, F ′ ) is not homologically consistent (that is, ker p ∗ ker q ∗ ), then thismay lead to ambiguity in determining f ∗ from F . Consider the following objects(see Figure 3): X := Y := { [1 , , [2 , , [3 , } , A := B := { [1 , , [3 , } , F ([1 , { [1 , , [3 , } , F ([2 , { [1 , , [2 , , [3 , } , F ([3 , { [3 , } . Let (
F, F ′ ) be the geometric realization of F : ( X , A ) −→→ ( Y , B ). Then H ( X, A ) ∼ = R and H ( F, F ′ ) ∼ = R ⊕ R , and the homology of these spaces at the levels differentfrom 1 is trivial. Let u be a generator of H ( X, A ) = H ( Y, B ) corresponding fg v wu u Figure 3.
A sample representation F of two continuous maps f and g such that f ∗ = g ∗ , described in Example 5.8. The homologygenerators discussed in the text correspond to the segments labeled u , v and w . Note that ker p ∗ = h v i 6⊂ h w i = ker q ∗ .to the segment [2 , v , w of H ( F, F ′ ) so that v correspondsto { } × [2 ,
3] and w corresponds to [2 , × { } (see Figure 3). Then p ( v ) =0 and p ( w ) = u , and thus ker p = h v i . On the other hand, q ( v ) = u and q ( w ) = 0, so ker q = h w i . Obviously, ker p ∗ ker q ∗ . Indeed, it is possible toshow two continuous maps such that f ( u ) = u and g ( u ) = 0; in particular, f ∗ is a nontrivial homomorphism, and g ∗ is trivial. Graphs of such sample maps areshown in Figure 3. Example . For our third example, we consider a combinatorial map which ishomologically consistent but not homologically complete. We analyze time seriesgenerated by the H´enon map(5.2) ( x, y ) (1 − ax + y, bx )at the classical parameter values a = 1 . b = 0 .
3. We produce a neighborhoodof the attractor from a time series generated of the form ( x n ) Nn =100 , N = 100 , x = (0 , − , into 256 ×
256 grid elements. We set X to be the set of grid elements thatcontain at least one of the points from the time-series and set A = ∅ . See Figure 4.The combinatorial map F is defined by the property that η ∈ F ( ξ ) whenever x i ∈ | ξ | and x i +1 ∈ | η | for some i ≥ F, F ′ ) associated with F , the homology of theplanar set X corresponding to X , and the induced maps on homology p ∗ and q ∗ for the projections p, q from ( F, F ′ ) to ( X, A ). (Note: F ′ = A = ∅ .) We findthat H ∗ ( F, F ′ ) = ( Z , Z , , , · · · ) and H ∗ ( X, A ) = ( Z , Z , , , · · · ). The specificentries of the matrices p ∗ and q ∗ is of limited importance so we only report thefollowing information: (i) we obtain homological consistency, and (ii) on the firstlevel p ∗ is rank deficient; in particular rank p = 15 <
18. Since we see that p isnot surjective, we conclude that F is not homologically complete.It is worth exploring this example further. As is indicated in Henon’s originalwork [11] at the chosen parameter values (5.2) is invertible and there exists a sim-ply connected polygonal trapping region. Thus, at first glance it is reasonable toexpect that the neighborhood of the attractor characterized by X should be simplyconnected. In particular, the 18 1-cycles of H ( X, A ) (these are visible in Figure
NDUCING A MAP ON HOMOLOGY FROM A CORRESPONDENCE 13
Figure 4.
The grid X computed in the planar region [ − , toouter approximate the time series points approaching the H´enonattractor. The bounds of the image are [ − . , . × [ − . , . X are gray. Note the 1-cycles produced by theover-approximation.4 as interior holes of the gray region) are ‘artifacts’ due to the outer approxima-tion via the cubical grid X . The map p ∗ is not surjective since for at least a fewof these artifacts there is no corresponding cycle in the complex corresponding tothe combinatorial map F . We can intuitively understand how this happens. Anartifact occurs in H ∗ ( X, A ) when two disconnected parts of the H´enon attractorare present in the same grid element and allow a spurious cycle to form. Howeverif those disconnected parts of the attractor map to sufficiently separated pointsthen we will not see a corresponding spurious cycle formed in the combinatorialmap. We might think to eliminate these artifacts by refining our grid, but due tothe fractal-like nature of the H´enon attractor one expects to compute artifactualhomology at every level of resolution. Thus a lack of homological completeness is ageneral phenomenon that should be expected when analyzing time series on strangeattractors.An interesting idea is to use the induced maps on homology of projections fromthe combinatorial map to clean up at least some of the artifacts in the homology ofan outer approximation of an attractor. Once we do this, we replace H ∗ ( X, A ) withim p ∗ and recover homological completeness. In the current example, doing thisproduces a map q ∗ ◦ p − ∗ . We checked and found that this map was nilpotent. Thisagrees with what we expect, as Conley theory tells us that the Conley index of asimply connected attractor on the 1st level ought to be shift equivalent to the zeromatrix. It is not clear what the conditions must be in order to rigorously justifythis procedure; we leave the precise development of this idea to future work. Acknowledgements
The authors gratefully acknowledge the support of the Lorenz Center whichprovided an opportunity for us to discuss in depth the work of this paper. Re-search leading to these results has received funding from Fundo Europeu de Desen-volvimento Regional (FEDER) through COMPETE—Programa Operacional Fac-tores de Competitividade (POFC) and from the Portuguese national funds throughFunda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) in the framework of the researchproject FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008), as well as from the People Programme (Marie Curie Actions) of the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) under REA grant agree-ment no. 622033 (supporting PP). The work of KM and SH has been partiallysupported by NSF grants NSF-DMS-0835621, 0915019, 1125174, 1248071, and con-tracts from AFOSR and DARPA. The work of HK was supported by Grant-in-Aidfor Scientific Research (No. 25287029), Ministry of Education, Science, Technology,Culture and Sports, Japan.
References
1. Zin Arai, William Kalies, Hiroshi Kokubu, Konstantin Mischaikow, Hiroe Oka, and Pawe lPilarczyk,
A database schema for the analysis of global dynamics of multiparameter systems ,SIAM J. Appl. Dyn. Syst. (2009), no. 3, 757–789. MR 2533624 (2011c:37034)2. Raoul Bott and Loring W. Tu, Differential forms in algebraic topology , Graduate Texts inMathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304 (83i:57016)3. Felix E. Browder,
The fixed point theory of multi-valued mappings in topological vector spaces ,Math. Ann. (1968), 283–301. MR 0229101 (37
Combinatorial-topological framework for the analysis of globaldynamics , CHAOS (2012), no. 4, 047508.5. Justin Bush and Konstantin Mischaikow, Coarse dynamics for coarse modeling: An examplefrom population biology , Entropy (2014), no. 6, 3379–3400.6. CAPD, Computer Assisted Proofs in Dynamics , http://capd.ii.uj.edu.pl/.7. CHomP,
Computational Homology Project. Homology Software , http://chomp.rutgers.edu/Software/Homology.html.8. Samuel Eilenberg and Deane Montgomery,
Fixed point theorems for multi-valued transforma-tions , Amer. J. Math. (1946), 214–222. MR 0016676 (8,51a)9. Shaun Harker, Generalized homology of maps. Supplemental materials , http://chomp.rutgers.edu/Archives/Computational Homology/GeneralizedHomologyOfMaps/SupplementalMaterials.html.10. Shaun Harker, Konstantin Mischaikow, Marian Mrozek, and Vidit Nanda,
Discrete Morsetheoretic algorithms for computing homology of complexes and maps , Found. Comput. Math. (2014), no. 1, 151–184. MR 316071011. M. H´enon, A two-dimensional mapping with a strange attractor , Communications in Mathe-matical Physics (1976), no. 1, 69–77.12. Tomasz Kaczynski, Konstantin Mischaikow, and Marian Mrozek, Computational homology ,Applied Mathematical Sciences, vol. 157, Springer-Verlag, New York, 2004. MR 2028588(2005g:55001)13. Konstantin Mischaikow and Marian Mrozek,
Conley index , Handbook of dynamical systems,Vol. 2, North-Holland, Amsterdam, 2002, pp. 393–460. MR 1901060 (2003g:37022)14. Konstantin Mischaikow, Marian Mrozek, and Pawe l Pilarczyk,