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Class 8:ˇcech Homology, Homology of Relations, Relative Homology

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Class 8:ˇcech Homology, Homology of Relations, Relative Homology Class 8: Cechˇ Homology, Homology of Relations, Relative Homology & Their Applications Undergraduate Math Seminar - Elementary Applied Topology Columbia University - Spring 2019 Adam Wei & Zachary Rotman 1 Cechˇ Homology Computing homologies is computationally intensive, and a way of computing the boundary of a complex easily is through Cechˇ Homology. Let X be a topological space and U = fUαg a locally-finite collection of open sets whose union is X. The Cechˇ complex of U is the chain complex C(U) = (Cn;@) generated by non-empty intersections of the cover U. This is also known as the nerve of the cover. The nerve of the cover is generated by associating to each open cover a vertex and associating an edge between these two vertices if there is an intersection between the two open covers. Figure 1: Cechˇ complex built from the nerve of three intersecting covers The nerve of a set of open covers is a simplicial complex and therefore we can compute homologies on this complex. The boundary operator for this chain complex is given by: k X i @(Uj) = (−1) UDi;j i=0 where Di is the face map that removes the i-th vertex from the list of vertices. Furthermore, there will also exist a homological equivalent to the Nerve Lemma. Reminder, the Nerve Lemma says that for if U is a finite collection of open contractible subsets of X with all non-empty intersections of sub-collections of U contractible, then N(U) is homotopic to [αUα. Theorem 1. If all non-empty intersections of elements of U are acyclic (H~n(Uj) = 0 for all non- empty Uj, you can also think about this as the cycles inn these intersections must always be boundaries), then the Cechˇ homology of U agrees with the singular homology of X. 1.1 Example Suppose we want to study the homology of the topological space X which is simply a disc. Instead of studying X directly, we can look at the Cechˇ complex built from the nerve of the covers. Since we see that all the intersections are acrylic, this means that we can apply the homologic equivalent ∼ to the Nerve Lemma, which means that Hn(U) = Hn(X) 2 Homology of Relations Definition 1. A relation between two sets X and Y is a subset R ⊂ X × Y . A point x 2 X is said to be related to a point y 2 Y if and only if (x; y) 2 R We can build homologies for these relations. We can define the chain complexes as follows: Ck(X; R) has as basis unordered (k + 1)-tuples of points in X that are all related to certain points in Y . You can then build a dual complex for C(Y; R) from columns of R-unordered tuples of points of Y related to some fixed x 2 X. The boundary operator of these two complexes mimics the boundary operator of a simplicial complex. ∼ Theorem 2. Dowker's Theorem: Hn(R) := Hn(X; R) = Hn(Y; R) This result was originally applied to prove an equivalence between Cechˇ and Vietoris homology theories, in the case that X is a metric space, Y is a dense net of points in X, and R records which points in X are within an arbitrary distance of points in Y . Instead of studying the homology directly, we can study the Dowker Complexes instead. These complexes, RX and RY , are defined by the following process. Rx is the nerve complex of the cover of Y by columns of R and Ry is the nerve complex of the cover of X by rows of R. We can also think of the simplex of Rx as the collection of points of X witnessed by some common y 2 Y via the relation R. The set of vertices of Rx would then only be X. This is built in the same way as a witness complex where the points being witnessed are the landmarks points and the point cloud are the others. 2.1 Example: Transmitters and Receivers Let X be a finite set of transmitters and Y a finite set of receivers. The transmitters in X broadcast their identities in an unspecified domain D. The resulting system of transmitters and receivers can be encoded in a relation R ⊂ X × Y where (xj; yj) 2 R if and only if a device yj receives a signal from device xj. The common domain D in this case can be approximated using a Dowker Complex. 2 Figure 2: Witness Complex. Points in black are the landmark points, points in white are part of the point cloud. 3 Relative Homology In relative homology groups, we introduce the idea that homology makes sense for spaces AND subspaces. 3.1 Exact Sequences Definition 2. Suppose we have the following sequence: @n+1 @n ···! An+1 −−−! An −! An−1 !··· This sequence is exact if we have Ker@i = Im@i+1 We can also relate the concept of exact sequences to some other basic algebraic notions: • Injection: Consider the sequence 0 ! A −!@ B. If @ is injective () Ker@ = 0 then the sequence is exact • Surjection: Consider the sequence A −!@ B ! 0. If @ is surjective () Im@ = B, then the sequence is exact. • Isomorphism: Consider the sequence 0 ! A −!@ B ! 0. If @ is an isomorphism, then the sequence is exact. Using the notions above, consider the following sequence: 0 0 ! A −!@ B −!@ C ! 0 If this sequence is exact, then it is called the Short Exact Sequence. The following needs to be true in order for this sequence to be exact: (1) @ is injective, (2) @0 is surjective, (3) Ker@0 = Im@ so that @0 induces an isomorphism C ≈ B=Im@, which can also be written as C ≈ B=A. 3 3.2 Relative Homology Groups Let X be a complex and A be a sub-complex of X. Consider the following sequence: 0 ! A ! X ! X=A ! 0 We want to look at the homology group relations between Hn(A), Hn(X) and Hn(X=A). Given a chain complex (Cn;@n), we define a subcomplex (An;@n) where for each n, An is a subspace of Cn and @n is the boundary map of (Cn;@n) restricted to An. From such a subcomplex, 0 0 we can define a relative chain complex (Dn;@n) as Dn = Cn=An and @n as the map induced by @n on the quotient. Now, taking the uotient corresponds to ignoring all topological features in A, and hence we expect to see a relationship between the relative complex and the quotient X=A, which will be made explicit with the following theorem: Theorem 3. If X is a space and A is a closed non-empty subspace that is a deformation retract of some neighborhood in X, then there is an exact sequence: @ @ ··· −! Hn(A) ! Hn(X) ! Hn(X; A) −! Hn−1(A) !···! H0(X; A) ! 0 This, paired with the excision theorems allow us to calculate homologies much more easily. Theorem 4. Let U ⊂ A ⊂ X such that the closure of U is contained within the interior of A. ∼ Then Hn(X − U; A − U) = Hn(X; A). This implies that for the pair (X; A) what happens inside of A is irrelevant. It then follows that this is true: ∼ Corollary 1. For A ⊂ X a closed sub-complex of a cell complex X, Hn(X; A) = Hn(X=A) 3.2.1 Example n n Consider the homology of a closed disc D relative to its boundary @D . This homology has n n n ∼ n a non trivial generator D as a relative n-cycle. Because we know that D =@D = S , using the n n ∼ n corollary above, we find that Hn(D ;@D ) = Hn(S ) 4 An Application of Relative Homology: Path Clustering Path clustering consists of partitioning sets of paths into smaller subsets of similar paths. Using these sets of paths, and combining it with a set of learned motion primitives (pre-computed motions that a robot can take), we can control a robot without preprogramming it. 4 4.1 Traditional Path Clustering (DTW Path Clustering) Traditionally, path clustering is done by looking at data points in a fixed dimension vector space. Clustering is usually done by comparing distances between paths and applying a clustering parameter based only on distance. This is usually done through a process called Dynamic Time Warping (DTW). How this usually works, is that for a given path, it will assign a slow in a matrix for each point of the path and compare to every single other point in the path in order to find the best match. This is extremely computational intensive since this must be repeat for all possible paths on the map. This process is also used in voice identification for example. Figure 3: Map modeled by a simplicial complex. Red region is the goal region, blue and red are paths. In traditional path clustering, the distance is calculated by calculating the distance between these two paths. The problem with traditional path clustering is that it does not take into account the topology of a space when calculating the distance between two paths and they are usually very expensive to compute and impossible to scale onto larger data sets. 4.2 Path Clustering with Relative Homology Path Clustering can also be done using a relative homology approach. In this approach, the distance between the paths is modeled by the area between the two paths instead. 4.2.1 Simplicial Homology n Given a topological space X ⊆ R , we need to be able to run calculations on this topological space.
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