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 = {Uα} 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 ∈ X is said to be related to a point y ∈ Y if and only if (x, y) ∈ 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 ∈ 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 ∈ 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) ∈ 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. Remember that geometrically, we represent a simplex by taking the convex hull of its vertices. Using this, we approximate the topological space X using simplicial homology by taking a discrete set X˜ ⊆ X and using a triangulation T of this set. Using this method you obtain a simplicial complex. It is much easier to compute areas on simplicial complexes.
5 4.2.2 Relative Homology
Through the use of relative homology, computation can be limited to only paths that have the same beginning and end point, drastically reducing the number of computations to be made. These paths constitute loops and the homology of each of these loops can be calculated to check for obstacles. If there is a hole, then the distance between the two paths is deemed infinite. Otherwise, the distance is just regularly calculated. In the figure below, we can see the representation of the quotient map on a simplicial complex. The red area, which are the ”goal regions” are compressed into a single point.
Figure 4: Transforming the map into a quotient space using relative homology
4.2.3 Results
By using this method, not only have they managed to find a way to make path clustering better, but computing it is much more efficient than it used to be:
5 Sensor Network Coverage
Imagine a future in which physical surroundings ”wakes up”, being endowed with sensory data, networked and responsive. This is one of the goals of the rapidly-developing field of pervasive sensor networks. The ability to fabricate increasingly small sensing devices, along with the computational scaling implicit in Moore’s Law and the parallel advances in wireless technology, foretell a world in which walls, furniture, roads, vehicles, grocery bags, and clothes are seemingly alive with data.
5.1 Problem we’re trying to solve
Given a collection of nodes X in a bounded domain D of the plane, assume that each node can sense, broadcast to, or otherwise cover a region of fixed coverage radius about the node. The most basic form of coverage problem is the simple query: given the nodes, does the collection of coverage discs at X cover the domain D?
6 If the nodes have no knowledge of their coordinate locations or of the locations of their neighbors, how can they determine whether there are gaps in coverage? Imagine a cocktail party full of people, blindfolded, and constrained to do nothing but whisper their identities and listen for the names of their immediate neighbors (with one ear, to prevent bearing data). Can such a crowd determine the topology or geometry of the party?
What is the appropriate mathematics for sensor networks? Most of the problems in networked sensing are of the local-to-global variety. As sensors shrink in size and increase in multiplicity, one goes from having a small number of “global” sensors to a dense array of “local” sensors. Sensing problems are increasingly about integrating localized data into a global picture of the environment. Among the many applicable tools that mathematicians have discovered, one branch of mathematics is outstanding in its fine-tuned ability to turn local data into global data: algebraic topology.
5.2 Assumptions
Some assumptions on the sensor modality and the communication protocols must be made. Weak assumptions would be as follows. Assume nodes lie in the plane, at unknown locations. Assume that any collection of K nodes that are in pairwise communication have coverage regions which contain the convex hull of these points in the plane. So, if three nodes communicate pairwise, then the abstract triangle they form in the plane lies within the coverage of the network. These are reasonable assumptions for a stable network where communication links are based on proximity.
Note while not knowing where the location of immobile the node may not seem like a common problem, it actually is. Many sensors have gotten so small that once they are placed they often moved because of external factors such as weather.
More formally we have the following assumptions for our example:
We assume a complete absence of localization capabilities. Nodes can determine neither distance nor direction. Only connectivity data between nodes is used. The only strong assumption we make is on the fence nodes set up along the boundary of the domain. This strong degree of control along the boundary is not strictly required, but it simplifies the statements and proofs of theorems dramatically • A1: Nodes χ broadcast their unique ID numbers. Each node can detect the identity of any node within broadcast radius rb. √ • A2: Nodes have radially symmetric covering domains of cover radius rc ≥ rb/ 3. 2 • A3: Nodes χ lie in a compact connected domain D ⊂ R whose boundary ∂D is connected and piecewise-linear with vertices marked fence nodes χf .
• A4: Each fence node v ∈ χf knows the identities of its neighbors on ∂D and these neighbors both lie within distance rb of v.
To summarize, the sensor data for each node consists of a list of node ID numbers within signal detection range, as well as a binary flag denoting whether or not it is a marked fence node.
7 We claim that, surprisingly, such coarse coordinate-free data is sufficient to rigorously verify coverage in many instances.
5.3 Communication graph and Rips Complex
Theorem (The Cechˇ Theorem). If the sets {Uα} and all nonempty finite intersections are con- tractible, then the union ∪αUα has the homotopy type of the Cechˇ complex C.
This would appear to be exactly what one wants for sensor networks. Unfortunately, it is highly nontrivial to compute the Cechˇ complex from the network graph. We have two radii to contend with: the broadcast radius rb and the coverage radius rc. For no (physically realistic) choice of these radii can the radius rc Cechˇ complex be derived from the radius rb network graph.
Figure 5: Changing the positions of nodes can change the topology of the radius rc cover without changing the radius rb network graph.
On the other hand, with the bound on coverage and broadcast radii in A2, it follows that for any triple of nodes which are in pairwise communication distance, the convex hull of these nodes 2 in R is contained in the cover. The limiting case, in which all three nodes are at pairwise distance 2 rb yields√ an equilateral triangle in R which is covered by balls at the nodes of radius rc only if rc ≥ rb/ 3.
This motivates the following construction. We consider the network graph as the 1-dimensional skeleton of a larger simplicial complex. Denote by R the largest simplicial complex whose 1-skeleton is the network graph. That is, for every collection of k nodes which are pairwise within distance rb, we assign an abstract k − 1 simplex. By assumption A4 the boundary ∂D can be represented as a 1-dimensional fence cycle F ⊂ R which is canonically identified with ∂D. This construction is equivalent to the Rips complex.
Definition of Rips Complex Given a set of points χ = {xα} in a metric space and a fixed > 0, the Rips complex of Xf ,R(X), is the abstract simplicial complex whose k-simplices correspond to unordered (k + 1)-tuples of points in X which are pairwise within distance of each other.
2 Unfortunately, the radius-rb Rips complex of a set of nodes in R does not always capture the topology of the union of radius-rc balls centered on these nodes.
8 Figure 6: The coverage criterion is an algebraic-topological formulation of the intuition of ‘filling in’ the fence cycle F of the communication graph [left] with 2-simplices of the Rips complex R [center] so as to triangulate the domain D [right].
5.4 Homological Criterion for Coverage
The intuition behind the coverage criterion is very straightforward. Based on the communication graph alone, it is difficult to ‘see’ potential holes in coverage. However, upon completing the graph to the Rips complex R, potential large holes in coverage would show up in the abstract complex. One might guess that showing there are no such holes in R implies coverage. This condition would be translated into algebraic topological terms as H1(R) = 0, or, that any cycle in the communication graph can be realized as the boundary of a surface built from 2-simplices of R, each of which indicates a coverage region thanks to assumption A2.
Figure 7: In a sensor network with a sufficiently large hole in coverage [left], the communication graph [center] has a cycle that cannot be ‘filled in’ by triangles. The filled in Rips complex [right] ‘sees’ this hole, even as an abstract complex devoid of sensor node location data.
Main Criterion We use a slightly different criterion than H1(R) = 0: one which is more robust to extensions and which yields stronger information about the actual cover. The union of the radius rc discs contains D if there is a element of the relative homology H2(R,F ) whose boundary is nonvanishing.
Definition of relative cycles
• Elements of Hn(X,A) are represented by relative cycles: n-chains α ∈ Cn(X) such that ∂α ∈ Cn−1(A)
• A relative cycle α is trivial in Hn(X,A) iff it is a relative boundary: α = ∂β + γ for some β ∈ Cn+1(X) and γ ∈ Cn(A) • Further example (Adam already did some): Cylinder and disk
9 •
Intuitively, 2-chain α has the appearance of “filling in” D with triangles composed of projected 2-simplices from R.
Figure 8: The coverage criterion is an algebraic-topological formulation of the intuition of ‘filling in’ the fence cycle F of the communication graph [left] with 2-simplices of the Rips complex R [center] so as to triangulate the domain D [right].
Note that relative homology H2(R,F ) captures the second homology of the quotient space R/F , in which all simplices in F are identified. This can be done by adding a “super node” to the complex. If the Rips complex is hole-free, then the topology of this quotient space is that of a sphere, and therefore, the second relative homology H2(R,F ) has a non-trivial generator. On the other hand, if the 1-cycles defined over subcomplex F are not boundaries of any 2-chain, then the relative homology has no generator with non-zero values on the boundary.
Figure 9: If the first homology of R is non-trivial, then the second relative homology H2(R,F ) has no generator with values on the boundary. Conversely, if the second homology relative to the boundary has a non-trivial generator with a non-vanishing boundary, then H1(R) = 0
10 Note that the dimension of the second relative homology H2(R,F ) may be greater than one. This can happen if there exists a 2-cycle which is a generator of H2(R) as well as H2(R,F ). Such 2-cycles do not represent a true relative class, as they may still exist even if the fence cycle F is not the boundary of any 2-chain. Hence, the coverage criterion requires the existence of a relative 2-cycle alpha with a non-zero boundary.
Figure 10: Eight faces of the octahedron form a non-trivial 2-cycle α such that [α] ∈ H2(R). However, α has a vanishing boundary, and does not correspond to a true relative 2-cycle.
5.4.1 Not a Sharp Criterion
This is not a sharp criterion. It is clearly possible to have the criterion always fail for injudicious choice of rc. For example, if rc is much larger than the bound in Assumption (A2), then there will be many instances of coverage without a homological forcing. This being said, we note that even if one chooses the minimal acceptable bounds from Assumption (A2), it is still possible to arrange the points to cover D − C without the homological criterion detecting this.
Figure 11: Examples of two covers. The homological criterion holds for one [left] but not for the other [center], because of a 1-cycle in R [right]. Note the fragility of the cover [center] within the 1-cycle: a small perturbation of the nodes creates a hole.
5.5 What if the nodes move?
Here it makes more sens to not verify for complete coverage at every time t but verify ”dynamic coverage”. Intuitively one can think of dynamic coverage as: ”could someone move through the domain without being sensed?” or is there an evasion path?.
The algebraic tools to verify this criterion are more complicated but use the same tools: Rips complex and homology.
Here are some of the steps to verify dynamic coverage:
11 • Since nodes are moving, take a ”snapshot” of the moving sensors at different time periods. We can them form Rips complexes for each snapshot. • Now we glue in prisms along corresponding triangles on the complexes. This is called the Stacked Rips complex denoted SR. We can then look at the boundary of each complex and identically glue together each triangle in successive slices, this is called Stacked Fence complex denoted SF • We then apply the Homology information of SR and SF to the moving sensors in order to determine the coverage has occured within the time iterval.
Figure 12: Subsequent Rips complexes [left] are attached via prisms between matching simplices [center] to capture the topology of the mobile cover [right].
5.5.1 Power-Saving and computation
The coverage criterion guarantees that the covering discs in fact cover the desired area. For reasons of power conservation, one would like to know which nodes could be “turned off” without impinging upon the coverage integrity.
Given the above, it is easy to see that the minimal cover is simply the sparsest generator of the second homology class of R relative to F . Therefore, one can formulate the problem of finding the sparsest cover over D as an optimization problem,simply by extending the results of the previous section to a higher dimension.
The algorithm used to compute the sparsest generator uses techniques of persistent homology. We won’t go over the specific math of it but we will mention the tool used in this case because we will look at it in future classes.
Definition Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters.To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets.
12 Figure 13: A typical simulation: [top] the locations of 212 nodes in D; [center] the image of the Rips complex R projected to D; [bottom] a simple generator of H2(R,F ) extracts 101 nodes which are guaranteed to cover D, leaving 111 nodes to be safely put into sleep mode.
5.5.2 Cool application of Persistent Homology: Neuro-science
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