Class 8: Cech Homology, Homology of Relations,Relative Homology & Their Applications

Undergraduate Math Seminar - Elementary Applied TopologyColumbia University - Spring 2019

Adam Wei & Zachary Rotman

1 Cech Homology

Computing homologies is computationally intensive, and a way of computing the boundary of acomplex easily is through Cech Homology.

Let X be a topological space and U = {Uα} a locally-finite collection of open sets whose unionis X. The Cech complex of U is the chain complex C(U) = (Cn, ∂) generated by non-emptyintersections of the cover U . This is also known as the nerve of the cover. The nerve of the coveris generated by associating to each open cover a vertex and associating an edge between these twovertices if there is an intersection between the two open covers.

Figure 1: Cech 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 homologieson this complex. The boundary operator for this chain complex is given by:

∂(Uj) =k∑i=0

(−1)iUDi,j

where Di is the face map that removes the i-th vertex from the list of vertices. Furthermore, therewill also exist a homological equivalent to the Nerve Lemma. Reminder, the Nerve Lemma saysthat for if U is a finite collection of open contractible subsets of X with all non-empty intersectionsof 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 (Hn(Uj) = 0 for all non-empty Uj , you can also think about this as the cycles inn these intersections must always beboundaries), then the Cech 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. Insteadof studying X directly, we can look at the Cech complex built from the nerve of the covers. Sincewe see that all the intersections are acrylic, this means that we can apply the homologic equivalentto 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 issaid 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 pointsin Y . You can then build a dual complex for C(Y,R) from columns of R-unordered tuples of pointsof Y related to some fixed x ∈ X. The boundary operator of these two complexes mimics theboundary 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 Cech and Vietoris homologytheories, in the case that X is a metric space, Y is a dense net of points in X, and R records whichpoints in X are within an arbitrary distance of points in Y .

Instead of studying the homology directly, we can study the Dowker Complexes instead. Thesecomplexes, RX and RY , are defined by the following process. Rx is the nerve complex of the coverof 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 commony ∈ Y via the relation R. The set of vertices of Rx would then only be X. This is built in the sameway as a witness complex where the points being witnessed are the landmarks points and the pointcloud 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 Xbroadcast their identities in an unspecified domain D. The resulting system of transmitters andreceivers can be encoded in a relation R ⊂ X × Y where (xj , yj) ∈ R if and only if a device yjreceives a signal from device xj . The common domain D in this case can be approximated using aDowker Complex.

2

Figure 2: Witness Complex. Points in black are the landmark points, points in white are part ofthe point cloud.

3 Relative Homology

In relative homology groups, we introduce the idea that homology makes sense for spaces ANDsubspaces.

3.1 Exact Sequences

Definition 2. Suppose we have the following sequence:

· · · → An+1∂n+1−−−→ An

∂n−→ 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→ A∂−→ B

∂′−→ C → 0

If this sequence is exact, then it is called the Short Exact Sequence. The following needs to betrue in order for this sequence to be exact: (1) ∂ is injective, (2) ∂′ is surjective, (3) Ker∂′ = Im∂so that ∂′ 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 asubspace of Cn and ∂n is the boundary map of (Cn, ∂n) restricted to An. From such a subcomplex,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, andhence we expect to see a relationship between the relative complex and the quotient X/A, whichwill 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 retractof 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 ofA 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

Consider the homology of a closed disc Dn relative to its boundary ∂Dn. This homology hasa non trivial generator Dn as a relative n-cycle. Because we know that Dn/∂Dn ∼= Sn, using thecorollary above, we find that Hn(Dn, ∂Dn) ∼= Hn(Sn)

4 An Application of Relative Homology: Path Clustering

Path clustering consists of partitioning sets of paths into smaller subsets of similar paths. Usingthese sets of paths, and combining it with a set of learned motion primitives (pre-computed motionsthat 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 vectorspace. Clustering is usually done by comparing distances between paths and applying a clusteringparameter based only on distance. This is usually done through a process called Dynamic TimeWarping (DTW). How this usually works, is that for a given path, it will assign a slow in a matrixfor each point of the path and compare to every single other point in the path in order to find thebest match. This is extremely computational intensive since this must be repeat for all possiblepaths 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 arepaths.

In traditional path clustering, the distance is calculated by calculating the distance between thesetwo paths. The problem with traditional path clustering is that it does not take into account thetopology of a space when calculating the distance between two paths and they are usually veryexpensive 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, thedistance between the paths is modeled by the area between the two paths instead.

4.2.1 Simplicial Homology

Given a topological space X ⊆ Rn, we need to be able to run calculations on this topologicalspace. Remember that geometrically, we represent a simplex by taking the convex hull of itsvertices. Using this, we approximate the topological space X using simplicial homology by takinga discrete set X ⊆ X and using a triangulation T of this set. Using this method you obtain asimplicial 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 havethe 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 forobstacles. 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 thequotient map on a simplicial complex. The red area, which are the ”goal regions” are compressedinto 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 sensornetworks. The ability to fabricate increasingly small sensing devices, along with the computationalscaling implicit in Moore’s Law and the parallel advances in wireless technology, foretell a world inwhich 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 cansense, broadcast to, or otherwise cover a region of fixed coverage radius about the node. The mostbasic form of coverage problem is the simple query: given the nodes, does the collection of coveragediscs 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 oftheir immediate neighbors (with one ear, to prevent bearing data). Can such a crowd determinethe topology or geometry of the party?

What is the appropriate mathematics for sensor networks? Most of the problems in networkedsensing are of the local-to-global variety. As sensors shrink in size and increase in multiplicity, onegoes from having a small number of “global” sensors to a dense array of “local” sensors. Sensingproblems 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 mathematicsis 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. Weakassumptions would be as follows. Assume nodes lie in the plane, at unknown locations. Assumethat any collection of K nodes that are in pairwise communication have coverage regions whichcontain the convex hull of these points in the plane. So, if three nodes communicate pairwise, thenthe abstract triangle they form in the plane lies within the coverage of the network. These arereasonable 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 commonproblem, it actually is. Many sensors have gotten so small that once they are placed they oftenmoved 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 distancenor direction. Only connectivity data between nodes is used. The only strong assumption we makeis on the fence nodes set up along the boundary of the domain. This strong degree of controlalong the boundary is not strictly required, but it simplifies the statements and proofs of theoremsdramatically

• A1: Nodes χ broadcast their unique ID numbers. Each node can detect the identity of anynode within broadcast radius rb.

• A2: Nodes have radially symmetric covering domains of cover radius rc ≥ rb/√

3.

• A3: Nodes χ lie in a compact connected domain D ⊂ R2 whose boundary ∂D is connectedand piecewise-linear with vertices marked fence nodes χf .

• A4: Each fence node v ∈ χf knows the identities of its neighbors on ∂D and these neighborsboth lie within distance rb of v.

To summarize, the sensor data for each node consists of a list of node ID numbers within signaldetection 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 verifycoverage in many instances.

5.3 Communication graph and Rips Complex

Theorem (The Cech Theorem). If the sets {Uα} and all nonempty finite intersections are con-tractible, then the union ∪αUα has the homotopy type of the Cech complex C.

This would appear to be exactly what one wants for sensor networks. Unfortunately, it is highlynontrivial to compute the Cech complex from the network graph. We have two radii to contendwith: the broadcast radius rb and the coverage radius rc. For no (physically realistic) choice ofthese radii can the radius rc Cech 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 withoutchanging the radius rb network graph.

On the other hand, with the bound on coverage and broadcast radii in A2, it follows that forany triple of nodes which are in pairwise communication distance, the convex hull of these nodesin R2 is contained in the cover. The limiting case, in which all three nodes are at pairwise distancerb yields an equilateral triangle in R2 which is covered by balls at the nodes of radius rc only ifrc ≥ rb/

√3.

This motivates the following construction. We consider the network graph as the 1-dimensionalskeleton of a larger simplicial complex. Denote by R the largest simplicial complex whose 1-skeletonis the network graph. That is, for every collection of k nodes which are pairwise within distancerb, we assign an abstract k − 1 simplex. By assumption A4 the boundary ∂D can be representedas a 1-dimensional fence cycle F ⊂ R which is canonically identified with ∂D. This construction isequivalent to the Rips complex.

Definition of Rips Complex Given a set of points χ = {xα} in a metric space and a fixed ε > 0,the Rips complex of Xf , Rε(X), is the abstract simplicial complex whose k-simplices correspond tounordered (k + 1)-tuples of points in X which are pairwise within distance ε of each other.

Unfortunately, the radius-rb Rips complex of a set of nodes in R2 does not always capture thetopology 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 ‘fillingin’ 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 communicationgraph alone, it is difficult to ‘see’ potential holes in coverage. However, upon completing the graphto the Rips complexR, potential large holes in coverage would show up in the abstract complex. Onemight guess that showing there are no such holes in R implies coverage. This condition would betranslated into algebraic topological terms as H1(R) = 0, or, that any cycle in the communicationgraph can be realized as the boundary of a surface built from 2-simplices of R, each of whichindicates a coverage region thanks to assumption A2.

Figure 7: In a sensor network with a sufficiently large hole in coverage [left], the communicationgraph [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 robustto extensions and which yields stronger information about the actual cover. The union of theradius rc discs contains D if there is a element of the relative homology H2(R,F ) whose boundaryis 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 someβ ∈ 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 projected2-simplices from R.

Figure 8: The coverage criterion is an algebraic-topological formulation of the intuition of ‘fillingin’ 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 thecomplex. If the Rips complex is hole-free, then the topology of this quotient space is that of asphere, and therefore, the second relative homology H2(R,F ) has a non-trivial generator. On theother hand, if the 1-cycles defined over subcomplex F are not boundaries of any 2-chain, then therelative 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 theboundary 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 ). Such2-cycles do not represent a true relative class, as they may still exist even if the fence cycle F isnot the boundary of any 2-chain. Hence, the coverage criterion requires the existence of a relative2-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 injudiciouschoice of rc. For example, if rc is much larger than the bound in Assumption (A2), then there willbe many instances of coverage without a homological forcing. This being said, we note that evenif one chooses the minimal acceptable bounds from Assumption (A2), it is still possible to arrangethe 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 theother [center], because of a 1-cycle in R [right]. Note the fragility of the cover [center] within the1-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 ”dynamiccoverage”. Intuitively one can think of dynamic coverage as: ”could someone move through thedomain without being sensed?” or is there an evasion path?.

The algebraic tools to verify this criterion are more complicated but use the same tools: Ripscomplex 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 theStacked Rips complex denoted SR. We can then look at the boundary of each complexand identically glue together each triangle in successive slices, this is called Stacked Fencecomplex denoted SF

• We then apply the Homology information of SR and SF to the moving sensors in order todetermine 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. Forreasons of power conservation, one would like to know which nodes could be “turned off” withoutimpinging upon the coverage integrity.

Given the above, it is easy to see that the minimal cover is simply the sparsest generator of thesecond homology class of R relative to F . Therefore, one can formulate the problem of finding thesparsest cover over D as an optimization problem,simply by extending the results of the previoussection 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 wewill look at it in future classes.

Definition Persistent homology is a method for computing topological features of a space atdifferent spatial resolutions. More persistent features are detected over a wide range of spatialscales and are deemed more likely to represent true features of the underlying space rather thanartifacts of sampling, noise, or particular choice of parameters.To find the persistent homology ofa space, the space must first be represented as a simplicial complex. A distance function on theunderlying space corresponds to a filtration of the simplicial complex, that is a nested sequence ofincreasing subsets.

12

Figure 13: A typical simulation: [top] the locations of 212 nodes in D; [center] the image of theRips complex R projected to D; [bottom] a simple generator of H2(R,F ) extracts 101 nodes whichare guaranteed to cover D, leaving 111 nodes to be safely put into sleep mode.

5.5.2 Cool application of Persistent Homology: Neuro-science

13