realization problems in persistent homology

Realization problems in persistent homology


Hans Riess

Under the supervision of Dr. Justin CurryDepartment of Mathematics

Duke University

Texas Undergraduate Topology and Geometry Conference


Outline

Introduction to realization problems and persistence

Persistent homology of curvature filtrations

Realization of Morse functions on a compact orientable smooth2-fold



Review of simplicial homology

I A simplicial complex is a collection K of finite subsets of aset V such that if τ ∈ K is nonempty and σ ≤ τ , thenσ ∈ K .

I Note that we can view K as a partially ordered set whichhas the structure of a category.

I Given a field k , let Ci(K ; k) be the free k -vector spacegenerated by all the i-dimensional simplices of K . Then,there is a chain complex

· · · ∂4−→ C3(K )∂3−→ C2(K )

∂2−→ C1(K )∂1−→ C0(K )

∂0−→ 0

I Define Hn(K ; k) to be the vector space ker ∂n/ im ∂n−1



Warmup realization problems

QuestionGiven a vector space V and n > 0, find a topological space Xsuch that that Hn(X ) = V.

Take X =∨dim(V )

i=1 Sn, the bouquet of spheres. We call thisassignment the Moore functor M(V ,n) := X .

QuestionGiven vector spaces V and W, n > 0 and T : V →W, findtopological spaces X and Y and f : X → Y such thatHn(X ) = V, Hn(Y ) = W, and Hn(f ) = T

Take X =∨dim(V )

i=1 Sn, Y =∨dim(W )

i=1 Sn. Up to a change in basis,T sends rank(T ) basis elements to itself and the rest to 0. Sodefine f to identically map rank(T ) spheres and crushdim(V )− rank(T ) spheres down to the basepoint.



Most general statement of homological realizationI Let (I,≤) be a partially ordered set (e.g the integers Z, the

reals R, or a simplicial complex K )I This is a category: ob(I) = I; a→ b iff a ≤ b.I A diagram of topological spaces is a functor

Dgm : (I,≤)→ Top.I Consider a diagram of vector spaces, or a quiver

representation, Rep : (I,≤)→ VecI Consider the homology functor Hn : Top→ Vec.

QuestionGiven n ≥ 0, find a diagram of topological spaces Dgm suchthat Rep = Hn ◦ Dgm and H̃i ◦ Dgm = 0 for i 6= n.

I We add various constraints: allowed diagram types,constraints on maps, constraints on spaces



Persistent homologyI Consider a real-valued function on a topological space

f : X → RI We can obtain a sublevelset filtration of X to obtain a

diagram of the type

· · · ↪→ f−1(−∞, ti−1] ↪→ f−1(−∞, ti ] ↪→ f−1(−∞, ti+1] ↪→ · · ·I We apply Hn to get the quiver representation

· · · →→ Hn(f−1(−∞, ti ])→ Hn(f−1(−∞, ti+1])→ · · ·I Fact: every quiver representation of the form

· · · → Vi−1 → Vi → Vi+1 → · · ·can be written uniquely as a direct sum of indecomposable

interval modules II(t) =

{k t ∈ I0 t /∈ I

with maps k id−→ k



I We record the intervals in this decompositiondecomposition in a multiset B called a barcode

I Thus, we can associate a barcode B to a sublevelsetfiltration f : X → R for a given homological degree

I An interval is essential if it is unbounded. Fact: if X iscompact and Hn(X ) ∼= kp, then B has p essential intervals.

I Barcodes generalize to levelset filtrations by the inducedmaps

· · · → Hn(f−1(Ui))← Hn(f−1(Ui∩Ui+1)→ Hn(f−1(Ui+1))← · · ·

I Fact: quiver representations of this form decompose in thesame way into interval indecomposables

I Barcodes defined for levelset persistence



PERSISTENT HOMOLOGY — A SURVEY 3

x

x 2

1

Figure 1: A single variable function with three local minima and three local maxima. The criticalpoints are paired and each pair is displayed as a point in the persistence diagram on the right.

method we define the persistence of the pair to be f(y) − f(x). Persistence is coded inthe persistence diagram by mapping each pair to the point (f(x), f(y)) whose coordinatesare the corresponding critical values. In the diagram, all points live in the half space abovex1 = x2, and the persistence is easily visible as the vertical distance to this diagonal line.For reasons that will appear later, we usually adjoin the diagonal to the persistence diagram.The remainder of this paper extends these ideas beyond single variable functions.

Specifically, we extend the domain first to manifolds and then to general triangulable topo-logical spaces. The algorithms compute homology and persistence for nested sequencesof simplicial complexes which we think of as piecewise constant or piecewise linear ap-proximations of functions defined on their underlying spaces. At the same time we extendfeatures beyond connected components using homology which we introduce next. To gofrom homology to persistence we are guided by the following property we observe for thecomponents of the sublevel sets of the single variable function f : R → R. Let s < t andconsider the sublevel sets Rs ⊆ Rt. Going from s to t, components of Rs may merge andnew components may be born and possibly merge with each other or with components ofRs. We let βs,t

0 be the number of components that are born at a finite time at or before sthat belong to distinct components in Rt. The pairing of critical points we described hasthe property that βs,t

0 is equal to the number of pairs (x, y) with f(x) ≤ s < t < f(y). Noother pairing satisfies this property for all s < t. As indicated by the shading in Figure 1,βs,t

0 is also the number of points in the upper left quadrant defined by (s, t).

Homology. LetK be a simplicial complex. The Z/2Z vector space generated by thep-dimensional simplices of K is denoted Cp(K). It consists of all p-chains, which areformal sums c =

!j γjσj , where the γj are 0 or 1 and the σj are p-simplices in K . The

boundary,∂(σj), is the formal sum of the (p−1)-dimensional faces of σj and the boundaryof the chain is obtained by extending ∂ linearly,

∂(c) ="

j

γj∂(σj),

where we understand that addition is modulo 2, i.e. 1 + 1 = 0. It is not difficult to checkthat ∂ ◦ ∂ = ∂2 = 0. The p-chains that have boundary 0 are called p-cycles. They forma subspace Zp of Cp. The p-chains that are the boundary of (p + 1)-chains are called p-boundaries and form a subspace Bp of Cp. The fact that ∂2 = 0 tells us that Bp ⊆ Zp. The

Figure: from Herbert Edelsbrunner and John Harer, “PersistentHomology – a survey.”



Filtrations by Gauss curvatureI Let M be a compact connected codimension-1 orientable

smooth k -manifoldI M has a smooth orientation map (i.e Gauss map)ν : M → Sk

I Then the Gauss curvature of M at a point x is defined

K (x) = det(−Dxν)

I For k = 1, Gauss curvature is simply referred to as thecurvature, κ, of the planar curve

I We can filter M by sublevel sets of Gauss curvatureK : M → R to produce a barcode B

I We hope that B encodes geometric information aboutM



The converse of the Four Vertex Theorem andgeneralization

Lemma (Converse of Four Vertex Theorem)Let f : S1 → R be a continuous function that is either a nonzeroconstant or else has at least two local maxima and at least twolocal minima. Then there is an embedding c : S1 → R2 suchthat f (x) is the curvature of c at x.

Lemma (Generalized Minkowski Theorem)Let f : Sk → R for k ≥ 2 be a continuous strictly positivefunction. Then, there exist an embedding s : Sk → Rk+1 whoseGauss curvature at s(x) is f (x).



Realization of barcodes by functions f : S1 → Rversus realization by curvature

PropositionGiven a barcode B with one essential class and any number ofhalfopen intervals, there is a function f : S1 → R such that Bis the barcode of the filtration by f in degree 0 homology.

PropositionGive a barcode B with exactly one essential interval and atleast one other interval, there exists an embeddingc : S1 → R2 such that B is the barcode of the filtration bycurvature κ of c in degree 0 homology.


Realization of Morse functions on a compact orientable smooth 2-fold

Reeb graphs of Morse functions

I Let M be a compact connected smooth manifoldI We say a smooth function f : M → R is Morse if every

critical point is nondegenerateI Given a Morse function f : M → R, we define an

equivalence relation ∼ on M where x ∼ y if x and y lie inthe same connected component of f−1(t) for each t ∈ im f .

I Then the Reeb graph Γ of f is the quotient space M/ ∼with the obvious map Γ→ R.

I We can give Γ the structure of a simplicial complex byplacing vertices at critical values



Figure: from Vin de Silva, Elizabeth Munch, Amit Patel, “CategorifiedReeb Graphs”



Useful facts about the fibers of f : M → RLemma (Fiber lemma)Let M be a compact smooth k-fold. If f : M → R is a smoothmap and t ∈ R is a regular value, then f−1(t) is a compactmanifold of dimension k − 1

CorollaryHi(f−1(t)) ∼= Hk−i−1(f−1(t)). In particular, if M is a compactorientable 2-fold, f−1(t) =

⊔S1 and H1(f−1(t)) ∼= H0(f−1(t))

Lemma (Handle attachment)Suppose f−1([t − ε, t + ε]) is compact and contains a criticalpoint p of index i, thenf−1(−∞, t + ε] ∼= f−1(−∞, t − ε] ∪ (i -handle) where the i-handleis Di × Dk−i attached along ∂Di × Dk−i .



Realization of Morse function on compact orientable2-folds via Reeb graphs

I An R-graph is a graph Γ with a height function h : Γ→ R

TheoremGiven an R-graph with valences either 1 or 3, there exists asmooth compact connected orientable 2-fold M with a Morsefunction f : M → R such that Γ is the Reeb graph of f .



Homotopy colimitsI Given f : X → Y , the mapping cylinder

Mf = (X × [0,1])∐

Y/ ∼ where (1, x) ∼ f (x) for all x ∈ X .I In general, the categorical colimit of a diagram is not the

same if we replace spaces by their weak equivalencesI Intuitively, given a diagram of topological spaces, the

homotopy colimit of the diagram is constructed by gluingtogether mapping cylinders of each map in the diagram

I Reference: Daniel Dugger, “A primer on homotopycolimits.”

ExampleThe diagram

D : ∗ ← Sk → ∗has hocolim(D) ∼= Sk+1



Simplicial cosheaf

DefinitionGiven a simplicial complex K which can be viewed as apartially ordered set and thus as a category, a simplicialcosheaf over K is a functor

F : K op → Vec

ExampleGiven a space X and a map f : X → |K |, the i th Leray simplicialcosheaf of f is given by L(σ) = Hi(f−1(star(|σ|)). In the casethat |K | is an interval, this is levelset persistence.



Example

Figure: We begin with an R-graph Γ with valency at most 3, andwithout trivial or extremal nodes



Figure: (Left) If we view Γ as simplicial complex, KΓ, we construct thesimplicial cosheaf F : K op

Γ → Vec by the formulas F(vertex)= kval(vertex)−1 and F(edge) = k with the obvious restriction maps.(Right) Apply the Moore functor M(_,1) to get a diagram Dgm.



Figure: The homotopy colimit hocol(Dgm) is homeomorphic by aheight-preserving smoothing to a manifold M whose height functionf : M → R is Morse and realizes Γ as the Reeb graph of f .



Reeb graphs and barcodes: main theorem

Lemma (Parameterized Alexander Duality)(Kalisnik) Let f : M → R. Then the barcode of levelsetpersistence in degree i is related to the levelset barcode indegree i − 1 by swapping open and closed intervals andswapping the closed and open ends of halfopen intervals.

TheoremGiven a barcode B with one long closed interval containingevery other interval, and any number of open or halfopenintervals with all distinct open endpoints, there exist acompact orientable 2-fold M and a Morse functionf : M → R such that the B is the barcode under degree 0levelset persistence. The Lemma determines the degree 1barcode.



S2

CorollaryThe number of open intervals of the degree 0 barcodedetermine the genus of the surface.

CorollaryGiven a barcode B with one long closed interval containingevery other interval, and any number of halfopen intervals withdistinct open endpoints, there exists a Morse functionf : S2 → R that realizes B.

CorollaryGiven a barcode B satisfying the same conditions as above,then there is an embedding of the sphere s : S2 → R3 such thatB is the degree 1 levelset persistent barcode of the Gausscurvature K : S2 → R of c.


Thank you!

Any questions welcomed!

realization problems in persistent homology

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