TDA

If U, V are q-tame persistence modules over \(\mathbb{R}\), then \(d_b(DgmU, DgmV) = d_I(U, V)\)

Let A, B ⊆ X be compact sets. Then the Hausdorff distance between A and B is defined as \(d_H(A, B) := max\{max_{a∈A} d(a, B), max_{ b∈B} d(b, A)\}\).

\(d_b(Dgm_p(\mathbb{C}(P)), Dgm_p(\mathbb{C}(Q))) ⩽ d_H(P, Q)\) for all p ⩾ 0. (Proof p.72)

Let X be some topological space, and f a function \(f : X → \mathbb{R}\). Two points x, y are called equivalent (x ∼ y), iff f(x) = f(y) = α and x and y are in the same path-connected component of \(f^{−1} (α)\). The Reeb graph \(R_f \)is the quotient space X/ ∼. 

We define the augmented Reeb graph of a simplicial complex with a piecewise-linear-function, as the discretization of the Reeb graph by placing a vertex for every connected component in all the levelsets for the values a for which f(v) = a for some vertex of the simplicial complex.

Given a simplicial complex K with m faces and a piece-wise linear function \(f : |K| → \mathbb{R}\) on it, we can compute the augmented Reeb graph \(R_f\) of K with respect to f in time O(m log m). (proof p.78)

The vertical homology group of X with respect to f is the group \(\overset{\vee}{H_p}(X) := H_p(X)/\overline{H_p(X)}\)

A p-th homology class \(h ∈ H_p(X)\) is called horizontal if there is a finite set of values \(A = \{a_1, . . . , a_k\}\) such that \( h ∈ im(ι_∗)\) where \( ι_∗\) is the map \(H_p( \bigcup_{ a∈A} X_a) → H_p(X)\) induced by inclusion, where \(X_a = f ^{−1} (a).\)

An ϵ-thickening \(X_ϵ \) of some topological space X is given by \(X_ϵ := X × [−ϵ, +ϵ].\)

For a Reeb graph \(R_f\) consider a function\( f_ϵ : (R_f)_ϵ → \mathbb{R}\) such that \((x, t) \mapsto f(x) + t\). The ϵ-smoothing of \(R_f\), denoted by \(S_ϵ(R_f)\) is the Reeb graph of \((R_f)_ϵ\) with regards to \(f_ϵ\).

Two Reeb graphs \(R_f\), \(R_g\) are ϵ-interleaved, if there exists a pair of function preserving maps \(φ : R_f → S_ϵ(R_g)\), \(ψ : R_g → S_ϵ(R_f)\) such that the diagram commutes. (see diagram p.82)

Let X, Z be topological spaces. Then we call f : X → Z well-behaved if for all path-connected open sets U ⊆ Z,\( f ^{−1} (U)\) has finitely many path-connected components.

Let f : X → Z be well-behaved, and U be a (finite) open cover of Z. Then the Mapper is defined as \(M(\mathcal{U}, f) := N(f ^∗ (\mathcal{U}))\). (N is ther nerve)

see p. 84/85

see page 85/86

1.  The discrete topology \((X, \mathcal{P}(X))\), where \(\mathcal{P}(X)\) denotes the family of all subsets of \(X\). 
2. The trivial topology \((X,(X,\emptyset))\), where the whole set and the empty set are the only opens.
3. The Euclidean space \(X=\mathbb{R}^d\), where the open sets \(T\) are defined as we know from calculus.

A function \(f : X → Y\) is continuous if for every open set \(U ⊆ Y\), its pre-image \(f^{-1} (U) ⊆ X\)  is open.

A homeomorphism is a bijective map \(f : X → Y \) whose inverse is also continuous. Two topological spaces are homeomorphic, if there is a homeomorphism between them. We also write \(X ≃ Y\) to say that \( X, Y\) are homeomorphic.

Let \(g, h\) be maps\( X → Y\). A homotopy connecting \(g\) and \(h\) is a map \( H : X × [0, 1] → Y \) such that \(H(·, 0) = g\) and \(H(·, 1) = h.\)

Two spaces \(X, Y\) are homotopy equivalent if there exist maps \(g : X → Y\) and \(h : Y → X\) such that:

1. \(h ◦ g\) is homotopic to \(id_X\) 

2. \(g ◦ h\) is homotopic to \(id_Y\)

or 

\(X, Y\) are homotopy equivalent if and only if there exists a space\( Z\) such that \(X\) and \( Y\) are deformation retracts of \(Z\).

Let \(A ⊆ X\). A deformation retract of \(X\) onto \(A\) is a map \(R : X×[0, 1] → X\), such that

1. \(R(·, 0) = id_X\)

2. \(R(x, 1) ∈ A, ∀x ∈ X\)

3. \(R(a, t) = a, ∀a ∈ A, t ∈ [0, 1]\)

A group \((G, +) \) is a set \(G\) together with a binary operation “+” such that

1. \(∀a, b ∈ G: a + b ∈ G \)

2. \(∀a, b, c ∈ G: (a + b) + c = a + (b + c)\) (Associativity)

3. \(∃0 ∈ G: a + 0 = 0 + a = a ∀a ∈ G \)

4. \(∀a ∈ G∃ − a ∈ G: a + (−a) = 0 \)

A map \(h : G → H\) between abelian groups \( (G, +)\) and \((H, ⋆)\) is a homomorphism if \(h(a + b) = h(a) ⋆ h(b), ∀a, b ∈ G\).

A bijective homomorphism is called an isomorphism.

We first prove this for \(ker(h)\).

1. \(a, b ∈ ker( h) ⇒ h(a) = h(b) = 0\). By definition of homomorphism, \(h(a + b) = h(a) ⋆ h(b) = 0 ⋆ 0 = 0\), and thus by definition of \( ker(h)\), \(a + b ∈ ker(h)\). We conclude that \(ker(h)\) is closed under addition.

2. Associativity follows from associativity of + in G, since \( ker(h) ⊆ G\).

3. \( ∀a ∈ G : h(0) ⋆ h(a) = h(0 + a) = h(a)\), and thus \( h(0) = 0\), from which \(0 ∈ ker(h)\) follows.

4. Let \( a ∈ ker(h)\). Then, \(0 = h(0) = h(a − a) = h(a) ⋆ h(−a) = 0 ⋆ h(−a) = h(−a)\), and thus \(−a ∈ ker(h)\).

A k-simplex in \(\mathbb{R}^d\) is the convex hull of k+1 affinely independent points in \(\mathbb{R}^d\). 

A geometric simplicial complex is a family K of simplices such that

1. if \(τ ∈ K\) and \(σ\) is a face of \(τ\), then \(σ ∈ K\), and

2. for \(σ, τ ∈ K\), their intersection \(σ ∩ τ\) is a face of both.

An abstract simplicial complex \(K\) is a family of subsets of a vertex set \(V(K)\) such that if \(τ ∈ K\) and \(σ ⊆ τ\), then \(σ ∈ K\).

Every k-dimensional simplicial complex has a geometric realization in \(\mathbb{R}^{ 2k+1}\) 

Place the vertices as distinct points on the moment curve in \(\mathbb{R}^{ 2k+1 }\), which is the curve given by \(f(t) = (t, t^2 , . . . , t^{2k+1} )\). This way, any 2k+2 of the placed points are affinely independent. Thus, any two faces with disjoint vertex sets will not intersect in the realization, showing that the realization is indeed an embedding.

A map\( f : K_1 → K_2\) is called simplicial if it can be described by a vertex map \(g : V(K_1) → V(K_2)\) such that for every simplex \({v_0, . . . , v_k}\) we have\( f({v_0, . . . , v_k}) = {g(v_0), . . . , g(v_k)}\). Since f maps to \(K_2\) we must have that \(f({v_0, . . . , v_k})\) is a simplex in \(K_2\). A simplicial map can also be seen as a map on the underlying spaces \( f : |K_1| → |K_2|.\)

We consider simplicial maps to be the analogue of continuous maps in the world of simplicial complexes.