TDA

A p-chain c (in K) is a formal sum of p-simplices added with some coefficients from some ring R. A p-chain c can thus be written as \(c = \sum^{m_p}_{i=1} α_iσ_i\), where \(α_i ∈ R\) and \(σ_i ∈ K\) are p-simplices.

Let \(σ = {v_0, . . . , v_p}\) be a p-simplex. Then, \( δ_p(σ)\) is defined by \((v_1, . . . , v_p) + (v_0, v_2, . . . , v_p) + . . . + (v_0, . . . , v_{p−1}) = \sum^{p}_{i =0} (v_0, . . . , \hat{v}_i, . . . , v_p)\).

A p-chain c is a p-boundary if \(∃c ′ ∈ C_{p+1}(K)\) such that \(δ(c ′ ) = c\). \(B_p(K)\) is the p-th boundary group, consisting of all p-boundaries of K.

A p-chain c is a p-cycle if \( δ(c) = 0\). \(Z_p(K)\) is the p-th cycle group, consisting of all p-cycles of K.

Recall that homology is intended as a tool to count holes in objects. Intuitively, a hole is a cycle that is not a boundary, that is, not filled by something higher-dimensional.

The p-th homology group \(H_p(K)\) is the quotient group \(Z_p(K)/B_p(K)\).

The idea of singular homology is to remove the need for a fixed triangulation by looking at all possible simplices at once. A singular p-simplex is a map \(σ : ∆ p → X\). We now define \(C_p(X)\) the same way as before, but now on the family of all singular p-simplices, which in general makes the group uncountably infinite. The rest is analouge as for the simplicial homology.

Let X be a topological space, K a triangulation of X. Then we have \(Hp(X) \cong Hp(K)\) for all p ⩾ 0.

For any \(d > 0\), we have \(H_p(S^d ) = \begin{cases}\mathbb{Z}_2 &p\in\{0,d\}\\ 0 &else\end{cases}\). (Proof on p.40)

Let \(f\) be a simplicial map and \( f_\#\) its induced chain map. This induces a homomorphism \(\begin{split}f∗: &H_p(K_1) → H_p(K_2) \\&[c] = c + B_p \mapsto f_\#(c) + B_p(K_2) = [f_\#(c)]\end{split}\)

Let \( f : K_1 → K_2\) be a simplicial map. This induces a chain map \(f_\# : C_p(K_1) → C_p(K_2)\)\(f_\#(c) :=\sumα_iτ_i, where \, τ_i = \cases{f(σi) &if $f(σ_i)$ a p-simplex in $K_2$\\0 & otherwise}\).

Let \(f : B^d → B^d\) be continuous. Then, \(f\) has a fixed point, that is, \(∃x ∈ B_d\) such that \( f(x) = x\). (Proof p.45)

 A filtration is a nested sequence of subspaces \(F : X_0 ⊆ X_1 ⊆ X_2 ⊆ . . . ⊆ X_n = X\).

A filtration is a nested sequence of subspaces\( F : X_0 ⊆ X_1 ⊆ X_2 ⊆ . . . ⊆ X_n\) For each i ⩽ j, we have the inclusion map \( ι_{i,j}: X_i ,→ X_j \).

The p-th persistent homology group \(H_{p}^{i,j}\) is defined by \(H_{p}^{i,j}:= im( h_{p}^{ i,j} )= Z_p(K_i)/(B_p(K_j) ∩ Z_p(K_i))\) . This definition characterizes the cycles that that are present already in \(K_i\)and that are not boundaries even in \(K_j\) .

persistence pairing algorithm and matrix reduction algorithm, algorithms on page 52, 54

Given a metric space \((M, d)\), a finite point set \(P ⊆ M\), and a real number radius r > 0, the Čech complex \(C^ r (P) \)is defined as the nerve of the set of balls \(B(p, r) = \{x \in M | d(p, x) ⩽ r\}\) for all \(p ∈ P\).

Given a finite metric space \((P, d)\) and a real number radius r > 0, the Vietoris-Rips complex \(VR^r (P)\) is defined as the simplicial complex containing a simplex σ if and only if \(d(p, q) ⩽ 2r\) for every pair p, q ∈ σ.

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.