Learning #28 – Topology & Manifolds: Homotopy, Knot Theory, Homology and TDA

Two shapes are topologically equivalent (homeomorphic) if one can be continuously deformed into the other without tearing or gluing. A coffee mug is homeomorphic to a donut (both have genus 1), but neither is homeomorphic to a sphere (genus 0). Beyond this pop-science starting point lies a rich hierarchy of invariants — fundamental groups, homology groups, knot polynomials, and persistent diagrams — that have found unexpected applications in neuroscience, robotics, condensed matter physics, and machine learning.

1. Topological Spaces and Continuity

The fundamental object of topology is a set X together with a collection τ of subsets (called open sets) satisfying three axioms: ∅ and X are open, arbitrary unions of open sets are open, and finite intersections of open sets are open. The pair (X, τ) is a topological space.

Topological space (X, τ): τ = collection of open sets on X
  Axioms: ∅, X ∈ τ;  unions of τ-sets ∈ τ;  finite intersections ∈ τ

Metric topology: open ball B(x, r) = {y : d(x,y) < r} generates τ
  Every metric space is also a topological space

Continuous map f: X → Y (topological definition):
  f is continuous iff preimage of every open set in Y is open in X

Homeomorphism: bijective continuous map with continuous inverse
  Homeomorphic spaces are topologically identical

T2 / Hausdorff condition:
  For all x ≠ y, there exist disjoint open sets U∋x, V∋y
  Ensures limits are unique; all metric spaces satisfy this

Compactness (generalised closed + bounded):
  X is compact iff every open cover has a finite subcover
  [0,1] is compact; (0,1) is not

Connectedness:
  X is connected iff it cannot be written as disjoint union of two non-empty open sets
  Path-connected: any two points joined by a continuous path
          

2. Homotopy and the Fundamental Group

Homotopy asks: when are two continuous maps “essentially the same”? This leads to the fundamental group π₁(X, x₀) — the group of loops based at x₀ up to homotopy equivalence — which captures the one-dimensional “holes” in a space.

Homotopy: continuous deformation H: X × [0,1] → Y with
  H(x,0) = f(x),  H(x,1) = g(x)   (f and g are homotopic: f ∼ g)

Loop: path γ with γ(0) = γ(1) = x_0 (basepoint)
Fundamental group π_1(X, x_0):
  Elements = homotopy classes of loops at x_0
  Group operation = loop concatenation

Examples:
  π_1(S¹) = ℤ   (number of times a loop winds around the circle)
  π_1(S²) = {e}   (all loops on sphere are contractible)
  π_1(T²) = ℤ × ℤ   (torus: two independent winding numbers)
  π_1(ℝP²) = ℤ/2   (projective plane; non-orientable)

Van Kampen's theorem:
  If X = A ∪ B with A, B, A∩B path-connected open sets and x_0 ∈ A∩B:
  π_1(X) = π_1(A) *_(π_1(A∩B)) π_1(B)   (amalgamated free product)

Covering spaces:
  p: &Xtilde; → X is a covering map if every x has a neighbourhood evenly covered
  Universal cover &Xtilde; is simply connected; π_1(X) acts on it by deck transformations
  π_1(S¹) = ℤ via covering ℝ → S¹, t ↦ e^(2πit)
          

3. Surfaces and Their Classification

The classification theorem for compact surfaces is one of the gems of nineteenth-century mathematics: every compact surface without boundary is homeomorphic to a sphere, a connected sum of tori, or a connected sum of projective planes. The Euler characteristic and orientability together determine the class.

Triangulation: decompose surface into vertices V, edges E, faces F
Euler characteristic: χ = V − E + F   (triangulation-independent)

Orientable surfaces (genus g):
  χ = 2 − 2g
  g=0: S² (sphere),  g=1: T² (torus),  g=2: double torus, ...

Non-orientable surfaces:
  ℝP² (projective plane): χ = 1,  no boundary embedding in ℝ³
  Klein bottle: χ = 0,  connected sum of two ℝP²
  ℝP² # ℝP² # ... (k times): χ = 2 − k

Classification theorem:
  Every compact connected surface is homeomorphic to exactly one of:
  S²  |  T²#T²#…#T² (g connected tori)  |  ℝP²#ℝP²#… (k projective planes)

Connected sum # M#N:
  Remove a disk from each, glue boundary circles together
  χ(M#N) = χ(M) + χ(N) − 2
          

4. Knot Theory

A knot is a closed loop embedded in ℝ³ (or S³). Two knots are equivalent if one can be continuously deformed into the other without passing through itself. Determining knot equivalence algorithmically is a solved but computationally expensive problem; polynomial invariants provide fast partial tests that have also found applications in DNA biology and quantum field theory.

Knot diagram: planar projection with crossing information (over/under)

Three Reidemeister moves (generate all isotopies):
  RI:   twist ↔ untwist a strand
  RII:  slide one strand over another at 2-crossing site
  RIII: slide strand through 3-crossing site (triangle move)
  Any knot invariant must be invariant under all three moves

Alexander polynomial Δ_K(t)  (1928):
  Trefoil: Δ(t) = 1 − t + t²
  Figure-eight: Δ(t) = −t + 3 − t−¹
  Computed from Seifert matrix of the knot

Jones polynomial V_K(t)  (Jones 1984, Fields Medal 1990):
  More powerful than Alexander; distinguishes chirality
  Trefoil and its mirror have different V_K(t)
  Satisfies skein relation:
  t−¹V_L+ − tV_L- = (t^(1/2) − t−^(1/2)) V_L0

HOMFLY-PT polynomial: generalises both Alexander and Jones
  P_K(v, z) with two variables

Applications to DNA topology:
  DNA replication leaves strands catenated; type II topoisomerase
  introduces transient double-strand cuts to unlink them
  Action = crossing change = Reidemeister II move
  Topoisomerase inhibitors (e.g. etoposide) trap cut complexes → anticancer drugs
          

5. Homology Groups

While the fundamental group captures one-dimensional loops, homology generalises this to detect holes of all dimensions. A k-dimensional hole corresponds to a non-trivial element of the k-th homology group H_k. The rank of H_k is the k-th Betti number β_k.

Simplicial complex K: vertices, edges, triangles, tetrahedra, ...
  k-simplex: convex hull of (k+1) affinely independent points

Chain group C_k(K): free abelian group on k-simplices
  Elements: formal sums ∑ a_i σ_i  with a_i ∈ ℤ

Boundary operator ∂_k: C_k → C_{k-1}
  ∂([v_0,...,v_k]) = ∑_i (−1)^i [v_0,...,v^_i,...,v_k]
  Fundamental property: ∂_k ∘ ∂_{k+1} = 0  (boundary of boundary = 0)

Homology groups:
  Z_k = ker ∂_k   (cycles: chains with no boundary)
  B_k = im ∂_{k+1} (boundaries: chains that are boundaries)
  H_k = Z_k / B_k

Betti numbers β_k = rank H_k:
  β_0 = number of connected components
  β_1 = number of independent 1-cycles (loops)
  β_2 = number of enclosed 2-cavities (voids)

Euler formula revisited:
  χ = V − E + F = β_0 − β_1 + β_2   (Euler-Poincaré formula)

Examples:
  Sphere S²: β_0=1, β_1=0, β_2=1  → χ=2 ✓
  Torus T²: β_0=1, β_1=2, β_2=1  → χ=0 ✓
  Klein bottle: β_0=1, β_1=1, β_2=0  (ℤ/2 torsion in H_1)
          

6. Topological Data Analysis

Topological Data Analysis (TDA) applies homological machinery to finite point-cloud data. The key idea: build a filtered simplicial complex that grows as a scale parameter ε increases, then track which topological features (connected components, loops, voids) are “born” and “die” as ε changes. The resulting persistence diagram is a robust data descriptor.

Vietoris-Rips complex VR(X, ε):
  Vertex set = data points X
  Include k-simplex [x_0,...,x_k] iff d(x_i, x_j) ≤ ε for all i,j
  Filtration: VR(X,ε_1) ⊂ VR(X,ε_2) for ε_1 ≤ ε_2

Persistent homology:
  Track birth and death of each H_k generator as ε grows:
    born at ε_b when feature first appears
    dies at ε_d when feature merges/fills

Persistence diagram PD_k:
  Set of points (ε_b, ε_d) in the plane
  Points far from diagonal (long-lived features) = signal
  Points near diagonal = noise

Bottleneck distance d_b(PD, PD'):
  max_{point p in PD} min_{point q in PD' ∪ diagonal} |p − q|_∞
  Stability theorem: d_b(PD(f), PD(g)) ≤ ||f − g||_∞

Persistence barcodes:
  Horizontal bars [ε_b, ε_d) for each generator
  H_0 bars: component merges  |  H_1 bars: loop fills  |  H_2 bars: cavity fills

Mapper algorithm (Singh-Mémoli-Carlsson 2007):
  1. Cover function f: X → ℝ with overlapping bins
  2. Cluster each preimage bin
  3. Connect clusters from adjacent bins sharing points
  Output: 1-complex (graph) summarising high-dimensional shape

Applications:
  Cancer genomics (Nicolau 2011): Mapper found genomically distinct breast cancer subgroup
  Materials science: persistent H_1 loops characterise ring statistics in silicate glasses
  Neuroscience: H_2 voids detected in neural firing patterns above chance baseline
          

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