Algebraic Topology

Algebraic topology uses tools from abstract algebra to study topological spaces. The goal is to find algebraic invariants that classify topological spaces up to homeomorphism or homotopy equivalence.

Euler Characteristic and Simplicial Complexes

A simplicial complex is a space built from “building blocks” called simplices (points, line segments, triangles, tetrahedra). The Euler Characteristic $\chi$ is a topological invariant defined for a finite simplicial complex as: $\chi = V - E + F$ where $V, E, F$ are the numbers of vertices, edges, and faces respectively. More generally: $\chi = \sum_{n=0}^{\infty} (-1)^n c_n$ where $c_n$ is the number of $n$-simplices.

def calculate_euler(v, e, f):
  return v - e + f

# Tetrahedron: 4 vertices, 6 edges, 4 faces
print(f"Tetrahedron chi: {calculate_euler(4, 6, 4)}")

# Cube: 8 vertices, 12 edges, 6 faces
print(f"Cube chi:        {calculate_euler(8, 12, 6)}")

# Torus (triangulated): chi should be 0
v, e, f = 9, 27, 18
print(f"Torus chi:       {calculate_euler(v, e, f)}")

Homology Groups

Homology is a way of associating a sequence of abelian groups $H_n(X)$ to a topological space. Informally, $H_n(X)$ represents the ” $n$-dimensional holes” in $X$.

• $H_0(X)$ counts path-connected components.
• $H_1(X)$ counts “loops” or 1D holes.
• $H_2(X)$ counts “voids” or 2D holes.

The Fundamental Group

The fundamental group $\pi_1(X, x_0)$ consists of equivalence classes of loops based at $x_0$ under homotopy. For a circle $S^1$, $\pi_1(S^1) \cong \mathbb{Z}$, representing the number of times a loop winds around the circle.

def winding_number(points):
  """Simple discrete winding number calculation around origin."""
  import numpy as np
  angles = np.arctan2([p[1] for p in points], [p[0] for p in points])
  diffs = np.diff(angles)
  # Adjust for wrap-around
  diffs[diffs > np.pi] -= 2*np.pi
  diffs[diffs < -np.pi] += 2*np.pi
  return round(np.sum(diffs) / (2*np.pi))

# A circle loop
theta = np.linspace(0, 2*np.pi, 100)
circle_loop = [(np.cos(t), np.sin(t)) for t in theta]
print(f"Winding number (1 loop): {winding_number(circle_loop)}")

# A double loop
double_loop = [(np.cos(2*t), np.sin(2*t)) for t in theta]
print(f"Winding number (2 loops): {winding_number(double_loop)}")