Homology Groups

Discover how homology groups count topological features: connected components (H₀), loops (H₁), and voids (H₂).

Counting Holes with Homology

Homology is an algebraic tool for measuring the "holes" in a space. The key insight is that holes are cycles that don't bound anything: a loop around a donut hole can't be filled in, but a loop on a sphere can be contracted to a point.

The Betti numbers count these holes in each dimension: β₀ counts connected components, β₁ counts loops/tunnels, and β₂ counts voids/cavities. These are topological invariants — they don't change under continuous deformation.

Betti Number Calculator

Adjust ε to see how the Betti numbers change as the complex fills in. When ε is small, there are many components (high β₀). As edges connect points, components merge (β₀ decreases) and loops may form (β₁ increases).

Betti Numbers

β₀ = 0(expected: 1)
Connected Components
Number of separate pieces
β₁ = 0(expected: 1)
Loops / Holes
Number of 1-dimensional holes
β₂ = 0
Voids / Cavities
Number of 2-dimensional voids
Simplex Counts
0
V
0
E
0
T
χ = V - E + T = 0

Betti numbers count topological features that persist across the filtration. Adjust ε to see how they change as the complex fills in.

Cycles, Boundaries, and Homology

The formal definition of homology involves three concepts:

Chains (Cₖ)

Formal sums of k-simplices with coefficients in Z/2Z (or Z). A 1-chain is a collection of edges; a 2-chain is a collection of triangles.

Cycles (Zₖ = ker ∂ₖ)

Chains with zero boundary. A 1-cycle is a closed loop of edges. Every boundary is a cycle (∂² = 0), but not every cycle is a boundary.

Boundaries (Bₖ = im ∂ₖ₊₁)

Chains that are the boundary of a (k+1)-chain. The boundary of a triangle is a 1-cycle consisting of its three edges.

Hₖ = Zₖ / Bₖ = (cycles) / (boundaries)
Homology = cycles that aren't boundaries = genuine holes

Holes in Each Dimension

H₀

Connected Components

Every 0-chain is a 0-cycle (there are no (-1)-chains to map to); the 0-boundaries are differences of edge endpoints. Quotienting identifies vertices joined by edge paths, so H₀ counts the separate pieces of the space.

Example: 3 disjoint points → β₀ = 3
H₁

Loops / Tunnels

1-cycles are closed loops; 1-boundaries are loops that bound a surface. H₁ counts loops that can't be filled in.

Example: Circle → β₁ = 1, Torus → β₁ = 2
H₂

Voids / Cavities

2-cycles are closed surfaces; 2-boundaries are surfaces that bound a volume. H₂ counts enclosed cavities.

Example: Sphere → β₂ = 1, Torus → β₂ = 1

The Euler Characteristic

The Euler characteristic χ is a fundamental topological invariant that relates the simplex counts to the Betti numbers:

χ = V - E + T - ... = β₀ - β₁ + β₂ - ...

This is a powerful constraint: if you know any three of {V, E, T, χ}, you can compute the fourth. For a connected planar graph:

χ = 2
Sphere
χ = 0
Torus
χ = 1
Disk

Key Takeaways

  • Homology Hₖ = Zₖ/Bₖ — cycles modulo boundaries
  • Betti numbers βₖ = rank(Hₖ) — count k-dimensional holes
  • β₀ = connected components, β₁ = loops, β₂ = voids
  • Euler characteristic χ = Σ(-1)ᵏβₖ — topological invariant
  • ∂² = 0 guarantees that boundaries are cycles