Betti Numbers

Description

This repository computes Betti numbers of a topological space. The Betti numbers of a topological space can be computed by first triangulating the space by a simplicial complex, then representing this complex as a collection of boundary matrices, and lastly performing some reduction on these boundary matrices.

The original purpose of this repository was to implement in code a general tool for solving the exercises given in Edelsbrunner and Harer's Computational Topology: An Introduction. Given there as exercises are a computation of the Betti numbers of the 2-dimensional Klein bottle, as well as a triangulation of the dunce cap and a verification of its Betti numbers. (The latter is posed as a high difficulty problem and indeed is significantly more involved, see examples.py.)

Betti Numbers

Betti numbers can be thought of as a count of p-dimensional holes in a topological space for positive dimension p. The 0th Betti number of a space is rather better thought of as a count of its disconnected components. The Betti numbers of a space are computed by finding the rank of the p-th homology group H_p of a triangulating simplicial complex. Such a complex can be represented as a collection of boundary matrices describing the (p-1)-dimensional boundaries of each p-simplex in the complex.

Boundary Matrices

The p-th boundary matrix of a complex is a relatively simple construct. Given a dimension p and an arbitrary indexing of the p-simplices and (p-1)-simplices of the complex, the p-th boundary matrix has a_i^j = 1 if the i-th (p-1)-simplex is a face of the j-th p-simplex and a_i^j = 0 otherwise. By computing the Smith normal form of the boundary matrices of a complex, one can easily compute the ranks of its p-th cycle groups Z_p and (p-1)-th boundary groups B_p-1. These appear as the zero columns and non-zero rows, respectively. If these ranks are computed for all dimensions p, then the ranks of the p-th homology groups (i.e. the Betti numbers of the space) can be computed by the differences rank(H_p) = rank(Z_p) - rank(B_p).

Euler Characteristic

Computing the Betti numbers of a space also enables us to compute its Euler characteristic. By the Euler-Poincaré theorem, the Euler characteristic of a topological space is merely the alternating sum of its Betti numbers. A few examples of Euler characteristic are given in examples.py.

Reduced Betti Numbers

Reduced Betti numbers give a somewhat more intuitive and pleasing result when interpreting Betti numbers as p-dimensional holes. In brief, they are equivalent to the standard Betti numbers except in the case of p=0, where the reduced 0th Betti number is given by the 0th Betti number minus one. This guarantees that the 0th reduced Betti number counts gaps between disconnected vertices (i.e. 0-dimensional holes). Techinically, the procedure of reducing boundary matrices of a triangulating complex produces the reduced Betti numbers of a space. The SimplicialComplex class here stores the standard Betti numbers after their computation, however both forms are retrievable through their respective class methods.

Limitations

Unfortunately, this approach to representing topological spaces and computing their Betti numbers is untenable for large complexes, as the boundary matrices of a complex are generally sparse and quite large. A sparse-matrix implementation would yield greater efficiency for larger complexes.

Example

To find the Betti numbers of the 3-dimensional ball, we can triangulate it with a single tetrahedron and find each p-th boundary matrix describing this tetrahedron.

'''
Compute the Betti numbers of the 3-ball triangulated by a single tetrahedron.
'''
ball = SimplicialComplex()

# Add vertices
ball.add_boundary_matrix(0, np.array([
    [1,1,1,1]
]))

# Relate vertices (rows) to edges (columns)
ball.add_boundary_matrix(1, np.array([
    [1,1,1,0,0,0],
    [1,0,0,1,1,0],
    [0,1,0,1,0,1],
    [0,0,1,0,1,1]
]))

# Relate edges (rows) to faces (columns)
ball.add_boundary_matrix(2, np.array([
    [1,1,0,0],
    [1,0,1,0],
    [0,1,1,0],
    [1,0,0,1],
    [0,1,0,1],
    [0,0,1,1]
]))

# Relate faces (rows) to solid interior (column)
ball.add_boundary_matrix(3, np.array([
    [1],
    [1],
    [1],
    [1]
]))

# Expected: [1,0,0,0]
print(f'3-Ball: {ball.get_betti_numbers()}')

When executed, this code outputs:

3-Ball: [1, 0, 0, 0]

These are the expected Betti numbers of the 3-ball. All of the p-th Betti numbers vanish for positive p, which matches our intuition that a solid ball has no holes.

Additional examples regarding the following spaces are provided in examples.py:

• The 2-dimensional sphere
• The klein bottle
• The torus
• The mobius strip
• The cylinder
• The dunce cap

Drawing Tool Instructions

The drawing tool allows a user to draw up to 2-dimensional simplicial complexes. To draw a simplicial complex:

1. Click to add vertices to the complex.
2. Click existing vertices to select them (up to 3), and press SPACE to add the corresponding simplex.
3. Right click to remove vertices.
4. Select vertices and press SPACE to remove the corresponding simplex if it already exists.
5. Press R to reset the drawing area.