Brouwer, Alexander, and the invariance of homology under subdivision (1911-1915)

Is homology a property of the space or of the chosen triangulation? Brouwer’s simplicial approximation (1911) answers it. Any continuous map is close to a simplicial one after subdividing finely enough. Alexander (1915) used this to prove that homology is a topological invariant. Barycentric subdivision replaces every simplex by the chains of its faces. The simplex counts grow about sixfold per round for surfaces, while the Betti numbers do not change.

import matplotlib.pyplot as plt

from mathematicskit.topology import barycentric_subdivision, betti_numbers, projective_plane, simplex, torus
from mathematicskit.topology.visualizers import plot_complex

for name, K in (("torus", torus(3, 3)), ("projective plane", projective_plane())):
    for level in range(3):
        print(f"{name:>16}, sd^{level}: f = {K.f_vector!s:>18}  chi = {K.euler_characteristic}  Betti = {betti_numbers(K)}  mod 2: {betti_numbers(K, field=2)}")
        if level < 2:
            K = barycentric_subdivision(K)

           torus, sd^0: f =        (9, 27, 18)  chi = 0  Betti = (1, 2, 1)  mod 2: (1, 2, 1)
           torus, sd^1: f =     (54, 162, 108)  chi = 0  Betti = (1, 2, 1)  mod 2: (1, 2, 1)
           torus, sd^2: f =    (324, 972, 648)  chi = 0  Betti = (1, 2, 1)  mod 2: (1, 2, 1)
projective plane, sd^0: f =        (6, 15, 10)  chi = 1  Betti = (1, 0, 0)  mod 2: (1, 1, 1)
projective plane, sd^1: f =       (31, 90, 60)  chi = 1  Betti = (1, 0, 0)  mod 2: (1, 1, 1)
projective plane, sd^2: f =    (181, 540, 360)  chi = 1  Betti = (1, 0, 0)  mod 2: (1, 1, 1)

Subdividing a triangle

triangle = simplex(2)
triangle.coordinates = [[0.0, 0.0], [1.0, 0.0], [0.5, 0.866]]
fig, axes = plt.subplots(1, 3, figsize=(12, 4))
K = triangle
for level, ax in enumerate(axes):
    plot_complex(K, ax=ax, face_values=list(range(len(K.simplices(2)))), cmap="tab20", alpha=0.6, vertex_size=6)
    ax.set_title(f"sd^{level}: f = {K.f_vector}, χ = {K.euler_characteristic}")
    ax.axis("off")
    K = barycentric_subdivision(K)
fig.suptitle("Barycentric subdivision changes the triangulation, not the space")

Text(0.5, 0.98, 'Barycentric subdivision changes the triangulation, not the space')