Klein’s bottle and torsion in homology (1882)

Felix Klein’s bottle glues a cylinder’s ends with a reflection. It is a closed surface with one side and no edge. It cannot sit in \(\mathbb{R}^3\) without crossing itself, as the figure-8 immersion below does. Its first homology group is \(\mathbb{Z} \oplus \mathbb{Z}/2\): a loop that is not a boundary but whose double is. The \(\mathbb{Z}/2\) torsion hides from rational Betti numbers and shows up mod 2.

import matplotlib.pyplot as plt

from mathematicskit.topology import betti_numbers, homology, is_orientable, klein_bottle, torus
from mathematicskit.topology.visualizers import plot_complex

K = klein_bottle(40, 24)
T = torus(6, 4)
for name, X in (("torus", T), ("Klein bottle", klein_bottle(6, 4))):
    print(f"{name:>12}: {homology(X)}; orientable = {is_orientable(X)}")
    print(f"{'':>12}  Betti over Q: {betti_numbers(X)}, over Z/2: {betti_numbers(X, field=2)}, over Z/3: {betti_numbers(X, field=3)}")

       torus: H_0 = Z, H_1 = Z^2, H_2 = Z; orientable = True
              Betti over Q: (1, 2, 1), over Z/2: (1, 2, 1), over Z/3: (1, 2, 1)
Klein bottle: H_0 = Z, H_1 = Z + Z/2, H_2 = 0; orientable = False
              Betti over Q: (1, 1, 0), over Z/2: (1, 2, 1), over Z/3: (1, 1, 0)

The figure-8 immersion

fig = plt.figure(figsize=(7, 6))
ax = fig.add_subplot(projection="3d")
heights = K.coordinates[:, 2]
plot_complex(K, ax=ax, face_values=[heights[list(t)].mean() for t in K.simplices(2)], alpha=0.85, vertex_size=0)
ax.view_init(elev=30, azim=-50)
ax.set_axis_off()
ax.set_title(r"Klein bottle: $H_1 = \mathbb{Z} \oplus \mathbb{Z}/2$")

Text(0.5, 1.0, 'Klein bottle: $H_1 = \\mathbb{Z} \\oplus \\mathbb{Z}/2$')