Quickstart

This page demonstrates a minimal end-to-end example using a classical circle bundle: the Hopf fibration \(S^3 \to S^2\). The Hopf fibration is a non-trivial orientable circle bundle with Euler number \(\pm 1\).

Minimal Working Example

import numpy as np
import circle_bundles as cb

# Sample points on S^3
n_samples = 3000
rng = np.random.default_rng(0)
s3_data = cb.sample_sphere(n_samples, dim=3, rng=rng)

# Hopf projection S^3 -> S^2
base_points = cb.hopf_projection(s3_data)

# Build an open cover of S^2
n_landmarks = 60
s2_cover = cb.get_s2_fibonacci_cover(
    base_points,
    n_vertices=n_landmarks,
)

bundle = cb.Bundle(X=s3_data, U=s2_cover.U)
triv_result = bundle.get_local_trivs(verbose = False)
class_result = bundle.get_classes()
print(class_result.summary_text)

=== Characteristic Classes ===
  Stiefel–Whitney:             w₁ = 0 (orientable)
  Euler number:                ±1 (not spin)