Sparse Chain Complexes & Simplicial Homology¶

The SparseChainComplex in algebrax.homology and SimplicialComplex in algebrax.homology formalize sequence spaces \(C_k\) and sparse boundary operators \(D_k: C_k \to C_{k-1}\) enforcing the fundamental nilpotency identity:

\[D_{k-1} \circ D_k = \mathbf{0}\]

Topological Betti Numbers¶

Betti numbers \(\beta_k\) count topological holes: - \(\beta_0\): Number of connected components - \(\beta_1\): Number of 1D circular loops - \(\beta_2\): Number of 2D enclosed voids

\[\beta_k = \dim(\ker D_k) - \text{rank}(D_{k+1})\]

Python Example: Simplicial Complex Betti Numbers¶

import algebrax as ax

# 1. 1D Hollow Ring Topology (S^1)
ring_edges = [(0, 1), (1, 2), (2, 3), (0, 3)]
sc_ring = ax.homology.SimplicialComplex(ring_edges)

# Verify Nilpotency D0 o D1 == 0
assert sc_ring.verify_nilpotency(k=1)

# Compute Betti Numbers
betti_ring = sc_ring.betti_numbers(max_k=1)
print(f"1D Ring Betti Numbers: beta_0={betti_ring[0]}, beta_1={betti_ring[1]}")
# Output: 1D Ring Betti Numbers: beta_0=1, beta_1=1

# 2. 3D Solid Tetrahedron Topology
sc_tet = ax.homology.SimplicialComplex([(0, 1, 2, 3)])
betti_tet = sc_tet.betti_numbers(max_k=2)
print(f"Solid Tetrahedron Betti Numbers: beta_0={betti_tet[0]}, beta_1={betti_tet[1]}, beta_2={betti_tet[2]}")
# Output: Solid Tetrahedron Betti Numbers: beta_0=1, beta_1=0, beta_2=0