EP-0110: Simplicial Homology, Betti Numbers & Persistent Barcodes¶

Abstract¶

This proposal specifies algebrax.homology, building upon the foundational SparseChainComplex (EP-0101). It introduces SimplicialComplex for \(k\)-simplices \((v_0, \dots, v_k)\), calculates topological hole invariants (Betti numbers \(\beta_0, \beta_1, \beta_2\)), and provides persistent homology filtration analysis for Topological Data Analysis (TDA).

Specification¶

1. Simplicial Complex Construction¶

class SimplicialComplex(SparseChainComplex):
    """
    Simplicial Complex built on SparseChainComplex (EP-0101).
    """
    def add_simplex(self, simplex: tuple[int, ...]) -> None:
        ...

2. Topological Betti Numbers \(\beta_k\)¶

def betti_numbers(complex: SimplicialComplex, max_k: int = 2) -> dict[int, int]:
    """
    Compute Betti numbers [beta_0, beta_1, ..., beta_max_k] for the complex.
    """
    ...

Deliverables¶

1. Core Implementation: src/algebrax/homology.py (SimplicialComplex, betti_numbers, persistent_homology).
2. Unit Tests: tests/algebrax/test_homology.py (verifying Betti numbers for spheres, tori, and point clouds).
3. Use Case Recipe: recipes/topological_homology_betti.py & .ipynb.
4. Graphical Laboratory View: View 21 (view_topological_homology_group) in recipes/lab.py.