Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Relative Homology, Excision, and Long Exact Sequences

3 minute read

Published:

TL;DR: Relative homology H_n(X, A) treats the subspace A as "collapsed to a point" — cycles in A become trivial and only topology that "escapes" A is counted. The long exact sequence of a pair connects H_n(A), H_n(X), and H_n(X,A), enabling powerful decomposition arguments. Extended persistence uses relative homology to capture topology that would otherwise be missed.

Relative Chain Groups

Given a simplicial pair \((K, L)\) with \(L \subseteq K\), the relative chain group is the quotient:

$$C_n(K, L) = C_n(K) / C_n(L)$$

Chains in \(L\) are set to zero — we ignore simplices entirely inside \(L\). The boundary map on \(K\) descends to a well-defined boundary map on the quotient, giving a relative chain complex and hence relative homology groups \(H_n(K, L)\).

Intuitively: \(H_n(K, L)\) detects \(n\)-dimensional holes in \(K\) that are not already present in \(L\).

The Long Exact Sequence of a Pair

The short exact sequence \(0 \to C_*(L) \to C_*(K) \to C_*(K,L) \to 0\) induces the fundamental tool of algebraic topology — the long exact sequence:

\[\cdots \to H_n(L) \xrightarrow{i_*} H_n(K) \xrightarrow{j_*} H_n(K,L) \xrightarrow{\partial_*} H_{n-1}(L) \to \cdots\]

where \(i_*\) is induced by inclusion, \(j_*\) by projection, and \(\partial_*\) is the connecting homomorphism (which lowers degree by 1). This sequence is exact: the image of each map equals the kernel of the next.

Excision Theorem

Theorem (Excision): If \(Z \subseteq A \subseteq X\) with \(\overline{Z} \subseteq \mathrm{int}(A)\), then:

\[H_n(X, A) \cong H_n(X \setminus Z, A \setminus Z)\]

Excision says that homology of a pair is insensitive to what happens in the interior of \(A\). This enables local computation: the topology of \(X\) relative to \(A\) only depends on what happens near the boundary of \(A\).

Mayer-Vietoris Sequence

For a space \(X = A \cup B\), the Mayer-Vietoris sequence relates the homology of \(A\), \(B\), \(A \cap B\), and \(X\):

\[\cdots \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \xrightarrow{\partial} H_{n-1}(A \cap B) \to \cdots\]

This is the main tool for computing homology of spaces built from simpler pieces.

Extended Persistence

Extended persistence (Cohen-Steiner, Edelsbrunner, Harer 2009) augments the standard filtration with a dual: after growing the complex from \(\emptyset\) to \(K\), one shrinks it back. The result is a pairing that includes:

  • Ordinary pairs: born in \(H_n(K^i)\), die entering \(H_n(K^j)\) (standard persistence).
  • Relative pairs: born in \(H_n(K^i, \partial K)\), die in \(H_n(K^j, \partial K)\) — using relative homology.
  • Extended pairs: one class from homology, one from relative homology.

Extended persistence captures features that would have infinite persistence in the standard setting — particularly useful for manifold-valued data where the “top” class never dies.

Key Insight: The Mayer-Vietoris sequence is the algebraic engine behind the Čech nerve theorem: if a cover $$\mathcal{U}$$ of $$X$$ has contractible intersections, the nerve of $$\mathcal{U}$$ is homotopy equivalent to $$X$$. This justifies approximating the topology of point clouds with Vietoris-Rips and Čech complexes.

References

  • H. Edelsbrunner & J. Harer, Computational Topology, Chapter IV.
  • D. Cohen-Steiner, H. Edelsbrunner, J. Harer, “Extending Persistence Using Poincaré and Lefschetz Duality,” Foundations of Computational Mathematics, 2009.

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