Alessio Borgi

Alessio Borgi

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

Homotopy, Contractibility, and Deformation Retracts

2 minute read

Published:

TL;DR: A homotopy is a continuous deformation between two maps; two maps are homotopic if one can be continuously deformed into the other. Homotopy equivalence of spaces preserves all homological invariants. A contractible space has the homology of a point — a fundamental concept when analysing what filtration steps change topologically.

Homotopy Between Maps

Let \(f, g: X \to Y\) be continuous maps. A homotopy from \(f\) to \(g\) is a continuous map:

$$H: X \times [0,1] \to Y \quad \text{with} \quad H(x,0) = f(x) \text{ and } H(x,1) = g(x)$$

We write \(f \simeq g\) and say \(f\) is homotopic to \(g\). Homotopy is an equivalence relation on continuous maps.

Intuitively: \(H\) continuously deforms \(f\) into \(g\) as the parameter \(t\) runs from 0 to 1. Every point traces a path in \(Y\).

Homotopy Equivalence of Spaces

Two spaces \(X\) and \(Y\) are homotopy equivalent (\(X \simeq Y\)) if there exist continuous maps \(f: X \to Y\) and \(g: Y \to X\) such that:

\[g \circ f \simeq \mathrm{id}_X \qquad \text{and} \qquad f \circ g \simeq \mathrm{id}_Y\]

This is weaker than homeomorphism (\(f\) and \(g\) need not be inverses, only homotopy inverses). But it is exactly the relation that preserves homology groups. Key examples:

  • \(\mathbb{R}^n \simeq \{*\}\) (a point) — any convex subset is contractible.
  • An annulus \(\simeq S^1\) — the inner boundary can be expanded to fill the hole.
  • \(\mathbb{R}^2 \setminus \{0\} \simeq S^1\) — the punctured plane deformation retracts onto any circle around the origin.

Contractible Spaces

A space \(X\) is contractible if \(X \simeq \{*\}\), i.e., the identity map \(\mathrm{id}_X\) is homotopic to a constant map. Contractible spaces have:

\[H_0(X) \cong \mathbb{F}, \qquad H_n(X) = 0 \text{ for all } n \geq 1\]

In filtrations: when a new simplex is added and the result is contractible, no topology is created or destroyed — this step has zero persistence and is topologically trivial.

Deformation Retracts

A deformation retract of \(X\) onto a subspace \(A \subseteq X\) is a homotopy \(H: X \times [0,1] \to X\) such that:

  • \(H(x,0) = x\) for all \(x \in X\),
  • \(H(x,1) \in A\) for all \(x \in X\),
  • \(H(a,t) = a\) for all \(a \in A\) and \(t \in [0,1]\).

Deformation retracts give homotopy equivalences: \(X \simeq A\). In TDA, deformation retracts appear when analysing how adding a simplex to a complex changes its homotopy type.

Key Insight: The Mayer-Vietoris sequence and excision theorems — powerful tools for computing homology — rely on homotopy equivalence. When a filtration step collapses a contractible subcomplex, the homology is unchanged. The persistence algorithm implicitly tracks exactly these topological changes via the boundary matrix reduction.

References

  • A. Hatcher, Algebraic Topology, Section 0. Free at pi.math.cornell.edu/~hatcher.
  • T. Bröcker & K. Jänich, Introduction to Differential Topology, Cambridge, 1982.

You May Also Enjoy

GAPE: Remember to Forget — Gated Adaptive Positional Encoding

4 minute read

Published:

GAPE is a drop-in RoPE augmentation that adds content-aware attention logit biases: a query-gate suppresses irrelevant distant context while a key-gate preserves salient distant tokens. Provably sharper attention and improved long-context robustness — no architecture changes needed.

PolyNSD: Polynomial Neural Sheaf Diffusion

4 minute read

Published:

PolyNSD replaces the NSD propagation operator with a degree-K Chebyshev polynomial in the normalised sheaf Laplacian, achieving SOTA on homo- and heterophilic benchmarks with only diagonal restriction maps and dramatically lower memory usage.

TDA in Materials Science: Topology of Structure and Phase

3 minute read

Published:

Persistent homology characterises the multi-scale structure of materials — from pore geometry in catalysts to glass transition in amorphous materials. H₀ captures connectivity, H₁ captures channels and rings, H₂ captures enclosed voids. These topological descriptors predict mechanical properties, adsorption capacity, and phase transitions more accurately than geometric averages.

TDA in Drug Discovery: Molecular Topology

3 minute read

Published:

Persistent homology provides multi-scale topological fingerprints of molecular structures — capturing rings, cavities, and protein pockets that traditional cheminformatics descriptors miss. Applications include binding site detection, ADMET property prediction, and protein-ligand interaction modelling.