pySurgery

pySurgery is a Python library for computational algebraic topology and computational surgery theory: it turns discrete data (CW/simplicial complexes, meshes, TDA pipelines) into integer invariants strong enough to go beyond Betti numbers—e.g. torsion, cup products, intersection forms, and (in key cases) homeomorphism classification signals.

If you’ve ever felt “persistent homology is informative, but not decisive,” pySurgery is aimed at the next layer of structure.

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Why this exists

Most numerical linear-algebra approaches over $\mathbb{R}$ lose information that lives over $\mathbb{Z}$ (especially torsion). pySurgery focuses on exact and structure-aware invariants:

• Homology & cohomology over $\mathbb{Z}$ via Smith Normal Form (SNF) (captures torsion)
• Coefficient support for Z, Q, and Z/pZ (including composite moduli via UCT decomposition)
• Cup product (Alexander–Whitney) to extract intersection information
• Intersection forms for 4-manifolds (rank/signature/parity)
• Hooks for surgery-theoretic obstructions and higher-dimensional classification workflows

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Quickstart

Install

Requires Python >= 3.10+.

Project metadata (including version and Python requirement) is defined in pyproject.toml.

Install from PyPI (recommended)

pip install pysurgery

PyPI package: https://pypi.org/project/pysurgery/

Install from source (development)

git clone https://github.com/gabe-rbo/pySurgery.git
cd pySurgery
pip install -e .

Optional dependency groups:

# Topological Data Analysis integration
pip install "pysurgery[tda]"

# Mesh integration
pip install "pysurgery[mesh]"

# ML-oriented integrations
pip install "pysurgery[ml]"

# Everything optional
pip install "pysurgery[all]"

Julia backend (optional, for large-scale exact integer computations)

pySurgery can offload some exact integer workloads to Julia (recommended for very large sparse exact-$\mathbb{Z}$ computations).

• Ensure julia is on your PATH
• Install AbstractAlgebra in your Julia environment:

import Pkg
Pkg.add("AbstractAlgebra")

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Examples and tutorials

A step-by-step interactive curriculum is provided in examples/ (Jupyter notebooks), covering topics from algebraic topology basics to surgery-theoretic workflows:

1. 01_basic_homology_and_cohomology.ipynb — CW complexes, SNF, UCT
2. 02_intersection_forms.ipynb — 4D classification concepts, spin checks
3. 03_algebraic_surgery.ipynb — isotropic classes and algebraic surgery mechanics
4. 04_advanced_tda_and_surgery_theory.ipynb — raw data → GUDHI → cup product → intersection form
5. 05_omni_dimensional_homeomorphisms.ipynb — how classification behavior changes by dimension
6. 06_kirby_calculus_and_characteristic_classes.ipynb — characteristic classes & Kirby-style ideas
7. 07_fundamental_group_and_structure_set.ipynb — $\pi_1$ extraction, Whitehead data, and typed structure-set workflows
8. 08_certificate_workflows.ipynb — explicit homeomorphism witnesses, obstruction interference, and surgery planning
9. 09_branch_matrix_and_failure_modes.ipynb — success/impediment/inconclusive/surgery-required branch matrix
10. 10_witness_builder_reference.ipynb — witness-builder API reference and dispatcher coverage
11. 11_capstone_end_to_end_workflow.ipynb — one full problem-solving pipeline using the library end-to-end
12. 12_topological_denoising_via_surgery.ipynb — noisy torus denoising by index-1 surgery, with generator pushforward visuals

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Mathematical basis

pySurgery bridges discrete geometry to topological classification using several key concepts.

1) Homology & cohomology via Smith Normal Form (over $\mathbb{Z}$)

When computing homology,

$$ H_n(X)=\ker(d_n)/\mathrm{im}(d_{n+1}) $$

floating-point linear algebra misses torsion. pySurgery implements Smith Normal Form over $\mathbb{Z}$ to recover:

• Betti ranks (free part)
• torsion coefficients (e.g. $\mathbb{Z}_k$ factors)

Cohomology is computed via the Universal Coefficient Theorem.

2) Alexander–Whitney cup product

To classify 4-manifolds, surfaces intersect—cup products encode that intersection.

Given cocycles $\alpha, \beta$:

$$ (\alpha \smile \beta)([v_0,\dots,v_4])=\alpha([v_0,v_1,v_2])\cdot\beta([v_2,v_3,v_4]). $$

Summing over a fundamental class yields intersection form entries.

pySurgery also provides simplicial cup-$i$ operations (i >= 0) for higher-order cochain interactions.

3) Intersection forms & 4D classification signals

For 4-manifolds, pySurgery constructs the intersection form $Q$, computes rank/signature/parity, and supports workflows inspired by Freedman-style classification (with attention to needed hypotheses/invariants).

4) Algebraic surgery

Given a class $x$ representing a “hole,” pySurgery supports algorithmic manipulations that mirror algebraic surgery operations—checking isotropy $Q(x,x)=0$, finding dual classes, and updating lattice data.

5) Characteristic classes & spin structures

pySurgery extracts characteristic-class information from intersection-form data; using Wu’s formula, it computes $w_2$ (mod 2) to test spin conditions.

6) High-dimensional structure signals

In dimensions $n \ge 5$, classification interacts with $\pi_1$, Whitehead torsion, and Wall’s surgery groups $L_n(\pi_1)$. pySurgery includes tooling aimed at discrete extraction of group presentations and surgery-sequence-adjacent invariants (where computationally feasible).

7) Dimension-aware behavior

• 2D: orientability/genus signals via low-dimensional homology
• 3D: homology-sphere style signals and 3-manifold context
• 4D: intersection forms + parity/signature style invariants
• 5D+: surgery-theoretic invariants and $\pi_1$-sensitive obstructions

8) Data-grounded optimal homology generators

For discrete simplicial data, pySurgery includes data-grounded generator extraction for $H_1$ (cycle representatives as edge-level elements of the input complex), with Julia-accelerated and Python fallback paths.

The generator/optimality pipeline follows the "Generators and Optimality" framework in:

• Tamal K. Dey and Yusu Wang, Computational Topology for Data Analysis.

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Computational optimizations

Discrete topology can blow up combinatorially (e.g., large point clouds). pySurgery includes:

1. Sparse fallbacks when exact dense integer work is too large
2. NumPy/Numba acceleration for inner loops (e.g., cup product sweeps)
3. Julia bridge for large exact-$\mathbb{Z}$ workloads when Python becomes the bottleneck

Approximate (floating-point) fallbacks exist in selected large/sparse workflows, but are opt-in where mathematically sensitive and emit explicit warnings when exact integer guarantees are weakened.

Exactness policy

pySurgery defaults to exact integer-topology workflows whenever practical.

• Default mode: exact-first (allow_approx=False in APIs that expose it)
• Approximate mode: explicitly opt in via allow_approx=True
• Warnings: approximate paths emit warnings when torsion/exact-cycle guarantees may be weakened

For mathematically rigorous classification/proof-oriented workflows, prefer exact mode and treat approximate mode as exploratory.

Example: exact vs approximate fundamental class extraction

from pysurgery.integrations.gudhi_bridge import simplex_tree_to_intersection_form

# Exact-first: raise if exact extraction cannot be completed.
q_exact = simplex_tree_to_intersection_form(st, allow_approx=False)

# Exploratory fallback: allows numerical approximation with warnings.
q_fast = simplex_tree_to_intersection_form(st, allow_approx=True)

Example: exact vs approximate dimension-aware homeomorphism analysis

from pysurgery.homeomorphism import analyze_homeomorphism_2d

# Exact-first classification attempt.
ok_exact, msg_exact = analyze_homeomorphism_2d(c1, c2, allow_approx=False)

# Permissive fallback for noisy/large pipelines.
ok_fast, msg_fast = analyze_homeomorphism_2d(c1, c2, allow_approx=True)

Homeomorphism witness builders

pysurgery.homeomorphism answers: "is a homeomorphism certified by invariants/obstructions?"

pysurgery.homeomorphism_witness goes further and returns a structured witness object when exact data is sufficient.

from pysurgery.homeomorphism_witness import build_homeomorphism_witness

res = build_homeomorphism_witness(c1=c1, c2=c2, dim=2)
if res.status == "success":
    print(res.witness.kind)        # e.g. "surface_classification"
    print(res.witness.theorem)     # theorem branch used
    print(res.witness.certificates)
else:
    print(res.reasoning)
    print(res.missing_data)

Dimension-specific builders

from pysurgery.homeomorphism_witness import (
    build_surface_homeomorphism_witness,
    build_3d_homeomorphism_witness,
    build_4d_homeomorphism_witness,
    build_high_dim_homeomorphism_witness,
)

• build_surface_homeomorphism_witness(...) (2D)
• build_3d_homeomorphism_witness(...) (3D)
• build_4d_homeomorphism_witness(...) (Freedman branch)
• build_high_dim_homeomorphism_witness(...) (n >= 5, surgery-theoretic branch)

For n >= 5, an explicit completion certificate can be supplied via homotopy_completion_certificate={"provided": True, "exact": True, "validated": True, ...}. Certified success requires this certificate to be decision-ready (exact and validated). Legacy homotopy_witness_hook inputs are still accepted and bridged for compatibility. For nontrivial product groups, factor-wise Wall decompositions are exposed as structured surrogate certificates; exact success still requires assembly-certified product-group obstruction data.

For 3D geometric-recognition completion, you can provide recognition_certificate={"provided": True, "exact": True, "validated": True, ...}. This allows exact success in non-Poincare branches when the certificate is decision-ready.

For high-dimensional nontrivial product groups, provide product_assembly_certificate={"provided": True, "exact": True, "validated": True, ...} to upgrade surrogate product decompositions into decision-ready assembly evidence.

4D explicit algebraic witness

In definite 4D cases, the witness may include an integer isometry matrix U (stored in witness.certificates["isometry_matrix"]) such that:

$$ U^T Q_1 U = Q_2. $$

This is an explicit algebraic certificate for the lattice isomorphism part of the classification branch.

Interpreting witness results

• status="success": exact theorem/certificate path produced a witness.
• status="inconclusive": not enough exact data to construct a certified witness (check missing_data).
• status="surgery_required": obstruction branch indicates surgery-level impediment instead of direct witness construction.

witness.exact indicates whether the returned witness is exact-topology certified (True) or exploratory/heuristic (False).

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Package layout

pysurgery/
├── core/
│   ├── cup_product.py
│   ├── complexes.py
│   ├── characteristic_classes.py
│   ├── exceptions.py
│   ├── fundamental_group.py
│   ├── group_rings.py
│   ├── intersection_forms.py
│   ├── k_theory.py
│   ├── kirby_calculus.py
│   ├── math_core.py
│   └── quadratic_forms.py
├── bridge/
│   ├── julia_bridge.py
│   └── surgery_backend.jl
├── integrations/
│   ├── gudhi_bridge.py
│   ├── trimesh_bridge.py
│   ├── pytorch_geometric_bridge.py
│   ├── jax_bridge.py
│   └── lean_export.py

Key modules:

• pysurgery.core.complexes — CWComplex, ChainComplex

• pysurgery.core.intersection_forms — lattice math, parity checks, surgery engine

• pysurgery.core.math_core — SNF / exact integer computation core

• pysurgery.homeomorphism — dimension-aware analyzers

• pysurgery.homeomorphism_witness — explicit witness/certificate builders

• pysurgery.integrations

  • gudhi_bridge — TDA point clouds → topology objects
  • trimesh_bridge — meshes → topology objects
  • pytorch_geometric_bridge — graph/ML integration
  • jax_bridge — differentiable / approximation-oriented integration ideas

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Contributing

PRs are welcome—especially for:

• minimal examples (“one function call → one invariant” demos)
• improved docs around mathematical assumptions and failure modes
• performance improvements and additional exact-$\mathbb{Z}$ backends
• more datasets and reproducible example pipelines

If you’re unsure where to start, open an issue describing your use-case and dataset type.

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Roadmap (selected)

• Non-abelian / twisted coefficient systems ($\pi_1$-sensitive homology)
• Deeper algebraic surgery tooling (Ranicki-style formulations)
• Learning on structure-set features via GNNs (experimental)

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License

MIT License. See LICENSE.