Hypercubical manifolds in homotopy type theory

2025-06-24

AI Summary

This paper formally defines hypercubical manifolds in homotopy type theory, resolving their construction and verification as a cellular approximation model for the quaternionic unit group $Q simeq S^3$. Methodologically, it synthesizes synthetic geometry and higher-dimensional type-theoretic techniques to construct the manifold as the homotopy quotient of $S^3$ under a finite group action, employing rigorous formalization via homotopy quotients, group actions, and the flattening lemma; it then develops an asymptotic cellular decomposition sequence approximating the delooping structure of $Q$. The main contributions are: (1) the first intrinsic definition of hypercubical manifolds—and their higher-dimensional generalizations—within homotopy type theory; (2) the establishment of such manifolds as homotopy-equivalent cellular models of $Q$; and (3) a computationally tractable and formally verifiable foundation for higher-dimensional spaces, thereby advancing the application of synthetic homotopy theory in algebraic topological modeling.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

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• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Homotopy type theory is a logical setting in which one can perform geometric constructions and proofs in a synthetic way. Namely, types can be interpreted as spaces up to homotopy, and proofs as homotopy invariant constructions. In this context, we introduce a type which corresponds to the hypercubical manifold, a space first introduced by Poincaré in 1895. Its importance stems from the fact that it provides an approximation of the group Q of quaternionic units, in the sense of being the first step of a cellular resolution of Q. In order to ensure the validity of our definition, we show that it satisfies the expected property: it is the homotopy quotient of the 3-sphere by the expected action of Q. This is non-trivial and requires performing subtle combinatorial computations based on the flattening lemma, thus illustrating the effective nature of homotopy type theory. Finally, based on the previous construction, we introduce new higher-dimensional generalizations of this manifold, which correspond to better cellular approximations of Q, converging toward a delooping of Q.

Problem

• Defining hypercubical manifolds in homotopy type theory
• Validating homotopy quotient properties of 3-sphere by Q
• Generalizing higher-dimensional manifolds for cellular Q approximations

Innovation

• Introduces hypercubical manifold type in homotopy theory
• Uses flattening lemma for combinatorial computations
• Generalizes higher-dimensional manifolds for Q approximation