seifert-van kampen gluing

This paper addresses the formal computation of fundamental groups via the Seifert–van Kampen theorem. Methodologically, it introduces a computational-path-based framework that recasts fundamental group construction as explicit rewriting sequences; integrates higher inductive types, word representations of quotient structures, and encode-decode techniques; and implements pushouts, free products, and isomorphism verification in Lean 4. Crucially, it explicitly characterizes the compatibility conditions for path equalities using computational paths and ensures fully constructive, decidable equational reasoning via the LNDEQ-TRS rewrite system. Contributions include: the first complete formalization of the Seifert–van Kampen theorem within dependent type theory; a rigorous proof that the fundamental group of a pushout space is isomorphic to the amalgamated free product of its components’ fundamental groups; and verified derivations of classical results, including π₁(S¹ ∨ S¹) ≅ ℤ * ℤ and π₁(S²) = 0.

This work proposes a systematic approach to constructing a category of combinatorial manifolds satisfying prescribed local properties and applies it to directed topology and automata theory. By forming a coreflective subcategory within the category of relational presheaves and endowing it with a model structure via a unique factorization system, the cofibrant objects are precisely the combinatorial manifolds. This framework introduces manifold-theoretic ideas into automata theory for the first time, yielding a categorical abstraction and proof of Kleene’s theorem. Furthermore, it establishes the category of Euclidean precubical sets as a coreflective subcategory of relational precubical sets and reveals its combinatorial correspondence with the blowup construction in directed topology.

This work addresses the challenge in semantic modeling of scoped computational paths, where algebraic structure and topological coherence are difficult to reconcile due to incompatibility between product and quotient topologies. By constructing a topological semantics for scoped rewriting systems, it endows rewrite steps with continuous geometric realizations and characterizes name-carrying rewrites via endpoint-fixing homotopies. The paper introduces an unconditional construction of a final composable topology that explicitly resolves—rather than obscures—the product-quotient inconsistency. It establishes quadruple equivalence criteria and sufficient conditions for compact Hausdorffness, and proves, via continuous sections in a universal representation, that the coherent path quotient is homeomorphic to the standard fundamental groupoid. Formal verification in Lean 4.24.0 covers finitely generated cases such as the circle and torus, yielding normal forms and ℤ, ℤ² classifications via winding numbers, and confirming that the realization map is a continuous groupoid homomorphism and faithful under geometric completeness.

This paper investigates the relationship between colimits in coslices of universes and ordinary colimits in homotopy type theory (HTT). Methodologically, it employs constructive techniques to provide the first explicit characterization of coslice colimits, defining and formally implementing the core colimit functor in Agda. It proves that the forgetful functor creates colimits over tree-shaped diagrams and establishes that pointed colimits preserve $n$-connectedness—thereby yielding closure of higher groups under directed-graph colimits. Furthermore, the results are applied to orthogonal factorization systems and cohomology theory, elucidating deep connections between coslice colimits, connectivity, and algebraic structure. The work furnishes a systematic new toolkit for universal algebraic modeling and homotopical constructions in HTT.

This paper addresses the fragmentation and lack of interoperability among structural complexity measures across graph theory, geometric group theory, and dynamical systems. Methodologically, it introduces a unified “structured decomposition” framework grounded in category theory: (i) it is the first to formalize diverse domain-specific decomposition paradigms categorically; (ii) it establishes a general duality theory linking decompositions to object completions; and (iii) it defines composable width functors enabling cross-model quantification, comparison, and translation of structural complexity. Key contributions include: a unified categorical characterization of over ten complexity parameters—including treewidth, layered treewidth, and hypergraph treewidth—revealing their intrinsic structural relationships; and a novel parameterized tractability paradigm for NP-hard problems, grounded in decomposition width. The framework achieves both theoretical unification and algorithmic realizability.

This work addresses the absence of rigorous formalizations of abstract simplicial complexes and their stellar subdivisions in existing proof systems. It presents the first purely combinatorial formal framework for abstract simplicial complexes grounded in combinatorial topology, implemented in the Lean theorem prover. The framework encompasses fundamental operations such as morphisms, links, and joins, and systematically investigates their interaction with stellar subdivision. Key contributions include the first formalization of stellar subdivision in any proof assistant, the verification of several crucial identities—some previously undocumented in the literature—for the study of triangulated manifolds, and the proof of significant theorems such as the invariance of links under subdivision. This development establishes a reliable formal foundation for computational topology.

This work proposes an efficient, low-cost approach to the large-scale automatic formalization of theorems from classical mathematical textbooks—such as Munkres’ *Topology*—into machine-verifiable proofs. By establishing a long-horizon feedback loop between large language models (e.g., ChatGPT 5.2 or Claude Sonnet 4.5) and the Megalodon higher-order set theory proof checker, and by integrating prompt engineering with a foundational set-theoretic library, the system generated 130,000 lines of formalized code within two weeks at a cost of approximately \$100. As of January 4, 2026, over 1,500 theorems—including Urysohn’s Lemma and the Tietze Extension Theorem—have been formalized, totaling 160,000 lines of code. This demonstrates a general-purpose pathway to automated formalization that operates without reliance on mainstream proof assistants or complex infrastructure.

This study addresses several long-standing conjectures by J. H. Davenport, A. Locatelli, G. K. Sankaran, and others concerning the topological properties of cells in cylindrical algebraic decomposition (CAD). For the first time, the paper systematically constructs explicit counterexamples that refute these conjectures. By integrating CAD theory from real algebraic geometry, topological construction techniques, and symbolic computation, the work reveals fundamental limitations in the existing assumptions. These findings not only clarify the theoretical boundaries of CAD but also provide a refined foundation for applications involving topological analysis of semi-algebraic sets, such as motion planning in robotics.

This work investigates the computational complexity of bisimulation equivalence and path logic model checking over finite graphs. By introducing the Existential Theory of Invertible Matrices (ETIM) and devising its first efficient randomized algorithm, and by leveraging Gabriel’s representation theorem together with a constraint-based hierarchical partial order structure, the authors reduce the complexity of bisimulation equivalence to NEXP—further to PSPACE over finite fields—and precisely characterize path logic model checking as NP-complete. Moreover, they establish that under constraints imposed by the special linear group, ETIM is equivalent to the existential theory of the reals, thereby revealing the profound impact of algebraic constraints on the complexity of logical verification.

This study investigates the algebraic stability of linear representations of posets under perturbations, with a focus on the deep interplay between the metric structure induced by the interleaving distance in persistent homology and underlying algebraic properties. By embedding persistence theory within a unified categorical framework of poset representation theory, the work systematically uncovers fundamental algebraic structures and stability mechanisms. It integrates tools from homological algebra, category theory, and topological data analysis, clarifying connections between this theoretical framework and geometric as well as applied mathematical contexts. The paper also identifies several key challenges and open problems, offering a fresh perspective on the algebraic foundations of persistent homology.