The Seifert-van Kampen Theorem via Computational Paths: A Formalized Approach to Computing Fundamental Groups

Abstract

The Seifert-van Kampen theorem computes the fundamental group of a space from the fundamental groups of its constituents. We develop a modular SVK framework within the setting of computational paths - an approach to equality where witnesses are explicit sequences of rewrites governed by the LNDEQ-TRS. Our contributions are: (i) pushouts as higher-inductive types with modular typeclass assumptions for computation rules; (ii) free products and amalgamated free products as quotients of word representations; (iii) an SVK equivalence schema parametric in user-supplied encode/decode structure; and (iv) instantiations for classical spaces - figure-eight (pi_1(S^1 v S^1) = Z * Z), 2-sphere (pi_1(S^2) = 1), and 3-sphere (pi_1(S^3) = 1) with Hopf fibration context. Recent extensions include higher homotopy groups pi_n via weak infinity-groupoid structure (with pi_2 abelian via Eckmann-Hilton), and pi_1 >= 1 in the 1-groupoid truncated setting; truncation levels connecting the framework to HoTT; automated path simplification tactics; basic covering space theory with pi_1-actions on fibers; fibration theory with long exact sequences; and Eilenberg-MacLane space characterization (S^1 = K(Z,1)). The development is formalized in Lean 4 with 41,130 lines across 107 modules, using 36 kernel axioms for HIT type-constructor declarations.