Loops of Infinite Order and Toric Foliations

The paper constructs a locally constant bundle epimorphism ρ: L → Z^l (via H1(M)/Tor) and proves that ker ρ_i yields nested lattice subbundles and hence nested toric foliations; in n=2, nontrivial ρ implies a trivial rank-1 subbundle and a compatible free S1-action. These are stated and proved as Theorems 18 and 19. The candidate solution follows the same structure: (A) it uses a non-torsion loop to ensure ρ has positive rank, puts ρ in local integer normal form, defines L(i)=ker(ρ1,…,ρi), and builds the foliations; (B) for n=2 it reduces transition maps to the stabilizer of (1,0) in SL(2,Z), producing a global primitive section and the free S1-action Θ([t],p)=Φ_{t·σ(qΦ(p))}(p). This is essentially the same proof strategy as the paper, differing mostly in presentation (more algebraic normal forms vs the paper’s monodromy-first route). Minor note: the candidate has a small typographical slip once writing L(i)=ker(ρ1,…,ρl) instead of ker(ρ1,…,ρi), but the construction elsewhere is consistent. Key ingredients match the paper’s results on ρ’s local constancy and epimorphism, ker ρ1’s smoothness and local splitting, the induced foliations, and the S1-action construction. See the paper’s Definition/Construction of ρ and local constancy (Theorem 14 and Corollary 15), the splitting (Theorem 16), the foliation (Corollary 17), and the main statements (Theorems 18–19), and the explicit S1-action formula in Proposition 12.