Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces

Let A denote the cylinder R × S¹ R × S 1 or the band R × I R × I , where I stands for the closed interval. We consider 2- of closed curves (“”) and compact surfaces (“”) in A , up to cobordisms that also are 2-moderate immersions in A × [0, 1] A × [ 0 , 1 ] of surfaces and solids. By definition, the 2-moderate immersions of curves and surfaces do not have tangencies of order ≥ 3 ≥ 3 to the fibers of the obvious projections A → S¹ A → S 1 , A × [0, 1] → S¹ × [0, 1] A × [ 0 , 1 ] → S 1 × [ 0 , 1 ] or A → I A → I , A × [0, 1] → I × [0, 1] A × [ 0 , 1 ] → I × [ 0 , 1 ] . These bordisms come in different flavors: in particular, we consider one flavor based on of doodles and blobs in A . We compute the bordisms of regular embeddings and construct many invariants that distinguish between the bordisms of immersions and embeddings. In the case of oriented doodles on A= R × I A = R × I , our computations of 2-moderate immersion bordisms OCⁱᵐᵐmoderate ≤ 2(A) OC moderate ≤ 2 imm ( A ) are near complete: we show that they can be described by an exact sequence of abelian groups $${aligned} 0 → {K} → OCⁱᵐᵐmoderate ≤ 2(A) /OCᵉᵐᵇmoderate ≤ 2(A) { { } ^{{I} ρ }} {Z} × {Z} → 0, {aligned}$$ 0 → K → OC moderate ≤ 2 imm ( A ) / OC moderate ≤ 2 emb ( A ) ⟶ I ρ Z × Z → 0 , where $$OCᵉᵐᵇmoderate ≤ 2(A) ≈ {Z} × {Z}$$ OC moderate ≤ 2 emb ( A ) ≈ Z × Z , the epimorphism $${I} ρ $$ I ρ counts different types of crossings of immersed doodles, and the kernel $${K}$$ K contains the group $$({Z})^∞ $$ ( Z ) ∞ whose generators are described explicitly.