Doodles and blobs on a lined page: convex quasi-envelops of traversing flows on surfaces

Abstract

Let $A$ denote the cylinder $\mathbb{R}\times S^1$ or the band $\mathbb{R}\times I$, where $I$ stands for the closed interval. We consider $2$-{\sf moderate immersions} of closed curves (``{\sf doodles}") and compact surfaces (``{\sf blobs}") in $A$, up to cobordisms that also are $2$-moderate immersions in $A\times [0,1]$ of surfaces and solids. By definition, the $2$-moderate immersions of curves and surfaces do not have tangencies of order $\ge 3$ to the fibers of the obvious projections $A\to S^1$,\; $A\times [0,1]\to S^1\times [0,1]$ or $A\to I$,\; $A\times [0,1]\to I\times [0,1]$. These bordisms come in different flavors: in particular, we consider one flavor based on {\sf regular embeddings} 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 $A=\mathbb{R}\times I$, our computations of $2$-moderate immersion bordisms ${\mathbf{OC}}_{\mathsf{moderate}\le 2}^{\mathsf{imm}}(A)$ are near complete: we show that they can be described by an exact sequence of abelian groups $$0\to \mathbf{K}\to {\mathbf{OC}}_{\mathsf{moderate}\le 2}^{\mathsf{imm}}(A)/{\mathbf{OC}}_{\mathsf{moderate}\le 2}^{\mathsf{emb}}(A)\stackrel{\mathcal{I}\rho }{⟶}\mathbb{Z}\times \mathbb{Z}\to 0,$$ where ${\mathbf{OC}}_{\mathsf{moderate}\le 2}^{\mathsf{emb}}(A)\approx \mathbb{Z}\times \mathbb{Z}$, the epimorphism $\mathcal{I}\rho$ counts different types of crossings of immersed doodles, and the kernel $\mathbf{K}$ contains the group $(\mathbb{Z}{)}^{\infty }$ whose generators are described explicitly.

42 pages, 12 figures