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

Abstract

Let A denote the cylinder

$${\mathbb {R}} \times S^1$$

R ×

S 1

or the band

$${\mathbb {R}} \times I$$

R × I

, where I stands for the closed interval. We consider 2-moderate immersions of closed curves (“doodles”) and compact surfaces (“blobs”) in A, up to cobordisms that also are 2-moderate immersions in

$$A \times [0, 1]$$

A × [ 0 , 1 ]

of surfaces and solids. By definition, the 2-moderate immersions of curves and surfaces do not have tangencies of order

$$\ge 3$$

≥ 3

to the fibers of the obvious projections

$$A \rightarrow S^1$$

A →

S 1

,

$$A \times [0, 1] \rightarrow S^1 \times [0, 1]$$

A ×

[ 0 , 1 ]

→

S 1

×

[ 0 , 1 ]

or

$$A \rightarrow I$$

A → I

,

$$A \times [0, 1] \rightarrow I \times [0, 1]$$

A × [ 0 , 1 ] → I × [ 0 , 1 ]

. These bordisms come in different flavors: in particular, we consider one flavor based on 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

$$A= {\mathbb {R}} \times I$$

A = R × I

, our computations of 2-moderate immersion bordisms

$$\textbf{OC}^{\textsf{imm}}_{\mathsf {moderate \le 2}}(A)$$

OC

moderate ≤ 2

imm

( A )

are near complete: we show that they can be described by an exact sequence of abelian groups

$$\begin{aligned} 0 \rightarrow {\textbf{K}} \rightarrow \textbf{OC}^{\textsf{imm}}_{\mathsf {moderate \le 2}}(A)\big /\textbf{OC}^{\textsf{emb}}_{\mathsf {moderate \le 2}}(A) {\mathop {\longrightarrow }\limits ^{{\mathcal {I}} \rho }} {\mathbb {Z}} \times {\mathbb {Z}} \rightarrow 0, \end{aligned}$$

0 → K →

OC

moderate ≤ 2

imm

( A )

/

OC

moderate ≤ 2

emb

( A )

⟶

I ρ

Z × Z → 0 ,

where

$$\textbf{OC}^{\textsf{emb}}_{\mathsf {moderate \le 2}}(A) \approx {\mathbb {Z}} \times {\mathbb {Z}}$$

OC

moderate ≤ 2

emb

( A )

≈ Z × Z

, the epimorphism

$${\mathcal {I}} \rho $$

I ρ

counts different types of crossings of immersed doodles, and the kernel

$${\textbf{K}}$$

K

contains the group

$$({\mathbb {Z}})^\infty $$

( Z )

∞

whose generators are described explicitly.