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 $\geq 3$ 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 $\mathbf{OC}^{\mathsf{imm}}_{\mathsf{moderate \leq 2}}(A)$ are near complete: we show that they can be described by an exact sequence of abelian groups $$0 \to \mathbf K \to \mathbf{OC}^{\mathsf{imm}}_{\mathsf{moderate \leq 2}}(A)\big/\mathbf{OC}^{\mathsf{emb}}_{\mathsf{moderate \leq 2}}(A) \stackrel{\mathcal I ρ}{\longrightarrow} \mathbb Z \times \mathbb Z \to 0,$$ where $\mathbf{OC}^{\mathsf{emb}}_{\mathsf{moderate \leq 2}}(A) \approx \mathbb Z \times \mathbb Z$, the epimorphism $\mathcal I ρ$ counts different types of crossings of immersed doodles, and the kernel $\mathbf K$ contains the group $(\mathbb Z)^\infty$ whose generators are described explicitly.

Figures (12)

• Figure 1: Diagram (a) shows doodles---an immersion $\beta: \mathcal{C} \to A$ of $3$ loops $\mathcal{C}$ in the surface $A = \mathbb R \times [0,1]$. Diagram (b) shows blobs---an immersion $\alpha: X \to A$ of two disks $X$ in $A$. The self-intersections of the curves $\beta(\mathcal{C})$ and of $\alpha(\partial X)$ and the points of tangency of $\beta(\mathcal{C})$ and of $\alpha(\partial X)$ to the vertical foliation $\mathcal{F}(\hat{v})$ on $A$ are marked. Thanks to the presence of figure "$\infty$", $\beta$ does not extend to an immersion $\alpha$ of any compact surface $X$ into $A$.
• Figure 2: A convex quasi-envelop $\alpha: X \to A$ of a $2$-moderately generic (actually, even traversally generic) vector field $\alpha^\dagger(\partial_u)$ on a compact surface $X$, the torus from which an open smooth disk is removed (the top diagram), and on a compact surface $X$, the closed surface of genus $2$ from which an open smooth disk is removed (the bottom diagram). In both examples, the cardinality of the fibers of the map $\theta \circ \alpha: \partial X \to \mathcal{T}(\hat{v})$ does not exceed $6$.
• Figure 3: Changing topology of slices $B^{-1}(A \times \{t\})$, as $t$ crosses a critical value $t_\star$ of the Morse function $f: \delta W \to [0, 1]$. Different shades correspond to different slices; each box is shown with $3$ slices. In $\mathsf{(a)}$ and $\mathsf{(b)}$, the portion of $B(W)$ over a small interval $[t_\star -\epsilon, t_\star +\epsilon]$ is a pair of solid pants. In $\mathsf{(c), (d)}$, this portion is a solid half-ball. In $\mathsf{(e)}$ and $\mathsf{(f)}$, it is the complements to such half-balls in the solid cube. The figure does not show the complements to solid pants, depicted in $\mathsf{(a)}$ and $\mathsf{(b)}$. Note the "parabolic locus" (an arc of which is dashed) in $B(\delta W)$, where the vector field $\hat{v}$ is quadratically tangent to the surface $B(\delta W)$.
• Figure 4: Eliminating a pair of dark-shaded "kidneys" with their horns facing each other by a surgery: the first surgery transforms the kidneys in the first slice into a ring in the second slice, and then the second surgery transforms the ring into an empty slice.
• Figure 5: Two portraits of a generator $\kappa \in \pi_1(\mathcal{F}^{\leq 2}, p)$, where the base point $p \in \mathcal{F}^{\leq 2}$ is modeled after the polynomial $p(u) = u^4- 1$ in diagram (a) and by $p(u) = u^4 +1$ in diagram (b). Diagrams (c) and (d) portray $2\kappa$. In diagrams (a) and (c) that represent the case $\mathcal{T}(\hat{v}) = S^1$, the left and the right edges of the rectangle should be identified so that the shaded regions match. Note the polarity $\oplus$ of the tangent $\hat{v}$-trajectories with the combinatorial pattern $\omega = (\dots 121 \dots)$, where the number of $1$'s that precede $2$ is odd.

Theorems & Definitions (43)

• Proposition 1.1
• Proposition 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Lemma 2.1