Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces
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40598_2024_Article_249.pdf
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1.54 MB
Format
Adobe PDF
Checksum (MD5)
e8b97ebf4e8d4bc2d676882c4380159d
Author(s)
Katz, Gabriel
Date Issued
May 16, 2024
Journal
Arnold Mathematical Journal
Publisher
Springer Science and Business Media LLC
Citation
Katz, G. Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces. Arnold Math J. (2024).
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s40598-024-00249-6
