Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces
Let A denote the cylinder $${\mathbb {R}} \times S^1$$ R × S 1...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Springer Science and Business Media LLC
2024
|
| Online Access: | https://hdl.handle.net/1721.1/155007 |
| _version_ | 1811090153147465728 |
|---|---|
| author | Katz, Gabriel |
| author_facet | Katz, Gabriel |
| author_sort | Katz, Gabriel |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T14:35:22Z |
| format | Article |
| id | mit-1721.1/155007 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:35:22Z |
| publishDate | 2024 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1550072024-09-18T04:29:03Z Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces Katz, Gabriel 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. 2024-05-20T18:46:12Z 2024-05-20T18:46:12Z 2024-05-16 2024-05-19T03:13:27Z Article http://purl.org/eprint/type/JournalArticle 2199-6792 2199-6806 https://hdl.handle.net/1721.1/155007 Katz, G. Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces. Arnold Math J. (2024). PUBLISHER_CC en 10.1007/s40598-024-00249-6 Arnold Mathematical Journal Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer Science and Business Media LLC Springer International Publishing |
| spellingShingle | Katz, Gabriel Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces |
| title | Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces |
| title_full | Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces |
| title_fullStr | Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces |
| title_full_unstemmed | Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces |
| title_short | Doodles and Blobs on a Ruled Page: Convex Quasi-envelops of Traversing Flows on Surfaces |
| title_sort | doodles and blobs on a ruled page convex quasi envelops of traversing flows on surfaces |
| url | https://hdl.handle.net/1721.1/155007 |
| work_keys_str_mv | AT katzgabriel doodlesandblobsonaruledpageconvexquasienvelopsoftraversingflowsonsurfaces |