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A Homotopy Invariant of Image Simple Fold Maps to Oriented Surfaces.

Record Type:	Language materials, manuscript : Monograph/item
Title/Author:	A Homotopy Invariant of Image Simple Fold Maps to Oriented Surfaces./
Author:	Kahmeyer, Liam.
Description:	1 online resource (77 pages)
Notes:	Source: Dissertations Abstracts International, Volume: 84-11, Section: B.
Contained By:	Dissertations Abstracts International84-11B.
Subject:	Mathematics. -
Online resource:	click for full text (PQDT)
ISBN:	9798379570118

A Homotopy Invariant of Image Simple Fold Maps to Oriented Surfaces.
Kahmeyer, Liam.

A Homotopy Invariant of Image Simple Fold Maps to Oriented Surfaces. - 1 online resource (77 pages)

Source: Dissertations Abstracts International, Volume: 84-11, Section: B.

Thesis (Ph.D.)--Kansas State University, 2023.

Includes bibliographical references

In 2019, Osamu Saeki showed that for two homotopic generic fold maps f, g : S3 → S2 with respective singular sets Σ(f) and Σ(g) whose respective images f(Σ) and g(Σ) are smoothly embedded, the number of components of the singular sets, respectively denoted #|Σ(f)| and #|Σ(g)|, need not have the same parity. From Saeki's result, a natural question arises: For generic fold maps f : M →N of a smooth manifold M of dimension m≥2 to an oriented surface N of finite genus with f(Σ) smoothly embedded, under what conditions (if any) is #|Σ(f)| a Z/2-homotopy invariant? The goal of this dissertation is to explore this question. Namely, we show that for smooth generic fold maps f : M → N of a smooth closed oriented manifold M of dimension m≥2 to an oriented surface N of finite genus with f(Σ) smoothly embedded, #|Σ(f)| is a modulo two homotopy invariant provided one of the following conditions is satisfied: (a) dimpMq " 2q for q ≥1, (b) the singular set of the homotopy is an orientable manifold, or (c) the image of the singular set of the homotopy does not have triple self-intersection points. Finally, we conclude with a few low-dimensional applications of the main results.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798379570118Subjects--Topical Terms:

527692
Mathematics.
Subjects--Index Terms:

GeometryIndex Terms--Genre/Form:

554714
Electronic books.

A Homotopy Invariant of Image Simple Fold Maps to Oriented Surfaces.
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500 $a Source: Dissertations Abstracts International, Volume: 84-11, Section: B.
500 $a Advisor: Sadykov, Rustam.
502 $a Thesis (Ph.D.)--Kansas State University, 2023.
504 $a Includes bibliographical references
520 $a In 2019, Osamu Saeki showed that for two homotopic generic fold maps f, g : S3 → S2 with respective singular sets Σ(f) and Σ(g) whose respective images f(Σ) and g(Σ) are smoothly embedded, the number of components of the singular sets, respectively denoted #|Σ(f)| and #|Σ(g)|, need not have the same parity. From Saeki's result, a natural question arises: For generic fold maps f : M →N of a smooth manifold M of dimension m≥2 to an oriented surface N of finite genus with f(Σ) smoothly embedded, under what conditions (if any) is #|Σ(f)| a Z/2-homotopy invariant? The goal of this dissertation is to explore this question. Namely, we show that for smooth generic fold maps f : M → N of a smooth closed oriented manifold M of dimension m≥2 to an oriented surface N of finite genus with f(Σ) smoothly embedded, #|Σ(f)| is a modulo two homotopy invariant provided one of the following conditions is satisfied: (a) dimpMq " 2q for q ≥1, (b) the singular set of the homotopy is an orientable manifold, or (c) the image of the singular set of the homotopy does not have triple self-intersection points. Finally, we conclude with a few low-dimensional applications of the main results.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2024
538 $a Mode of access: World Wide Web
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650 4 $a Applied mathematics. $3 1069907
653 $a Geometry
653 $a Homotopy
653 $a Invariant
653 $a Singularities
653 $a Topology
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710 2 $a Kansas State University. $b Department of Mathematics. $3 1474066
773 0 $t Dissertations Abstracts International $g 84-11B.
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