### Higher topological complexity of cohomologically determined spaces

- 발행기관 서강대학교 일반대학원
- 지도교수 조장현
- 발행년도 2024
- 학위수여년월 2024. 2
- 학위명 석사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/sogang/000000076857
- UCI I804:11029-000000076857
- 본문언어 영어
- 저작권 서강대학교 논문은 저작권 보호를 받습니다.

#### 초록

In this thesis, we first carry out a survey to provide a natural and uniform explanation of the properties of the Lusternik-Schnirelmann category and the higher topological complexity of spaces, under the perspective of seeing these as sectional categories of certain fibrations. In particular, we are interested in the cohomological lower bounds of the above invariants obtainable from this perspective, namely the cup-length and the zero-divisors cup-length of spaces. It is known that the behavior of the spaces whose Lusternik-Schnirelmann categories coincide with their cup-lengths is optimal in the sense that they have an additivity with respect to the product operation of spaces. Motivated by such phenomena, in the second part of this thesis, we investigate the spaces whose Lusternik-Schnirelmann categories coincide with their cup-lengths, and the spaces whose higher topological complexities coincide with their zero-divisors cup-lengths. Further inspecting such optimality, we show that there is an additivity of higher topological complexity with respect to the product operation in the latter class. Moreover, in the last part of this thesis, we introduce an efficient method of computing the higher topological complexity of a space.

more#### 목차

1 Introduction 1

2 Preliminaries 3

2.1 Fibration 3

2.2 Cohomology ring 6

3 Sectional Category 8

3.1 Sectional category 8

3.2 Properties of sectional category 10

3.3 Sectional category of a fibrational substitute 11

4 Lusternik-Schnirelmann Category 14

4.1 Lusternik-Schnirelmann category of a space 14

4.2 Properties of Lusternik-Schnirelmann category 16

4.3 Examples 18

5 Higher Topological Complexity 20

5.1 Higher topological complexity of a space 20

5.2 Properties of higher topological complexity 22

5.3 Examples 25

6 Cohomologically Determined Space 27

6.1 Algebraic cup-length and zero-divisors cup-length 27

6.2 Cup-length space 30

6.3 Zero-divisors cup-length space 32

Bibliography 37