Unit of study_

# MATH4311: Algebraic Topology

## Overview

One of the most important aims of algebraic topology is to distinguish or classify topological spaces and maps between them up to homeomorphism. Invariants and obstructions are key to achieve this aim. A familiar invariant is the Euler characteristic of a topological space, which was initially discovered via combinatorial methods and has been rediscovered in many different guises. Modern algebraic topology allows the solution of complicated geometric problems with algebraic methods. Imagine a closed loop of string that looks knotted in space. How would you tell if you can wiggle it about to form an unknotted loop without cutting the string? The space of all deformations of the loop is an intractable set. The key idea is to associate algebraic structures, such as groups or vector spaces, with topological objects such as knots, in such a way that complicated topological questions can be phrased as simpler questions about the algebraic structures. In particular, this turns questions about an intractable set into a conceptual or finite, computational framework that allows us to answer these questions with certainty. In this unit you will learn about fundamental group and covering spaces, homology and cohomology theory. These form the basis for applications in other domains within mathematics and other disciplines, such as physics or biology. At the end of this unit you will have a broad and coherent knowledge of Algebraic Topology, and you will have developed the skills to determine whether seemingly intractable problems can be solved with topological methods.

### Details

Academic unit Mathematics and Statistics Academic Operations MATH4311 Algebraic Topology Semester 2, 2021 Normal day Remote 6

### Enrolment rules

 Prohibitions ? None None None Familiarity with abstract algebra and basic topology, e.g., (MATH2922 or MATH2961 or equivalent), (MATH3961 or equivalent) and (MATH2923 or equivalent). Yes

### Teaching staff and contact details

Coordinator Laurentiu Paunescu, laurentiu.paunescu@sydney.edu.au

## Assessment

Type Description Weight Due Length
Oral exam

Final exam
Oral exam through Zoom.
50% Formal exam period 20 minutes (oral)
Outcomes assessed:
Assignment Assignment 1
Report
25% Week 06 3 to 5 questions
Outcomes assessed:
Assignment Assignment 2
Report
25% Week 12 3 to 5 questions
Outcomes assessed:
• Assignment 1: Work individually on about 5 questions regarding the content of the course from weeks 1-5. Type the answers into a pdf file to be submitted at the end of week 6.  Detailed information for each assessment can be found on Canvas in week 4.
• Assignment 2: Work individually on about 5 questions regarding the content of the course from weeks 1-10. Type the answers into a pdf file to be submitted at the beginning of week 12.  Detailed information for each assessment can be found on Canvas in week 10.

### Assessment criteria

High Distinction-85-100-

At HD level, a student demonstrates a flair for the subject as well as a detailed and comprehensive understanding of the unit material. A ‘High Distinction’ reflects exceptional achievement and is awarded to a student who demonstrates the ability to apply their subject knowledge and understanding to produce original solutions for novel or highly complex problems and/or comprehensive critical discussions of theoretical concepts.

Distinction-75-84

At DI level, a student demonstrates an aptitude for the subject and a well-developed understanding of the unit material. A ‘Distinction’ reflects excellent achievement and is awarded to a student who demonstrates an ability to apply their subject knowledge and understanding of the subject to produce good solutions for challenging problems and/or a reasonably well-developed critical analysis of theoretical concepts.

Credit-65-74

At CR level, a student demonstrates a good command and knowledge of the unit material.
A ‘Credit’ reflects solid achievement and is awarded to a student who has a broad general
understanding of the unit material and can solve routine problems and/or identify and
superficially discuss theoretical concepts.

Pass- 50-64-

At PS level, a student demonstrates proficiency in the unit material. A ‘Pass’ reflects satisfactory achievement and is awarded to a student who has threshold knowledge.

Fail-0-49-

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

### Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

• Deduction of 5% of the maximum mark for each calendar day after the due date.
• After ten calendar days late, a mark of zero will be awarded.

This unit has an exception to the standard University policy or supplementary information has been provided by the unit coordinator. This information is displayed below:

Without special circumstances which qualify for special consideration, late submissions will not be accepted.

### Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

The Current Student website provides information on academic integrity and the resources available to all students. The University expects students and staff to act ethically and honestly and will treat all allegations of academic integrity breaches seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic integrity breach. If such matches indicate evidence of plagiarism or other forms of academic integrity breaches, your teacher is required to report your work for further investigation.

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

## Weekly schedule

WK Topic Learning activity Learning outcomes
Week 01 Motivation and Foundations Lecture (3 hr)
Week 02 Fundamental group Lecture (3 hr)
Week 03 Groups and spaces Lecture (3 hr)
Week 04 Covering spaces Lecture (3 hr)
Week 05 Covering spaces Lecture (3 hr)
Week 06 Homology theory Lecture (3 hr)
Week 07 Homology theory Lecture (3 hr)
Week 08 Cellular and simplicity homology Lecture (3 hr)
Week 09 Cohomology Lecture (3 hr)
Week 10 Duality Lecture (3 hr)
Week 11 Duality Lecture (3 hr)
Week 12 Further topics Lecture (3 hr)
Weekly Weekly tutorial Tutorial (1 hr)

### Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Algebraic Topology by Allen Hatcher https://www.math.cornell.edu/~hatcher/AT/AT.pdf

The lecture notes will be available on the course page.

## Learning outcomes

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

• LO1. Demonstrate a coherent and advanced understanding of the key concepts of fundamental group, covering spaces, homology and cohomology.
• LO2. Apply the fundamental principles and results of algebraic topology to solve given problems.
• LO3. Distinguish and compare the properties of different types of topological spaces and maps between them.
• LO4. Formulate topological problems in terms of algebraic invariants and determine the appropriate framework to solve them.
• LO5. Communicate coherent mathematical arguments appropriately to student and expert audiences, both orally and through written work.
• LO6. Devise computational solutions to complex problems in algebraic topology.
• LO7. Compose correct proofs of unfamiliar general results in algebraic topology.

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

 GQ1 Depth of disciplinary expertise Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline. GQ2 Critical thinking and problem solving Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem. GQ3 Oral and written communication Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context. GQ4 Information and digital literacy Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies. GQ5 Inventiveness Generating novel ideas and solutions. GQ6 Cultural competence Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues. GQ7 Interdisciplinary effectiveness Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries. GQ8 Integrated professional, ethical, and personal identity An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context. GQ9 Influence Engaging others in a process, idea or vision.