## PURE MTH 7002 - Pure Mathematics Topic B

### North Terrace Campus - Semester 1 - 2022

This course is available for students taking a Masters degree in Mathematical Sciences. The course will cover an advanced topic in pure mathematics. For details of the topic offered this year please refer to the Course Outline.

• General Course Information
##### Course Details
Course Code PURE MTH 7002 Pure Mathematics Topic B Mathematical Sciences Semester 1 Postgraduate Coursework North Terrace Campus 3 Y Ongoing assessment, exam
##### Course Staff

Course Coordinator: Dr David Baraglia

##### Course Timetable

The full timetable of all activities for this course can be accessed from Course Planner.

• Learning Outcomes
##### Course Learning Outcomes
In 2022, the topic of this course is Algebraic Topology.

Outline

The aim of Algebraic Topology is to use algebraic structures and techniques to classify topological spaces up to homeomorphism. Algebraic objects are associated to topological spaces in such a way that "natural" operations on the former correspond to "natural" operations on the latter - continuous maps might correspond to group homomorphisms, homeomorphisms to isomorphisms, etc. In this way, it is often possible to distinguish between different topological spaces by demonstrating that certain associated algebraic objects are not isomorphic. It is rarely the case that the converse can be shown; i.e., that two topological spaces with the same associated algebraic objects are actually homeomorphic, but when this can be done, it is often regarded as a major triumph of the theory.

Within the realms of algebraic topology, there are several basic concepts that underly the theory and serve as the building blocks and models for subsequent generalisation, the algebraic topology of today being a very broad and highly generalised area that has pervaded much of contemporary mathematics. Such concepts include homotopy, homology and cohomology, and the course will be aimed at providing students with an introduction to these key ideas.

Learning Outcomes

On successful completion of this course, students will be able to:

1) understand the basic notions of homotopy theory such as homotopy of maps, homotopy equivalences, contractible spaces, deformation retracts,

2) define the fundamental group of a (path connected) topological space and be able to compute fundamental groups of some simple examples using for example the Seifert-van Kampen Theorem,

3) define the singular homology and cohomology groups of a topological space and their relative versions,

4) understand and work with basic concepts in homological algebra, including chain complexes and long exact sequences,

5) compute the homology and cohomology of some topological spaces using the Eilenberg-Steenrod axioms,

6) apply the topological invariants constructed in this course to the solution of various problems in topology, for instance, to prove that two spaces are not homeomorphic.

Prerequisites

It will be assumed that you have some familiarity with basic point-set topology at the level of Topology and Analysis III and familiarity with basic notions of abstract algebra (groups, rings, fields etc.) at the level of Groups and Rings III. However I will give a review of point-set topology in the first week

This course will provide students with an opportunity to develop the Graduate Attribute(s) specified below:

University Graduate Attribute Course Learning Outcome(s)

Attribute 1: Deep discipline knowledge and intellectual breadth

Graduates have comprehensive knowledge and understanding of their subject area, the ability to engage with different traditions of thought, and the ability to apply their knowledge in practice including in multi-disciplinary or multi-professional contexts.

all

Attribute 2: Creative and critical thinking, and problem solving

Graduates are effective problems-solvers, able to apply critical, creative and evidence-based thinking to conceive innovative responses to future challenges.

all
• Learning Resources
##### Required Resources
The course will make use of the following textbook, which is freely available to download from the author's webpage (https://pi.math.cornell.edu/~hatcher/AT/ATpage.html):

A. Hatcher, Algebraic Topology
##### Recommended Resources
In addition to Hatcher, the following books are fairly standard:

M. Greenberg and J. Harper, Algebraic topology: A first course, (515.14 G798a)

W. Massey, A basic course in algebraic topology, (515.14 M416b)

C. R. F. Maunder, Algebraic Topology, (513.83 M451A)

E. H. Spanier, Algebraic Topology, (513.83 S735)

The book by Hatcher is probably the best of all. The book by Greenberg and Harper used to be a very standard reference until the arrival of Hatcher's book. The books by Massey and Maunder are also good reference books that have good explanations. Spanier is comprehensive but very hard to digest.
##### Online Learning
The course will have an active MyUni website.
• Learning & Teaching Activities
##### Learning & Teaching Modes
Students are expected to read and engage with the assigned reading material. There will be a weekly workshop with a mix of lecturing, students working on problems, together and with guidance from the lecturer, and consulting. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress.

The information below is provided as a guide to assist students in engaging appropriately with the course requirements.

 Activity Quantity Workload hours Workshops 12 24 Assignments 5 50 Self-study 82 Total 156
##### Learning Activities Summary
1) understand the basic notions of homotopy theory such as homotopy of maps, homotopy equivalences, contractible spaces, deformation retracts,

2) define the fundamental group of a (path connected) topological space and be able to compute fundamental groups of some simple examples using for example the Seifert-van Kampen Theorem,

3) define the singular homology and cohomology groups of a topological space and their relative versions,

4) understand and work with basic concepts in homological algebra, including chain complexes and long exact sequences,

5) compute the homology and cohomology of some topological spaces using the Eilenberg-Steenrod axioms,

6) apply the topological invariants constructed in this course to the solution of various problems in topology, for instance, to prove that two spaces are not homeomorphic.
• Assessment

The University's policy on Assessment for Coursework Programs is based on the following four principles:

1. Assessment must encourage and reinforce learning.
2. Assessment must enable robust and fair judgements about student performance.
3. Assessment practices must be fair and equitable to students and give them the opportunity to demonstrate what they have learned.
4. Assessment must maintain academic standards.

##### Assessment Summary

Examination Summative Examination period 60% All
Homework assignments Formative and summative One week after assigned 40% All
##### Assessment Related Requirements
An aggregate score of 50% is required to pass the course.
##### Assessment Detail
There will be five homework assignments distributed approximately once every two weeks across the semester. There will also be a final
examination.
##### Submission
Homework assignments must be submitted on MyUni as pdf files. Failure to meet the deadline without reasonable and verifiable excuse may result in a significant penalty for that assignment.

Grades for your performance in this course will be awarded in accordance with the following scheme:

M10 (Coursework Mark Scheme)
FNS   Fail No Submission
F 1-49 Fail
P 50-64 Pass
C 65-74 Credit
D 75-84 Distinction
HD 85-100 High Distinction
CN   Continuing
NFE   No Formal Examination
RP   Result Pending

Further details of the grades/results can be obtained from Examinations.

Grade Descriptors are available which provide a general guide to the standard of work that is expected at each grade level. More information at Assessment for Coursework Programs.

Final results for this course will be made available through Access Adelaide.

• Student Feedback

The University places a high priority on approaches to learning and teaching that enhance the student experience. Feedback is sought from students in a variety of ways including on-going engagement with staff, the use of online discussion boards and the use of Student Experience of Learning and Teaching (SELT) surveys as well as GOS surveys and Program reviews.

SELTs are an important source of information to inform individual teaching practice, decisions about teaching duties, and course and program curriculum design. They enable the University to assess how effectively its learning environments and teaching practices facilitate student engagement and learning outcomes. Under the current SELT Policy (http://www.adelaide.edu.au/policies/101/) course SELTs are mandated and must be conducted at the conclusion of each term/semester/trimester for every course offering. Feedback on issues raised through course SELT surveys is made available to enrolled students through various resources (e.g. MyUni). In addition aggregated course SELT data is available.

• Student Support
• Policies & Guidelines
• Fraud Awareness

Students are reminded that in order to maintain the academic integrity of all programs and courses, the university has a zero-tolerance approach to students offering money or significant value goods or services to any staff member who is involved in their teaching or assessment. Students offering lecturers or tutors or professional staff anything more than a small token of appreciation is totally unacceptable, in any circumstances. Staff members are obliged to report all such incidents to their supervisor/manager, who will refer them for action under the university's studentâ€™s disciplinary procedures.

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