The University of Southampton

MATH6137 Homotopy and Homology

Module Overview

Homotopy theory is the study of continuous deformations. A geometric object may be continuously deformed by pulling, stretching, pressing or compressing, but not by tearing or puncturing (which are discontinuous). Two objects can then be regarded as equivalent if one can be continuously deformed into the other and vice-versa. The goal of homotopy theory is to determine which geometric objects are equivalent in this sense, or not. To do this, methods are needed which assign algebraic information to these geometric objects which are invariant (stay the same) under continuous deformations. Examples consider in the module are homotopy groups and homology groups.

Aims and Objectives

Module Aims

Description Students will gain an understanding of essential concepts in homotopy theory. They will be able to use logical and coherent arguments to prove basic results in homotopy theory and homology. They will be able to apply and test the theoretical results on a range of spaces, including spheres, real and complex projective spaces, and matrix groups. Students will be able to demonstrate a practical grasp of the material by being able to calculate an array of algebraic invariants involving homotopy groups; homotopy sets with group structure; homology groups.

Learning Outcomes


1. The Fundamental group and higher homotopy groups 2. Group structures on homotopy sets 3. Fibrations and cofibrations 4. Axiomatic homology theory 5. The equivalence of simplicial, singular and cellular homology 6. Examples and applications of homology

Learning and Teaching

Teaching and learning methods

Lectures and weekly problem sheets. Private study is also important for work on the problem sheets.

Independent Study114
Total study time150

Resources & Reading list

Allen Hatcher (2001). Algebraic Topology. 



MethodPercentage contribution
Coursework 100%


MethodPercentage contribution
Exam 100%

Repeat Information

Repeat type: Internal & External

Linked modules

Pre-requisites: MATH2003 AND MATH2049


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

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