MANIFOLDS AND TOPOLOGY - 2026/7

Module Overview

This module introduces the students to topological spaces and manifolds:  the core structures in pure mathematics.  The module begins with fundamental concepts of point-set topology, homeomorphisms and their invariants - the sigma-algebra of open sets, the Hausdorff property, connectedness and compactness. We then move on to algebraic topology, introduce topological surfaces as quotient spaces, and discuss their invariants - the Euler characteristic and orientability. Finally, we progress to smooth manifolds and diffeomorphisms, discuss their construction method, that include two fundamental results of analysis, and the concepts of tangent bundles and transversality.  

Module provider

Mathematics & Physics

Module Leader

VYTNOVA Polina (Maths & Phys)

Number of Credits: 15

ECTS Credits: 7.5

Framework: FHEQ Level 6

Module cap (Maximum number of students): N/A

Module Availability

Semester 1

Prerequisites / Co-requisites

None

Module content

Indicative content includes:

• Point-set Topology: Homeomorphisms, Hausdorff property, invariance of domain, connectedness, path-connectedness, compactness, topological boundary, quotient topology, projective spaces, topological groups; 
• Algebraic Topology: topological surfaces, orientability, Euler characteristic, classification of surfaces, homotopy group;   
• Differential Topology: Smooth manifolds, diffeomorphisms, tangent spaces, manifold with a boundary, immersions, embeddings, submersions, transversality, preimage theorem,Sard's theorem, Brower's fixed point theorem, Lie groups. 

Assessment pattern

Alternative Assessment

NA

Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of subject knowledge, and recall of key definitions and properties of topological spaces and manifolds.
• The ability to identify identify and use the appropriate techniques to solve problems in foundations of differentiable topology.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 20% of the module mark, corresponding to Learning Outcome 1 and 2.
• A synoptic examination (2 hours), worth 80% of the module mark, corresponding to all Learning Outcomes 1 to 3.

There will be bi-weekly in-class quizes (10 minutes), that are followed by immediate discussion.  

Feedback

The students will have an opportunity to submit solutions to weekly exercise sheets via SurreyLearn. All submissions will be marked for feedback.

Module aims

• Introduce students to topological spaces, and ideas such as Hausdorff topological spaces, homeomorphisms, connectedness and compactness.
• Introduce students to the core constructions of differential geometry, such as smooth manifolds, tangent spaces, orientability and transversality.
• Introduce students to a variety of mathematical constructions, including, but not limited to: basic topological surfaces, Cantor sets, fractals, knots, Lie groups, space-filling curves, topological groups, and their elementary properties.

Learning outcomes

Methods of Teaching / Learning

The learning and teaching strategy is designed to:

Introduce students to the core ideas in topology and differential geometry, focusing in particular on geometric approach, to develop conceptual thinking building on intuition, with the help of visual demonstrations in low dimensions. Key results in analysis such as invariance of domain theorem, inverse function theorem, implicit function theorem introduced in full generality.     

  The learning and teaching methods include:

• Three one-hour lectures for eleven weeks, with module notes provided to complement the lectures. The lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.
• Weekly one-hour tutorials. These tutorials provide an opportunity for students to gain feedback and assistance with the exercises which complement the module notes.
• Fortnightly quizes (10 minutes, in class, unassessed). They are designed to quickly ascertain understanding of key concepts taught during past 2 weeks. 
• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Reading list

https://readinglists.surrey.ac.uk
Upon accessing the reading list, please search for the module using the module code: MAT3009

Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The students will have an opportunity to engage in AI ¿ assisted learning.

Employability: The module MAT3009 equips students with skills which significantly enhance their employability. Students will be able to develop an essential skills of critical, independent thinking, and analytic reasoning. an ability to construct a solution to a problem from scratch, starting from a large variety of methods and tools. Students will learn to evaluate complex problems, break them into manageable components, and apply their knowledge and logical reasoning to arrive at solutions.. These are highly sought after skills in many professions.

Global and Cultural Capabilities: Student enrolled in MAT3009 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in tutorials and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and Resilience: MAT3009 is a module which demands the ability to visualise geometric problems, and formulate and solve these problems using the abstract mathematics of differential geometry. This rigorous branch of mathematics blends abstract mathematics with geometry, in an approach which stretches students¿ understanding and to which they will learn to adapt. Students will gain skills in analysing geometric problems using lateral thinking, and will complete assessments which challenge them and build resilience.

Sustainability: Topology forms part of the foundation of computed tomography, which has a broad variety of applications from geology to medicine. One or more case studies will be included in the module to illustrate applications in the context of sustainable decision-making.

Programmes this module appears in

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2026/7 academic year.