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Geometry and Topology 1

Stream Overview

This is an introductory course in Geometry and Topology intended both for students who did not specialise in these topics at undergraduate level and for specialists, i.e. graduate students in Geometry and Topology.

The stream consists of two independently-assessed modules:

  • Module 1 (semester 1) on General and Algebraic Topology
  • Module 2 (semester 2) on Diff erential Geometry and Manifolds.

In addition to setting regular assignments the lecturers will suggest readings (as much as possible available on the internet) to supplement the course material and to further ful l the SMSTC mission of broadening education in fundamental areas of mathematics.

Module Overview: Geometry and Topology 1 (semester 1 - this module)

This module is intended to give an overview of basic concepts, examples and techniques in algebraic topology.  Students will learn basic tools for the analysis of topological spaces, via CW-structures or bundle structures, and important topological invariants via homotopy groups and homology groups.  The course then connects these ideas to algebraic geometry via the theory of algebraic curves, and concludes with a summary of the classification of surfaces; i.e. 2-dimensional manifolds.

Basic topology and homology theory [Spiros Adams-Florou]

  • Lecture 1: Basic examples and constructions of topological spaces.
  • Lecture 2: Basic homotopy theory, homotopy groups and CW-complexes.
  • Lecture 3: Cofibrations, cell attachments and CW-complexes.
  • Lecture 4: Cellular approximation and relative homotopy groups.
  • Lecture 5: Fibre bundles, fibrations and the Hopf map.
  • Lecture 6: The definition of singular homology.
  • Lecture 7: Properties of singular homology: homotopy invariance and excision.
  • Lecture 10: Compuations and applications of singular homology.

Algebraic Topology [Vanya Cheltsov]

  • Lecture 8: Algebraic curves and Riemann surfaces I.
  • Lecture 9: Algebraic curves and Riemann surfaces II.


This module is assessed in 2 written assignments.

  • Assignment 1 is due on Friday of Week 9, December 2nd, 2016
  • Assignment 2 is due on Monday January 9th, 2017

A student taking this module requires a working knowledge of metric and topological spaces; linear algebra (vector spaces, linear maps and quotient vector spaces); group theory (groups and group actions).


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