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Department of Mathematical Sciences

# MATH3281 Topology III

Topology is a mathematical theory which explains many interesting natural phenomena: it helps to study equilibrium prices in economics, instabilities in behaviour of dynamical systems (for example, robots), chemical properties of molecules in modern molecular biology and many other important problems of science. Topological methods are used in most branches of pure and applied mathematics.

The course is an introduction to topology. We shall start by introducing the idea of an abstract topological space and studying elementary properties such as continuity, compactness and connectedness. A good deal of the course will be geometrical in flavour. We shall study closed surfaces and polyhedra, spend some time discussing winding numbers of planar curves and their applications. Orbit spaces of group actions will provide many interesting examples of topological spaces.

Elements of algebraic topology will also appear in the course. We shall study the fundamental group and the Euler characteristic and apply these invariants in order to distinguish between various spaces. We shall also discuss the concept of homotopy type; this material will be further developed in the course "Algebraic Topology IV".

## Outline of Course

Aim: To provide a balanced introduction to point set, geometric and algebraic topology, with particular emphasis on surfaces.

### Term 1

• Topological spaces and continuous functions: Topological spaces, limit points, continuous maps, homeomorphisms, compactness, product topology, connectedness, path-connected spaces, quotient topology, graphs and surfaces.
• Topological groups and group actions: Orthogonal groups, connected components of O(n), the concept of orientation, topological groups, quaternions, group actions, orbit spaces, projective spaces, lens spaces.

### Term 2

• Basic examples of spaces in low dimensions. Orientable and non-orientable surfaces. Connected sums of surfaces. Polyhedra, triangulations of topological spaces, the Euler characteristic, properties of the Euler characteristic. The topological classification of compact surfaces.
• Basic concepts of homotopy theory: The winding number of loops and applications. Homotopic maps and homotopy equivalence of spaces. Classification of graphs up to homotopy. The fundamental group. Van Kampen’s theorem.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.