# MATH5665 Algebraic Topology

MATH5665 is a Honours and Postgraduate coursework Mathematics course. See the course overview below.

Units of credit: 6

Prerequisites: No formal ones, but at least second year algebra is assumed

Exclusions: Third year number theory courses MATH3740, MATH3521

Cycle of offering: Not offered every year, probably offered every second year

Graduate attributes: The course will enhance your research, inquiry and analytical thinking abilities.

This recent course handout (pdf) contains information about course objectives, assessment, course materials and the syllabus.

The Online Handbook entry contains information about the course. (The timetable is only up-to-date if the course is being offered this year.)

If you are currently enrolled in MATH5665, you can log into UNSW Moodle for this course.

#### Course Overview

Algebraic topology is the study of knots, links, surfaces and higher dimensional analogs called manifolds with the understanding that continuous deformations do not change objects. So a doughnut (torus) and a coffee mug are essentially the same (homeomorphic) in this course.

For example, how does a creature living on a sphere tell that she is not on the plane, on the torus, or perhaps a two holed torus? Can one turn a sphere inside out without creasing it? What would it be like to live inside a three dimensional sphere? Can one continuously deform a trefoil knot to get its mirror image? Can the wind be blowing at every point on the earth at once? Can you tell if a graph is planar? Can you tell if a knot is trivial? Is there a list of all possible two dimensional surfaces? How about three dimensional ones? These are some of the motivating questions for the subject.

Algebraic topology attempts to answer such questions by assigning algebraic invariants such as numbers, or groups, to topological spaces.

Examples include the Euler number of a surface, the Poincare index of a vector field, the genus of a torus, the fundamental group and more fancy homology groups. The subject however has a very strong visual and spatial aspect, together with an often informal way of arguing.

Connections with geometry, in particular hyperbolic geometry, projective geometry and differential geometry are also important, and models are very helpful. The subject has its origins in complex analysis, with the study of Riemann surfaces.