Algebraic topology studies properties of topological spaces and maps between them by associating algebraic invariants (fundamental groups, homology groups, cohomology groups) to each space.

This course gives a solid introduction to fundamental ideas and results that are employed nowadays in most areas of mathematics, theoretical physics and computer science.

This course aims to understand some fundamental ideas in algebraic topology; to apply discrete, algebraic methods to solve topological problems; to develop  some intuition for how algebraic topology relates to concrete topological problems.

Topics to be covered include: 

• Fundamental group and covering spaces
• Brouwer fixed point theorem
• Fundamental theorem of algebra
• Homology theory and cohomology theory
• Jordan-Brouwer separation theorem
• Lefschetz fixed theorem
• Additional topics (Orientation, Poincare duality, if time permits)

Note: Graduate students attend joint classes with undergraduates but will be assessed separately.

On satisfying the requirements of this course, students will have the knowledge and skills to:

Assessment will be based on:

• Assignment 1 (10%: LO 1-4)
• Assignment 2 (10%; LO 1-4)
• Assignment 3 (10%; LO 1-4)
• Presentation (10%; LO 1-4)
• Final exam (60%; LO 1-4)