Course overview
One of the great insights of 20th century mathematics is that we may associate some algebraic structure (a group, or a ring) to a topological space (for example, a surface), in such a way that continuous changes made to the topological space do not change the algebraic structure. In that case we have an invariant algebraic structure. These algebraic invariants provide a powerful tool to study and classify topological spaces, which we will do in this course.
- Fundamental Group and Covering Spaces
- Simplicial and Singular Homology
- Singular Cohomology and Poincaré Duality
Course learning outcomes
- Explain the fundamental group, its properties, and its relationship with covering spaces
- Explain homology and cohomology groups, their properties, and relationship to each other
- Calculate algebraic invariants for suitable examples
- Apply the theory in the course to solve a variety of problems
- Demonstrate skills in communicating mathematics