Instructor: Dr. Boldizsár KALMÁR

Text:  Allan Hatcher: Algebraic Topology and class notes

Prerequisites:
Basic algebra: vector spaces, groups, factor groups, homomorphisms.
Basic analysis in \R^n: continuous maps, convergence, differentiable maps.

Topics:
The goal of the lectures is to give a wide and detailed view of the most important tools in algebraic and geometric topology. These connect the most natural "naive" and elementary geometric ideas and constructions with various fields of modern mathematics. We will see how deeply algebra and geometry interact in the study of a mathematical problem.
The course is useful to students who want to have a global picture of a big part of today's mathematics or to students who are interested in mathematical research. Although this course could be called "Introduction to algebraic and geometric topology" and the instructor provides as much explanation as necessary, the level of the lectures is not elementary and students are asked to work out details by themselves fairly often.

Syllabus:

1. CW complexes in Euclidean spaces, surfaces, manifolds, constructions: We construct topological spaces in \R^n, which are important in practice. We give many examples and study their basic properties.
2. Homology groups and functors: We define homology groups and see how to use them in solving topological problems.
3. Exact sequences: We learn an abstract algebraic tool to compute the homology groups and see many applications.
4. Knots, knot invariants, fundamental group, Jones polynomial: We study knots in \R^3, and see how to work with their basic invariants.
5. 3-manifolds, 4-manifolds and surgeries: We see how to use knots to understand low dimensional topology.
6. Morse functions, classification of surfaces: We see a geometric proof for the classification of surfaces. We extend real analysis to surfaces and manifolds.
7. Homotopy groups: We study a natural generalization of fundamental group, which helps to classify continuous maps.