By the end of the course the student will learn
- the definitions of homotopy, homotopy equivalence, retract, deformation retract
- techniques to recognize when two spaces are homotopy equivalent or when they are not homotopy equivalent
- examples of cell complexes, and the cell structure of some familiar spaces like spheres, projective spaces, tori, klein bottle etc.
- the basic constructions like products, cones, suspensions, joins, wedge sum, smash product on cell complexes and other spaces
- the definition of fundamental group and group operations
- the fundamental groups of spaces like spheres, projective space, torus etc.
- the Van Kampen's theorem and its applications to compute fundamental groups of cell complexes
- the definition of covering spaces and their lifting properties, to use them in lifting problems
- to methods to construct and classify covering spaces for known spaces, and for other spaces whenever it is possible,
- the relation between deck transforrmations and fundamental groups, and to use it in computation and classifications
- the definition of simplicial and sigular homology groups and their properties
- the methods to compute the simplicial and singular homology
- basic homologic techniques like exact sequences, diagram chasing, five lemma
- the axioms of homology and to use them in the computations of homology groups and in geometric applications
- to compute homology groups of cell complexes using cellular homology
- the relation between simplicial and singular homology groups
- the relation between singular homology and fundamental group
- the computation of homology with coefficients other then integers