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MATH537 ALGEBRAIC TOPOLOGY I

Course Objectives

At the end of this course the student will know

• the basic concepts about homotopy and homotopy type, fundamental group and covering spaces to use in his/her research and in other other areas like differential geometry, differential topology, geometric topology, algebraic geometry, physics etc.
• the basic concepts about Delta-Complexes and their simplicial homology groups to use in his/her research or in other related areas.
• the basic concepts about singular homology groups and some of their applications to use in his/her research or in other related areas.
• the techniques to compute the  fundamental groups, simplicial and singular homology groups of of some well known spaces and various cell complexes.

Course Content

Fundamental group, Van Kampen’s Theorem, covering spaces. Singular homology: Homotopy invariance, homology long exact sequence, Mayer-Vietoris sequence, excision. Cellular homology. Homology with coefficients. Simplicial homology and the equivalence of simplicial and singular homology. Axioms of homology. Homology and fundamental groups. Simplicial approximation. Applications of homology.

Course Learning Outcomes

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