MATH 537 Algebraic Topology I 

Description of the course:

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 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.

TextBook: Allen Hatcher, Algebraic Topology,  Cambridge University Press, 2002

                       (Available online at  http://www.math.cornell.edu/~hatcher/AT/ATpage.html)

Tentative Weekly Outline:
Week           Topic                                                                                   Relevant Reading             Assignments
   1             Homotopy and homotopy equivalence, cell complexes                                           Chapter 0                                     -

   2            Homotopy of paths, fundamental group basic constructions                                     Section 1.1                                   -
 
   3            Fundamental group of the circle, Induced maps                                                       Section 1.1                          Homework 1

   4           Van Kampen's Theorem and its applications to cell complexes                                 Section 1.2                                   -

   5           Covering spaces, lifting properties                                                                           Section 1.3                                   -

  6            The classification of covering spaces                                                                       Section 1.3                          Homework 2

  7            Deck transformations and group actions                                                                  Section 1.3                                   -

  8           Delta-complexes and  simplicial homology groups, Singular homology groups            Section 2.1                                   -

  9           Homotopy invariance, Exact sequences                                                                    Section 2.1                          Homework 3

10           Relative homology groups, long exact sequence of homology groups                         Section 2.1                                  -

11           Excision, The equivalence of simplicial and singular homology                                   Section 2.1                                   -

12           Degree, Cellular homology                                                                                       Section 2.2                          Homework 4

13           Homology groups of projective spaces, Euler characteristic, Split exact sequences,
               Mayer Vietoris sequence, Homology with coefficients                                                Section 2.2                                 -

14         Axioms for homology, Categories and functors, Homology and fundamental group       Section 2.3,  2.A                        -


Schedule:
       Mondays:    13:40-15:30 M-203
 Wednesdays:    13:40-15:30 M-203


Course Grading:
 The grading will be based on a midterm exam, a final (both will be held face to face) and 4 homework assignments.

Midterm Exam    30%
Homeworks         30%
Final Exam          40%

Course Policies:
The students are expected to attend at least %70 of the classes.
The students are strongly encouraged to participate discussions actively in the class.

Make up for Exams and Assignments:
Make-up will be given only if there exists a legitimate excuse.