Math 728- Homological Methods in Topology

IMPORTANT:

!The classes will be done online via the Zoom platform. Before each class, the Zoom link will be posted on Odtuclass. These lectures are intended for Math 728 students and sharing the link with third parties is strictly forbidden and lecture notes will be posted in ODTUclass. !
!Attendance of lectures is required.!
!If you want to follow the course as a guest student, please send me an email: pasemra@metu.edu.tr!

Course Code: 2360728

Schedule:

Content: Spectral sequences, cohomology operations, Steenrod algebra and their applications in transformation groups, group cohomology and homotopy theory.

Course Objectives:
This course is designed to provide the students with some of the necessary potential research  background from the homological algebra for reading and understanding research articles in algebraic topology. In particular, it is designed to equip the students with the essential computational technique of spectral sequences and Steenrod algebra, to give applications to spaces with a group action, to introduce some topics of algebraic topology such as group cohomology and transformation groups, which are  potential research areas for graduate  students.

Course Learning Outcomes:
Obtain knowledge of the facts and computational techniques of spectral sequences and Steenrod algebra.
Apply these computational methods to deal with problems in algebraic topology.
Compute homology and cohomology groups of spaces and groups.

Tentative Weekly Outline:

1-What is a spectral sequence? Definitions and basic properties.
2-Double complexes and filtrations.
3-Convergence of spectral sequences.
4-Fibrations.
5-Leray-Serre spectral sequence.
6-Cohomology operations
7-Steenrod Algebras
8-(Co)Homology of groups
9-Cohomology of  finite groups
10-Equivariant cohomology of G-CW complexes
11- Borel construction
12- Localization
13- Applications of Localizations
14- Applications to homotopy theory


Course Textbook(s):

1. User's Guide to Spectral Sequences, John Mcleary, Cambridge studies in Adv.    Math.,2001.
2. Cohomology Operations, N.E. Steenrod, Ann.Math. Stud., Princeton Univ.Press, 1962
3. Cohomology of Groups, K.S. Brown,Grad. Texts in Maths., Springer, 1982

Supplementary Readings :

1. Cohomology Operations and Applications in Homotopy Theory, R.E. Mosher and M.C. Tangora, Harper & Row Publishers, 1968.
2. Cohomological Methods in Transformation Groups, C. Allday and V. Puppe, Cambridge Stud. in Adv. Math., 1993.

Assessment of Student Learning/ Grading::
Assignments %70: Homework assignments will be given during the course.
Final Exam %30: Oral exam