Math 538 ~ Algebraic Topology - II   (SPRING 2018)

Schedule:

Mondays       11:00-12:30, room: M-215
Wednesdays  14:00-15:30, room: M-215

Instructor: Fırat Arıkan,    Office: M-130,    Email: farikan(at)metu.edu.tr,   

Office hours: Monday 10:00-10:50 & Wednesday 11:00-11:50 (or by appointment)

Announcements:  All course announcements will be posted here!

TAKE HOME FINAL EXAM (click)

*** HW1 is avaliable

*** HW2 is avaliable

*** HW3 is avaliable

*** HW4 is avaliable

Exams and Grading:
 
There will be Take-Home Final Exam (%50) and %50 other assignments

Content:
Comohology groups, Universal Coefficient Theorem, comohology of spaces. Products in comohology, Kunneth formula. Poincare duality. Universal coefficient theorem for homology. Homotopy groups.

Course textbook:  

Hatcher Allen,  Algebraic Topology, Cambridge University Press, 2002

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

Course Outline: (Section numbers refer to those in the 2nd. edition of the textbook  ''Algebraic Topology'' by A. Hatcher.

Week Topics Relavant Reading
1 Cohomology groups, Universal Coefficient Theorem for cohomology Section 3.1
2 Universal Coefficient Theorem (continuation); cohomology of spaces   Section 3.1
3 Relative cohomology groups; long exact sequence of a pair; homotopy invariance; excision Section  3.1
4 Axioms for cohomology; Mayer-Vietoris sequence; cellular cohomology Section 3.1
5 Universal coefficient theorem for homology  Section 3.A.
6 Cup product and cohomology ring  Section 3.2
7 Computation of cohomology ring of some spaces;  commutativity of the cohomology ring Section 3.2
8 Künneth Formula for homology and cohomology  Section 3.B.
9 Manifolds , orientations and homology Section 3.3
10 Fundamental class of a manifold, cap product  Section 3.3
11 Poincaré Duality Section 3.3
12 Homotopy Groups, definitions and basic constructions  Section 4.1
13 Relative homotopy groups, homotopy exact sequence of a pair Section 4.1
14 Homotopy exact sequence of a fiber bundle, Whitehead, Cellular Approximation, Hurewicz Theorems Sections 4.1, 4.2

 

Other books:

Bredon, Glen E., Geometry and Topology, Springer-Verlag, 1993

Rotman, Joseph J. , An introduction to Algebraic Topology, 1988

Spanier, Edwin H., Algebraic Topology, McGrow-Hill, 1966

Munkres, James R., Elements of Algebraic Topology, 1996