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!
*** 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