Lecturer

Sara Kalisnik Hintz email

Coordinator

Younghan Bae email

Lecture room

Wednesday 10:15~12:00 ML E 12
Friday 14:15~16:00 HG G 3
Office hours: Monday 17:15~18:00 LEE C114

Content

This is a continuation course to Algebraic Topology I. The course will cover more advanced topics in algebraic topology including: cohomology of spaces, operations in homology and cohomology, duality. See also Paul Brian's previous course

Exam information and preparation

We will have oral exam between 11.08.2023 and 19.08.2023. We will post will post a document with sample questions in late June.

• Sample question - Sample questions for the oral exam.

Prerequisites

General topology, linear algebra, singular homology of topological spaces (e.g. as taught in Algebraic topology I). Some knowledge of differential geometry and differential topology is useful but not absolutely necessary.

Background material

For previous course on algebraic topology I, see AT1.

• Lecture notes on Hom and Tensor

Lecture Contents

We will post lecture notes after every lectures on Wednesday and Friday.

• Tensor products - Lecture note for 02.24
See also Link 1 and Link 2.
• Homology with coefficients, part 1 - Lecture note for 01.03
• Change of coefficients, cellular homology with coefficients - Lecture note for 03.03
Here are some clicker questions: Question 1, Question 2, Question 3.
• Borsuk Ulam theorem - Lecture note for 08.03
• Applications of Borsuk Ulam theorem - Lecture note for 08.10
• Cohomology and the Universal Coefficient Theorem - Lecture note for 15.03
• Review of UCT -Lecture note for 17.03
• Examples of UCT, UCT for tensor products - Lecture note for 22.03
• UCT for homology - Lecture note for 24.03
• More on cohomology - Lecture note for 29.03
• Eilenberg-Zilber 1 - Lecture note for 31.03
• Eilenberg-Zilber 2 - Lecture note for 05.04
• Eilenberg-Zilber 3 - Lecture note for 19.04. See also a note by Chris Kottke
• Homological cross product and Kunneth formula - Lecture note for 21.04
• Cohomology cross product - Lecture note for 26.04
• Cup product - Lecture note for 28.04
• Examples of cup products - Lecture note for 03.05 and 05.05 Solution to the torus example
• Manifolds and orientability - Lecture note for 10.05 and 12.05
• Fundamental class - Lecture note for 17.05 and 19.05
• Cup products - Lecture note for 24.05
• Cohomology with compact supports - Lecture note for 26.05
• Proof of Poincare duality - Lecture note for 31.05
• Dual complexes and Poincare duality - Lecture note for 02.06

Exercises

Exercise sheets will be posted here. For the week after the exercise sheet is posted, we will discuss the solutions during the office hours. If possible, please ask questions in advance, via email. Solutions to the exercises will also appear in the weeks after the exercises are posted.

Office hours

We have office hours every two weeks starting from March 6th. Dates: 06.03, 20.03, 03.04, 17.04, 15.05, 29.05.

Literature

• Topology and Geometry, GTM 139, Springer-Verlag, 1997 by G. Bredon
• Algebraic Topology, Cambridge University Press. by A. Hatcher
• Algebraic Topology, Springer-Verlag by E. H. Spanier