Algebraic topology

### COURSE DESCRIPTION

Lecturer: Roland van der Veen, Assistants: Jorge Becerra and Sjabbo Schaveling. Contact us at: algebraictopology18@gmail.com

Lectures: 14-17, HFG 611AB (or MIN 201, weeks 38,41,44,45, In week 42 Ruppert D(80), In week 43 BBG 061 (64)).
Virtual Office Hours (experimental! starting sept 21) Fridays 14-15

Aim of the course This course is an introduction to Algebraic Topology. Its main topic is the study of homology groups of topological spaces. These homology groups provide algebraic invariants of topological spaces which can be computed in many examples of interest. In the first part of the course we will construct the singular homology groups of topological spaces and establish their basic properties, such as homotopy invariance and long exact sequences. In the second part of the course we will introduce CW-complexes. These provide a useful class of topological spaces with favorable properties, and we will explain how the homology of CW-complexes can be computed using cellular homology. We will also discuss some basic concepts from homotopy theory.

Final grade The final grade is mostly determined by the final exam on Wed Jan 23 14:00-16:30 KBG COSMOS, Utrecht. The exam is a written exam on all material treated in the course. You are not allowed to use notes. The regular homework assignments can either be handed in by email or in person before class. Homework is optional can only increase your final grade. More formally: $\mathrm{FinalGrade} = \min\{\mathrm{rank}(\pi_3(S^2))\ \mathrm{ExamGrade}+\sin(\frac{\pi}{20}\mathrm{Average Homework}),10\}$.
The retake exam is on Wed Feb 27, 14-17 in Ruppert 033. It is optional and may be used to replace your exam grade in the above formula, if your home institution agrees.

Prerequisites - Background in point-set topology: topological spaces, continuous maps, compactness, quotients and products, along the lines of these notes by A. Hatcher and maybe a first encounter with the fundamental group. - Knowledge about basic constructions with vector spaces and abelian groups. - Some familiarity with categories and functors is also helpful. For those who haven't seen this before, the "Intensive course on Categories and Modules" is recommended.

EXERCISES: Exercises by Steffen Sagave

VIDEO RECORDINGS (Thanks to Mike Daas):

Recordings of all the lectures will be available here, you'll need the password: Xz4F

$\pi_n(X,A,x_0)\cong H_n(X,A,\mathbb{Z})$
$\cdots \to H_n(\bar{C}) \to H_{n-1}(C')\to H_{n-1}(C)\to \cdots$
$\pi_3(S^2) = \mathbb{Z}$
$\partial_n = \sum_{i=0}^n (-1)^i d_i:C_n\to C_{n-1}$
SCHEDULE
Date Class material Exercises
Wed Sep 12Introduction, Lecture 1Homework due Sept 19 before class
Wed Sep 19 in MIN 201Lecture 3, homology of spheresHomework due Sept 26 before class: Exercise 1 from Exercises 19-9 and Exercise 2.4 from Exercises 2016 by Steffen Sagave.
Wed Sep 26Lecture 3, 4 more on Long Exact Sequence, proof of Homotopy invariance (without simplicial sets)Exercises 26-9, Exercises 1,2
Wed Oct 3 Lecture 6 proof of small simplices theorem (Following Hatcher p.119 prop 2.21)Homework due Oct 10 before class:Exercises 1,2 from Exercises 3-10
Wed Oct 10Lecture 5, proof of Excision theorem, Degree of a map, lect 7 cell attachmentExercise 1 from Exercises 10-10 and Exercise 7.4 from Exercises 2016 by Steffen Sagave
Wed Oct 17 (in Ruppert D)Lecture 7, Hawaiian earringExercises 1,2 from Exercises 17-10
Wed Oct 24 (in BBG 061)Lecture 8 CW complexesExercises 8.1, 8.4 from Exercises 2016 by Steffen Sagave. It may be helpful to also check out Jorge's comments on torsion for 8.4
Wed Oct 31 (in Minnaert 201)Lecture 9, Cellular homologyNo homework.
Wed Nov 7Lecture 10Homework, Sagave Exercise 10.1 and this exercise.
Wed Nov 14Lecture 10Homework: this exercise by Jorge
Wed Nov 21Lecture 11Homework: Sagave Exercise 11.4
Wed Nov 28Lecture 12Homework: Sagave Exercise 12.1
Wed Dec 5Lecture 13
Wed Dec 12Lecture 14Practice exam by Jorge
Wed Jan 23 EXAM 14:00-17:00 KBG COSMOSThe exam is a written exam on all material treated in the course. You are not allowed to use notes. Solutions to the exam
Wed Feb 27 RETAKE EXAM 14:00-17:00 Ruppert 033The exam is a written exam on all material treated in the course. You are not allowed to use notes.