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Algebraic Topology, WS 2021/22

Prof. Dr. Clara Löh / Matthias Uschold / Johannes Witzig

News

• Registrations for exams: The exams will be oral exams (25 mins) on zoom. Dates: 18.02., 11.03., 04.04.
  An additional option in March is be offered (11.03.).
  Algorithm:
  • Register with Ms Bonn for a date/timeslot for an exam (rosina.bonn@ur.de, +49/941/943-2766).
  • Register in FlexNow for the corresponding date
  In case you don't know what to expect from such an exam: The Fachschaft has a collection of minutes of exams from students.
  Technical prerequisites: You will need some device to share handwritten notes that you write during the exam, e.g., a graphics tablet, an ordinary tablet, a document camera, a mobile-phone-used-as-document-camera, ... There are many options!
• Registration for the Studienleistung: via FlewNow; please register before the deadline.
• Follow-up courses in SS 2022:
  • Algebraic Topology II (Pavel Sechin)
  • Geometric Group Theory (Clara Löh)
  • Seminar: Thompson Groups (Clara Löh)
• 10.01.2022: Until further notice:
  • We will continue the lectures in the hybrid format (i.e., in person lectures on campus with a live zoom stream).
  • The exercise class of Matthias will be held in hybrid format.
  • The exercise class of Johannes will be held in person; participants that want/need to attend online should join the zoom stream of the exercise class of Matthias.
• Exercise sheet 15 (of February 8) is online: pdf; bonus sheet; optional submission before February 15, 8:30 (via GRIPS)
• The lecture notes are updated: pdf (version of 11.02.2022)
• The results of the course evaluation can be found on GRIPS; thanks for all your feedback!
• Organisational matters
• The time/location changed! (In order to resolve the time conflict with Differential Geometry I.)
• If you plan to write a bachelor thesis under my supervision in SS 2022 (in Topology/Geometry), you should participate in a seminar in the Global Analysis and Geometry group before SS 2022.

Algebraic Topology

Algebraic topology studies topological spaces via algebraic invariants -- by modelling certain aspects of topological spaces in the realm of algebra, e.g., by groups and group homomorphisms. Classical examples include homotopy groups and (co)homology theories.


Algebraic topology has various applications, both in theoretical and in applied mathematics, for instance, through fixed point theorems, (non-)embeddability results, topological data analysis, and many more. For example, Nash's proof of existence of certain equilibria in game theory is based on a topological argument. Topics covered in this course include:

• What is algebraic topology?
• The fundamental group and covering theory
• The Eilenberg-Steenrod axioms
• Singular homology
• Cellular homology
• Classical applications of (co)homology.

This course will be complemented with the course "Geometric Group Theory" in the summer 2022. The course in SS 2022 can also be attended independently of the present course on Algebraic Topology. Moreover, there probably will also be a continuation of the Algebraic Topology Series.

If all participants agree, this course can be held in German; solutions to the exercises can be handed in in German or English.

Lecture notes:

• pdf;
• correction log.

Topics covered so far:

• Literature
• Introduction
• What is Algebraic Topology?
  
  • Topological Building Blocks
    [Construction: Subspaces, Construction: Products, Construction: Quotient Spaces and Glueings, The Homeomorphism Problem]
  • Categories and Functors
    [Categories, Functors Natural Transformations]
  • Homotopy and Homotopy Invariance
    [Homotopy, Homotopy Invariance, Using Homotopy Invariant Functors]
• Fundamental Group and Covering Theory
  
  • The Fundamental Group
    [The Group Structure on the Fundamental Group, Changing the Basepoint]
  • Divide and Conquer
    [The Fundamental Example, Products, Glueings, Computations, Ascending Unions]
  • Covering Theory
    [Coverings, Lifting Properties, The Universal Covering, The Fundamental Example, The Classification Theorem, Application: The Nielsen-Schreier Theorem ]
• Axiomatic Homology Theory
  
  • The Eilenberg-Steenrod Axioms
    [The Axioms, First Steps]
  • Homology of Spheres and Suspensions
    [Suspensions, Homology of Spheres, Mapping Degrees of Self-Maps of Spheres]
  • Glueings: The Mayer-Vietoris sequence
  • Classification of Homology Theories
• Singular Homology
  • Construction
    [Geometric Idea, Singular Homology, First Steps, The Long Exact Sequence of Pairs ]
  • Homotopy Invariance
    [Geometric Idea, Prism Decomposition, Proving Homotopy Invariance]
  • Excision
    [Geometric Idea, Barycentric Subdivision, Proving Excision]
  • Applications
    [Singular Homology as Ordinary Homology, The Jordan Curve Theorem, Invariance of Domain and Non-Embeddability, (Commutative Division Algebras), (Rigidity in Geometry)]
  • Singular Homology and Homotopy Groups
• Celluar Homology
  
  • CW-Complexes
  • Cellular Homology
    [Geometric Idea, The Construction, Comparison of Homology Theories]
  • The Euler Characteristic
    [Geometric Definition of the Euler Characteristic, A Homological Description, Divide and Conquer, Application: Nielsen-Schreier, Quantitatively]
• Appendix
  
  • Point-Set Topology
  • Homotopy Flipbooks:
    one example: pdf
    and another example: pdf
  • Cogroup Objects and Group Structures
  • Amalgamated Free Products
  • Group Actions
  • Basic Homological Algebra
    [Exact Sequences, Chain Complexes and Homology, Homotopy Invariance]
  • Homotopy Theory of CW-Complexes
    [Whitehead Theorem, Cellular Approximation, Subcomplexes and Cofibrations]

Notes from online teaching:

• Current lecture: pdf
• 14.12.2021: pdf
• 17.12.2021 pdf
• 21.12.2021 pdf
• 07.01.2022 pdf

Time/Location

Tuesday, 8:30--10:00, M 101,
Friday, 8:30--10:00, M 101.

Exercise classes

Time/Location:
Fr 10--12, M009, Matthias Uschold
Fr 12--14, M103, Johannes Witzig

• Please register before October 20, 10:00, via GRIPS for the exercise classes!
• The exercise classes start in the second week.
• Organisational matters

Exercise sheets

Solutions can be submitted in English or German and in teams of up to two people. Please do not forget to add your name to all your submissions!

Sheet 1,	of October 19, 2021,	submission before October 26, 2021 (08:30)	will be discussed in the exercise classes in the second week
Sheet 2,	of October 26, 2021,	submission before November 2, 2021 (08:30)	will be discussed in the exercise classes on 05.11.2021
			Projective Asteroids (proof of concept, by Johannes Witzig)
Sheet 3,	of November 2, 2021,	submission before November 9, 2021 (08:30)	will be discussed in the exercise classes on 12.11.2021
Sheet 4,	of November 9, 2021,	submission before November 16, 2021 (08:30)	will be discussed in the exercise classes on 19.11.2021
Sheet 5,	of November 16, 2021,	submission before November 23, 2021 (08:30)	will be discussed in the exercise classes on 26.11.2021
Sheet 6,	of November 23, 2021,	submission before November 30, 2021 (08:30)	will be discussed in the exercise classes on 03.12.2021
Sheet 7,	of November 30, 2021,	submission before December 7, 2021 (08:30)	will be discussed in the exercise classes on 10.12.2021
Sheet 8,	of December 7, 2021,	submission before December 14, 2021 (08:30)	will be discussed in the exercise classes on 17.12.2021
Sheet 9,	of December 14, 2021,	submission before December 21, 2021 (08:30)	will be discussed in the exercise classes on 07.01.2022
Sheet 10,	of December 21, 2021,	submission before January 11, 2022 (08:30)	will be discussed in the exercise classes on 14.01.2022
Sheet 11,	of January 11, 2022,	submission before January 18, 2022 (08:30)	will be discussed in the exercise classes on 21.01.2022
Sheet 12,	of January 18, 2022,	submission before January 25, 2022 (08:30)	will be discussed in the exercise classes on 28.01.2022
Sheet 13,	of January 25, 2022,	submission before February 1, 2022 (08:30)	will be discussed in the exercise classes on 04.02.2022
Sheet 14,	of February 1, 2022,	submission before February 8, 2022 (08:30)	will be discussed in the exercise classes on 11.02.2022
Sheet 15,	of February 8, 2022,	bonus sheet; optional submission before February 15, 2022 (08:30)

Études

These études help to train elementary techniques and terminology. These problems should ideally be easy enough to be solved within a few minutes. Solutions are not to be submitted and will not be graded.

Sheet 0,	of October 19 2021,	no submission
Sheet 1,	of October 22 2021,	no submission
Sheet 2,	of October 29 2021,	no submission
Sheet 3,	of November 5 2021,	no submission
Sheet 4,	of November 12 2021,	no submission
Sheet 5,	of November 19 2021,	no submission
Sheet 6,	of November 26 2021,	no submission
Sheet 7,	of December 3 2021,	no submission
Sheet 8,	of December 10 2021,	no submission
Sheet 9,	of December 17 2021,	no submission
Sheet 10,	of January 7, 2022,	no submission
Sheet 11,	of January 14, 2022,	no submission
Sheet 12,	of January 21, 2022,	no submission
Sheet 13,	of January 28, 2022,	no submission
Sheet 14,	of February 4, 2022,	no submission
Sheet 15,	of February 11, 2022,	no submission

Literature

This course will not follow a single book. Therefore, you should individually compose your own favourite selection of books.
A list of suitable books can be found in the lecture notes.

Prerequisites

All participants should have a firm background in Analysis I/II (in particular, basic point set topology, e.g., as in Analysis II in WS 2011/12), in Linear Algebra I/II, and basic knowledge in group theory (as covered in the lectures on Algebra).
Knowledge about manifolds as in Analysis IV is not necessary, but helpful.
Knowledge about basic homological algebra (as in the last two weeks of Kommutative Algebra) is not necessary, but helpful.

Exams

Please read the information on organisation and formalities of this course.

Last change: February 11, 2022