Algebraic Topology, WS 2018/19

Prof. Dr. C. Löh / D. Fauser / J. Witzig

News

The lecture notes are updated (version of February 7 (last lecture)).
The new etudes are online: Etudes 15 (sheet of February 7, no submission).
If you are interested in taking the course Group Cohomology, please register in HIS/LSF (before February 6)!
The new exercise sheet with serious topology is online: Sheet 15 (sheet of February 4, optional submission before Friday, February 8, 10:00);
20 bonus credits!.
Organisation of the exams: see below
Projective Asteroids (proof of concept, by Johannes Witzig)
Organisational matters
The seminar Topology vs. Combinatorics in WS 2017/18 accompanies this course; no previous knowledge of algebraic topology is required.
The organisational meeting is on Friday, July 6, 13:45, M 201.
If you plan to write a bachelor thesis under my supervision in SS 2019 (in Topology/Geometry), you should participate in a seminar in the Global Analysis and Geometry group before SS 2019.

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.

Moebius

Algebraic topology has various applications, both in theoretical and in applied mathematics, for instance, through fixed point theorems and (non-)embeddability results. 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 "Group Cohomology" in the summer 2019, where (co)homology of groups will be studied. The course in SS 2019 can also be attended independently of the present course on Algebraic Topology.

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. 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, Degrees of Self-Maps of Spheres]
- Glueings: The Mayer-Vietoris sequence
- Classification of Homology Theories
Singular Homology
- Construction
  [Geometric Idea, Construction, 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]

Time/Location

Monday, 10--12, M 102,
Thursday, 10--12, M 104.

Exercise classes

Do 14--16, H31
Fr 8--10, M009

The exercise classes start in the first week; in this first session, some basics on topological spaces and categories will be discussed (as on the sheet Etudes 0).
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 15, 2018,	submission before October 22, 2018 (10:00)	will be discussed in the exercise classes on October 25/26
Sheet 2,	of October 22, 2018,	submission before October 29, 2018 (10:00)	will be discussed in the exercise classes on November 2/?
			Projective Asteroids (proof of concept, by Johannes Witzig)
Sheet 3,	of October 29, 2018,	submission before November 5, 2018 (10:00)	will be discussed in the exercise classes on November 8/9
Sheet 4,	of November 5, 2018,	submission before November 12, 2018 (10:00)	will be discussed in the exercise classes on November 15/16
Sheet 5,	of November 12, 2018,	submission before November 19, 2018 (10:00)	will be discussed in the exercise classes on November 22/23
Sheet 6,	of November 19, 2018,	submission before November 26, 2018 (10:00)	will be discussed in the exercise classes on November 29/30
Sheet 7,	of November 26, 2018,	submission before December 3, 2018 (10:00)	will be discussed in the exercise classes on December 6/7
Sheet 8,	of December 3, 2018,	submission before December 10, 2018 (10:00)	will be discussed in the exercise classes on December 13/14
Sheet 9,	of December 10, 2018,	submission before December 17, 2018 (10:00)	will be discussed in the exercise classes on December 20/21
Sheet 10,	of December 17, 2018,	submission before January 7, 2019 (10:00)	will be discussed in the exercise classes on January 10/11
Sheet 11,	of January 7, 2019,	submission before January 14, 2019 (10:00)	will be discussed in the exercise classes on January 17/18
Sheet 12,	of January 14, 2019,	submission before January 21, 2019 (10:00)	will be discussed in the exercise classes on January 24/25
Sheet 13,	of January 21, 2019,	submission before January 28, 2019 (10:00)	will be discussed in the exercise classes on January 31/February 1
Sheet 14,	of January 28, 2019,	submission before February 4, 2019 (10:00)	will be discussed in the exercise classes on February 7/8
Sheet 15,	of February 4, 2019,	optional submission before February 8, 2019 (10:00)

Etudes

These etudes 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 15, 2018,	no submission,	will be discussed in the exercise classes on October 18/19.
Sheet 1,	of October 19, 2018,	no submission,
Sheet 2,	of October 25, 2018,	no submission,
Sheet 3,	of November 1, 2018,	no submission,
Sheet 4,	of November 8, 2018,	no submission,
Sheet 5,	of November 15, 2018,	no submission,
Sheet 6,	of November 22, 2018,	no submission,
Sheet 7,	of November 29, 2018,	no submission,
Sheet 8,	of December 6, 2018,	no submission,
Sheet 9,	of December 13, 2018,	no submission,
Sheet 10,	of December 20, 2018,	no submission,
Sheet 11,	of January 10, 2019,	no submission,
Sheet 12,	of January 17, 2019,	no submission,
Sheet 13,	of January 24, 2019,	no submission,
Sheet 14,	of January 31, 2019,	no submission,
Sheet 15,	of February 7, 2019,	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.

Dates for the oral exams (25 minutes):

Fri, February 22,
Wed, March 13,
Wed, April 3.

Registration details:

Please register on time in FlexNow for the oral exam.
Moreover, for the oral exam, you also need to register with Ms. Bonn (M 217) for a time slot.
If applicable: For the Studienleistung (successful participation in the exercise classes), please register on time in FlexNow.

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Last change: February 7, 2019