Veckodisposition
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This course is owned jointly by SU and KTH.
Algebraic Topology is a continuation of the idea of the fundamental group, i.e. of assigning a group (or other simple algebraic object) to a space in order to measure its properties. We will look at the homology and cohomology groups of a space, which are invariants defined by using the machinery of homological algebra. These groups are easier to compute than the fundamental group but nevertheless powerful invariants. Here are some striking results that are easy to prove with the basic tools of algebraic topology:
- There are always two opposite points on the earth with the exact same temperature and humidity.
- There is always a place on the earth with no wind.
- However messily made, any sandwich with bread, cheese, and tomato can be cut by a straight cut into two halves with the exact same amount of bread, cheese, and tomato in each half.
Algebraic topology is a major branch of pure mathematics with many interconnections to other areas such as geometry, algebra, physics, and even data science.
Course contents
- singular homology and cohomology of topological spaces
- exact sequences, chain complexes and homology
- homotopy invariance of singular homology
- the Mayer-Vietoris sequence and excision
- cell complexes and cellular homology
- the cohomology ring
- homology and cohomology of spheres and projective spaces
- applications such as the Brouwer Fixed Point theorem, the Borsuk-Ulam theorem and theorems about vector fields on spheres.
Prerequisites: abstract algebra (groups and rings), topology.
The first part of the course (period 3) is taught by Alexander Berglund (SU) and the second part (period 4) by Tilman Bauer (KTH). The teaching assistant for the course is Sylvain Douteau.
Schedule
Classes are on Mondays:
8:30-9:15 exercise session with Sylvain
9:30-11:15 lecture with Alexander/Tilman.
The first lecture is on January 24 (no exercise session this day).
The second half of the course will start on March 14 and be given by Tilman Bauer at KTH, room 3721. Classes will be in-person and no longer be streamed.
Examination
The examination is based on weekly short homework sets (given out on Mondays, due on the following Thursday), along with individual oral presentations of one or two of the homework problems at the end of the course. We will tell you in advance which problem we want you to present in this oral exam, which will be graded on a pass/fail scale.
Literature
Hatcher, Alan: Algebraic Topology, Cambridge University Press 2001. Freely available online and cheaply in print.
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Simplicial homology.Reminder on basic topology, simplices, Delta-complexes, simplicial homology.Reading: Hatcher p.97-107.
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Singular homology. Singular simplices, singular chain complex, functoriality, topological invariance of singular homology, path components and degree 0 homology, the fundamental group and degree 1 homology. Reminder on homotopic maps, homotopy equivalence.
Reading: Hatcher 108-113 (except proof of Theorem 2.10) and §2A (cursory).
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Homotopy invariance of singular homology. Chain complexes, exact sequences, chain maps, chain homotopies, proof of homotopy invariance (using increasing coordinates), applications of homotopy invariance.
Reading: Hatcher 110-114 (until Theorem 2.13).
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Relative homology. Long exact sequence in homology associated to a short exact sequence of chain complexes, diagram chasing, relative homology groups.
Reading: Hatcher 115-119 (until Excision).
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Excision. Small singular simplices, cone operator, barycentric subdivision, excision theorem.
Reading: Hatcher 119-126.
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Five Lemma. Mayer-Vietoris sequence. Equivalence of simplicial and singular homology. The five-lemma, Mayer-Vietoris sequence, simplicial vs. singular homology, skeletal filtration of a Delta-complex.
Reading: Hatcher 128-131, 149-153.
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Degree theory for spheres. Cellular homology. The degree of a self-map of a sphere and basic properties. Degrees of reflections and the antipodal map. The hairy ball theorem. Cell complexes and cellular homology.
Reading: Hatcher 134-147.
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The definition of cohomology of a chain complex, space, CW complex, or Delta-complex. Example: homology and cohomology of real projective space, the latter with coefficients in an arbitrary abelian group G.
Covariant and contravariant additive functors; left and right exact functors. Proof that Hom(-,G) is a left exact contravariant functor. Definition of Ext(A,G). Formulation of the universal coefficient theorem for cohomology.
No homework this week (because it's exam week for some). Suggested reading: Chapter III until p. 193 (up to and excluding Lemma 3.1); Chapter II.3 "Categories and Functors"
Suggested exercises: III.1.4, 6, 8, 9. Compute the (cellular) homology and cohomology of your favorite spaces (e.g. orientable and nonorientable surfaces, lens spaces, etc.)
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Derived functors of abelian groups with free resolutions. The fundamental lemma of homological algebra. The universal coefficient theorem in cohomology.
The definition of the cup product in cohomology with coefficients in a ring. Computation of the mod-2 cohomology of RP^n.
Reading: Hatcher 3.1, whole section. This section contains proofs of the long exact sequence, excision, Mayer-Vietoris etc. for cohomology, but these proofs are completely analogous to the homology setting, so I will skip them in class. Hatcher 3.2 up to p. 209.
Suggested exercises: 3.1.5, 8, 9; 3.2.1, 3, 7
Preparatory reading: read about tensor products and, if you have time, look at the universal coefficient theorem in homology (Hatcher 3A).
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Applications of the cohomology computation of RP^n: the Borsuk-Ulam and ham sandwich theorems.
The graded commutativity of the cup product in cohomolog.
Tensor products of abelian groups.
The Künneth theorem for the cohomology of a product of spaces.
Reading: Hatcher 3.2 up to p. 222
Supplemental reading: Hatcher 3.A: Derived functors of the tensor products and universal coefficients in homology; 3.B: The general Künneth formula.
Recommended preparatory reading: the definition of a manifold.
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Focus on exercises, with Sylvain.
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Introduction to duality; topological manifolds; orientations.
Hatcher Ch 3.3
tom Dieck Ch 16.1-2
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Colimits of functors into sets and abelian groups; filtered colimits.
Homology in the top dimension of a manifold and the bootstrap theorem.
tom Dieck, 16.3
Hatcher 3.3 up to p. 239
For colimits of functors, there is a plethora of good sources on- and offline. A classical one is Saunders Mac Lane, "Categories for the working mathematician".
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Cap product; cohomology with compact support; the proof of Poincaré duality
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You will be assigned a problem to present, a time slot and a place on Monday, May 2. Every oral exam consists of a 10 minutes presentation followed by up to 10 minutes of questions and discussions about the presented problem. The grading is on a P/F scale, and you will only be offered an oral exam slot if you have the required minimum number of points in the homework. You may take notes to the oral exams, but keep them short.
The grades are computed as follows, assuming that you passed the oral exam. Your three worst homework sets are dropped, and the sum of the remaining points determines your grade as follows:
90+ = A
80+ = B
70+ = C
60+ = D
50+ = E
48+ = Fx