# Inleiding in de algebraïsche topologie, najaar 2018

Dit is the webpagina voor het college Inleiding in de algebraïsche topologie van najaar 2018.

BELANGRIJK: dit college is bedoeld voor (derdejaars) bachelor-studenten. Master students interested in this topic are strongly recommended to take the mastermath course Algebraic Topology (M1).

In dit college zetten we de in het college Topologie aangevangen studie van de fundamentaalgroep en van overdekkingsruimten voort. We zullen zien dat er een nauw verband tussen de twee concepten bestaat, die wordt uitgedrukt via de zogenaamde monodromiewerking. Verdere onderwerpen die aan bod komen zijn bijvoorbeeld windingsgetallen en de stelling van Van Kampen. We zullen de stelling van Borsuk-Ulam bewijzen. Op een informeel niveau zegt deze stelling dat op elk moment er op de aarde twee antipodale plekken zijn waar zowel de luchtdruk als de temperatuur hetzelfde zijn. We komen ook toepassingen in de algebra tegen, bijvoorbeeld: een ondergroep van eindige index in een vrije groep is vrij.

Rooster: maandagen van 9--10:45, in zaal 174. LET OP: het college begint op maandag 10 september 2018. Het volledige rooster vind je hier.

Literatuur: we behandelen enkele hoofdstukken uit het onderstaande boek. Het boek is als e-boek verkrijgbaar op het universitaire netwerk, via SpringerLink. For the second part of the course, we will discuss parts of Sections 1--3 of the syllabus "Algebraic Topology -- an introduction" by Eduard Looijenga that you can find here.

• W. Fulton: Algebraic Topology: A First Course, Springer Graduate Texts in Mathematics 153.
• Benodigde voorkennis: Algebra 1 & 2, Lineaire Algebra 1 & 2, Topologie.

Eindcijfer: het eindcijfer wordt gebaseerd op zowel huiswerkopdrachten als een (mondeling of schriftelijk, afhankelijk van het aantal deelnemers) tentamen. Het huiswerk telt voor 25 % en het tentamen voor 75 %. Om te slagen voor dit vak moet voor zowel huiswerk als tentamen minstens een 5 zijn behaald.

Aanmelding voor het tentamen: indien je tentamen voor dit vak wilt doen moet je je daarvoor registreren in usis. Een aanvraag voor extra tijd op het tentamen dien je ruim van tevoren in bij de onderwijscoordinator Laura van Kempen-Helmsing.

Question hour: a question hour is planned on Friday, January 18, 13:30-15:30, in room Sn412.

Huiswerkopgaven (voorlopige planning):

• Homework set 1, issued 17 September, deadline 28 September.
• Homework set 2, issued 8 October, deadline 19 October.
• Homework set 3, issued 29 October, deadline 9 November.
• Homework set 4, issued 19 November, deadline 30 November.
• Homework set 5, issued 10 December, deadline 21 December.
• Vragenuur: the weekly question hour is cancelled. The teaching assistants are Wout Gevaert (woutgevaert*funny symbol*gmail.com), Mathé Hertogh (m.c.hertogh*funny symbol*gmail.com) and Stefan van der Lugt.

Behandelde stof en oefenopgaven:

College 1 (10 september):

• Inhoud: introduction, recap of fundamental group and covering spaces. Examples. A contractible space is simply connected. Functoriality of the fundamental group. A retraction induces an injection on the level of fundamental groups. Classification of all covering spaces of the circle with source path connected - to be proven later during the course.
• Literatuur: please have another look at the sections 11-15 of the Topologie syllabus (Fall 2017). Alternatively/additionally, from Fulton: sections 11a,b and 12a,b. Warning: in Fulton, for each topological space X, the empty set mapping to X is a covering map. With our definition (that we copy from the Topologie syllabus), if the empty set mapping to X is a covering map, then X itself is necessarily the empty set. One concludes that the definition of covering space is not uniform over all literature.
• Oefenopgaven: see here.

• College 2 (17 september):
• Inhoud: definition of covering, examples, properties, non-examples, monodromy action by fundamental group, even actions by automorphisms, an even action gives rise to a covering space by taking the quotient, G-coverings, examples, automorphisms of coverings, theorem about the unicity of lifts, corollary: if Y -> X is a G-covering with Y connected, then the natural inclusion of Y into Aut(Y/X) is an isomorphism.
• Literatuur: Fulton, 11a, b, c, d.
• Oefenopgaven: see the homework above.

• College 3 (24 september):
• Inhoud: fundamental group of S^n/\mu_2, theorem about the unicity of lifts, corollary: if Y -> X is a covering with Y connected then the action of Aut(Y/X) on Y is free. Moreover the cardinality of Aut(Y/X) is bounded by the cardinality of a fiber. Proof of theorem about unicity of lifts. Theorem about the existence of lifts. Definition of locally path connected spaces. Proof of theorem about the existence of lifts.
• Literatuur: Fulton, 11a, b, d, 13a. The theorem about unicity of lifts is Fulton, Lemma 11.5. The theorem about existence of lifts is Fulton, Proposition 13.5. (it contains a typo: the H should be f-tilde).
• Oefenopgaven: see here.

• College 4 (8 oktober):
• Inhoud: proof of Borsuk-Ulam. There is no homeomorphism from R^3 to R^2. Degree (or winding number) of a self-map of S^1. Basic properties of degrees. A self-map of S^1 that is the restriction to the boundary of a map from D^2 to S^1 has degree zero. An odd map has odd degree. Assume from now on that the base space of a covering is locally path-connected. If Y -> X is a connected covering then Aut(Y/X) acts evenly on Y. For every subgroup H of Aut(Y/X), the induced map Y/H -> X is a covering. If Aut(Y/X) acts transitively on each fiber then the induced map Y/Aut(Y/X) -> X is a homeomorphism. Notion of regular (aka Galois) covering. Universal coverings. Examples. Universal property.
• Literatuur: Fulton 4c, 11d, esp. Proposition 11.38. Our proof of Borsuk-Ulam for the 2-sphere is based on Fulton, pp. 185--186.
• Oefenopgaven: see the homework above.

• College 5 (15 oktober, lecture given by Stefan van der Lugt):
• Inhoud: assume from now on that the base space of a covering is connected and locally path-connected. Regular and universal coverings. Universal property of universal coverings. A pointed universal covering is unique up to a unique isomorphism of coverings, if it exists. Corollaries: the automorphism group of a universal covering acts transitively on each of its fibers, hence a universal covering is regular. The unique map as described in the universal property of universal covering is a covering, once the target is connected. Galois correspondence. Description of all connected coverings of the circle. Connection with monodromy action.
• Literatuur: Fulton 13b, c, d.
• Oefenopgaven: see the homework above.

• College 6 (22 oktober, lecture given by Stefan van der Lugt):
• Inhoud: further discussion of connection with monodromy action. Normal subgroups of the fundamental group correspond with regular coverings. For (Y,y) -> (X,x) a regular covering, description of a natural group homomorphism from fundamental group of (X,x) to Aut(Y/X). This is surjective, with kernel the fundamental group of (Y,y). If (X',x') -> (X,x) universal, then have a natural isomorphism of groups \pi_1(X,x) -> Aut(X'/X). Examples. Discussion of existence/construction of a universal covering. Discussion of semi-locally simply connected and locally simply connected spaces. Universal covering of the figure-eight, and preliminary discussion of its automorphism group (full description later, after Van Kampen is discussed).
• Literatuur: Fulton 13b, c, d.
• Oefenopgaven: see here.

• College 7 (29 oktober):
• Inhoud: intrinsic description of the universal covering space (for connected and locally path connected spaces, once one exists) as the 'space' of path homotopy classes of paths into the base space X with begin point the base point x. Recap of the Galois correspondence; regular pointed coverings correspond to normal subgroups of the fundamental group. Let G be a group. For (Y,y) -> (X,x) a connected pointed G-covering, description of a natural surjective group homomorphism from fundamental group \pi of (X,x) to G. For (Y,y) -> (X,x) a pointed G-covering, description of a natural group homomorphism from \pi to G. Example: the trivial G-covering gives the trivial group homomorphism. This sets up a bijection between G-isomorphism classes of pointed G-coverings and Hom(\pi,G). Description of the inverse map.
• Literatuur: Fulton 14a.
• Oefenopgaven: see the homework above.

• College 8 (12 november):
• Inhoud: pushout of groups. For (Y,y) -> (X,x) a pointed G-covering, with X good, description of a natural associated group homomorphism from \pi(X,x) to G. For \rho a group homomorphism from \pi(X,x) to G, description of a natural associated pointed G-covering of (X,x). This sets up a bijection between the set of G-isomorphism classes of pointed G-coverings, and Hom(\pi(X,x),G). Proof of this statement. This bijection is compatible with restriction to good subspaces U of X containing x, on the side of G-coverings, and with composition with i_* (where i is the inclusion of U in X), on the side of Hom-sets. Description of the space of pointed G-coverings of (X,x) when an open cover of X by good subspaces U, V is given such that the intersection of U, V is good and contains x. Corollary: the Van Kampen theorem, saying that the diagram of natural maps involving the fundamental groups of U \cap V, U, V and X (with base-point x) is a pushout diagram. Example: the fundamental group of a sphere of dimension two or higher is trivial.
• Literatuur: Fulton 14a, b, c.
• Oefenopgaven: see here.

• College 9 (19 november):
• Inhoud: glueing spaces together, glueing pointed G-coverings together, description of the space of pointed G-coverings of (X,x) when an open cover of X by good subspaces U, V is given such that the intersection of U, V is good and contains x. Corollary: the Van Kampen theorem, saying that the diagram of natural maps involving the fundamental groups of U \cap V, U, V and X (with base-point x; each of U \cap V, U, V and X is good, U, V are open and cover X) is a pushout diagram. Example: when U, V are simply connected, then so is X. Example: when U \cap V is simply connected, then \pi_1(X,x) is the free group on \pi_1(U,x) and \pi_1(V,x). The fundamental group of the figure-eight space is the free group on two generators. Discussion of universal covering space of the figure-eight space and its automorphism group. Discussion of some examples of connected coverings of the figure-eight space, and how they connect (via the Galois-correspondence) with subgroups of the free group on two generators. Discussion of connected topological graphs, and their coverings. Proof of the Nielsen-Schreier theorem: every subgroup of finite index of a free group on finitely many generators, is itself free on finitely many generators.
• Literatuur: Fulton 14c,d.
• Oefenopgaven: see the homework above.

• College 10 (26 november):
• Inhoud: affine span, standard simplex, singular p-simplex in a space, free abelian groups, group of singular p-chains in a space, boundary map, chain complexes, cycles and boundaries, homology groups, functoriality, homology groups of a one-point space.
• Literatuur: syllabus Looijenga, pp. 1--7.
• Oefenopgaven: see the homework above.

• College 11 (3 december):
• Inhoud: homotopies of chain maps, homotopic chain maps induce equal maps on homology, homotopic maps between spaces induce equal maps on singular homology (proof only sketched, but an example is worked out), a homotopy equivalence induces an isomorphism on homology, homology groups of a contractible space coincide with those of the one-point space, notion of (the complex of) small simplices, small simplices theorem: singular homology can be computed as homology of the complex of small simplices (proof only sketched, using the technique of barycentric subdivision).
• Literatuur: syllabus Looijenga, Thm. 2.4 and corollaries, Prop. 2.12.
• Oefenopgaven: see here.

• College 12 (10 december, lecture given by Stefan van der Lugt):
• Inhoud: long exact sequence associated to a short exact sequence of complexes, snake lemma, naturality of the connecting homomorphism, diagram chasing, Mayer-Vietoris long exact sequence, homology groups of the spheres.
• Literatuur: syllabus Looijenga, Thm. 2.2, and the section Applications (Ch. 3) up to and including Cor. 3.2. Note that we calculate the homology of spheres in a different way, though, using Mayer-Vietoris.
• Oefenopgaven: see homework V above.

• College 13 (17 december, lecture given by Stefan van der Lugt):
• Inhoud: S^n is not a retract of B^{n+1}, Brouwer fixed point theorem, S^n and S^m homotopy equivalent implies m=n, R^m and R^n homeomorphic implies m=n, degree of a self-map of S^n, examples, classical applications: a non-surjective self-map of S^n has degree zero, a self-map of S^n without fixed point has degree (-1)^{n+1}, the antipodal map on S^n is homotopic with the identity map if and only if n is odd, hairy ball theorem.
• Literatuur: syllabus Looijenga, the section Applications (Ch. 3) up to and including Cor. 3.9.
• Oefenopgaven: see homework V above.