Topology Spring 2020

	Date	Content	References	Extras
1	17.02.	Introduction to the course: resources, key objectives and the roadmap in front of us. Some advice on taking a Math course.	[GT] pp. 1-3	Study Guide
2	19.02.	Topological spaces: definition and examples. Partial order structure on the class of topologies on a given set. Open and closed sets. Interior, closure and boundary (via two equivalent perspectives).	[GT] pp. 3-7	Interior, closure, boundary
3	24.02.	Bases of a topological space: definition and examples. A criterion for a collection of subsets to generate a topology. Application: metric spaces. Different distances may determine different bases of the same topology.	[GT] pp. 7-10
4	26.02.	Topological subspaces, definition and equivalent characterisations. Metric subspaces. Discrete subspaces. Subspaces and topological operations. Continuous maps and continuity criteria. Homeomorphisms. Some examples.	[GT] pp. 10-13
5	02.03.	Topological products, (equivalent) definitions and examples. Projections, injections, and a continuity criterion for maps with target a product space.	[GT] pp. 13-16
6	04.03.	Connectedness: definition and examples. A subset of the real line is connected if and only if it is an interval. The continuous image of a connected set is connected. Path-connectedness: definition and examples. Path-connectedness implies connectedness (the converse is false). The continuous image of a path-connected set is path-connected. Cut points. Application: certain pairs of spaces cannot possibly be homeomorphic (counting cut points).	[GT] pp. 18-21
7	09.03.	Path-connected components, definition and basic facts. Examples of connected spaces that are not path-connected.	[GT] pp. 21-23
8	11.03.	Connected components, definition and basic facts. Review of the key example. Connected + locally path-connected implies path-connected.	[GT] pp. 23-25
		Compactness: definition and examples. Three results: a closed subspace of a compact space is compact; the continuous image of a compact space is compact; the product of two compact spaces is compact (weak Tychonoff). Three equivalent notions of compactness for subdomains of Euclidean spaces.	[GT] pp. 30-33
9	16.03.	Proof of the Heine-Borel theorem. Separation axioms: the Hausdorff property. Examples, basic facts. Compact subspaces of a Hausdorff space are closed. A criterion to ensure that a continuous bijection is a homeomorphism.	[GT] pp. 33-36	Notes - L09
10	18.03.	Normal spaces. A Hausdorff space that is not normal. Two classes of normal spaces: i) compact Hausdorff spaces, ii) metric spaces. The Lebesgue number of a cover. Compact metric spaces have a Lebesgue number. Application: uniform continuity for functions from/to metric spaces.	[GT] pp. 36-39	Notes - L10
11	23.03.	Infinite products. The box topology and its defects. The product topology and its virtues. Proof of the Tychonoff compactness theorem in the case of infinite products.	[GT] pp. 39-41	Notes - L11
12	25.03.	Quotient maps and quotient spaces. Examples. A continuity criterion for maps whose domain is a quotient space. The Klein bottle. The real projective plane.	[GT] pp. 44-52	Notes - L12
13	30.03.	The problem of openness for quotient maps. An example. The saturation of a set and an openness criterion.	-	Notes - L13
14	01.04.	Topological manifolds: definition, examples and basic facts. The problem of classifying n-dimensional compact topological manifolds: resolution of the case n=1, and presentation of the case n=2.	[GT] pp. 53-55	Notes - L14
15	06.04.	Some heuristics about categories and functors. Homotopy of maps (possibly relative to a set); the special case of paths with fixed endopoints. Homotopic equivalence of topological spaces. Contractible spaces.	[AT] pp. 1-4 and pp. 21-26	Notes - L15
16	08.04.	Paths and concatenation. Based loops. Construction of the fundamental group of a topological space. Some examples. Role of the basepoint. Functorial properties. Homeomorphic spaces have isomorphic fundamental groups.	[AT] pp. 26-28 and pp. 34-35	Notes - L16
17	20.04.	Homotopic invariance of the fundamental group. Contractible spaces are simply connected; some examples.	[AT] pp. 36-38	Notes - L17
18	22.04.	Covering maps, basic definitions and examples. Constacy of the cardinality of the fibers, degree of a covering. Lifts, uniqueness and existence. The fundamental group of the circle. The fundamental group of product spaces, and application to the case of tori.	[AT] pp. 29-30 and pp. 34	Notes - L18
19	27.04.	Proof of the existence of lifts of paths and homotopies. Corollary: the monodromy theorem.	[AT] pp. 30 and pp. 60	Notes - L19
20	29.04.	The free product of groups, basic facts. Free group over n generators. The universal property of free products. Statement of Van Kampen's Theorem, special cases. The fundamental groups of spheres in any dimension. The fundamental group of the wedge of circles.	[AT] pp. 40-43	Notes - L20
21	04.05.	Retracts and deformation retracts. The disc does not retract onto its boundary circle, and the Brouwer fixed point theorem. The topological wedge of an arbitrary collection of topological spaces and calculation of its fundamental group.	[AT] pp. 31-32, pp. 36 and pp. 43	Notes - L21
22	06.05.	Factorisation of loops, subordinate to an open cover. Equivalence of factorisations. Proof of Van Kampen's Theorem.	[AT] pp. 44-46	Notes - L22
23	11.05.	A covering map induces an injection of the fundamental groups. The index of the image subgroup equals the degree of the covering. Application: the fundamental group of real projective spaces. Definition of normal coverings.	[AT] pp. 56-61	Notes - L23
24	13.05.	The general criterion for lifting maps between arbitrary topological spaces. A description of the Galois correspondence between covers of a given space and subgroups of its fundamental group. Construction of universal covers of an arbitrary topological space.	[AT] pp. 62-65	Notes - L24
25	18.05.	Existence of covers (of an arbitrary topological space) with assigned fundamental group.	[AT] pp. 66	Notes - L25
26	20.05.	A uniqueness/classification result for covering spaces. Universal cover. Deck transformations, definition and basic examples. Normal covering spaces and their deck transformations. An example of a non-normal covering space.	[AT] pp. 67-68 and pp. 70-71	Notes - L26
27	25.05.	The geometric effects of conjugacy and the fundamental theorem on deck transformations for normal coverings.	[AT] pp. 71	Notes - L27
28	27.05.	An introduction to group actions, some examples.	[AT] pp. 71-74	Notes - L28
		The fundamental group of graphs and an algebraic application.	[AT] pp. 83-85

Final exam

The duration of the written exam (in any session) will always be 120 minutes. During the exam no written aids nor calculators or any other electronic device are allowed in the exam room. Mobile phones must be switched off and stowed away in your bag during the whole duration of the exam.

• Information about the Final Exam
• Advice for the Final Exam
• Probeprüfung: This is a practice exam with the same structure of the actual exam. We strongly advise that you try it only after a careful and complete study of the material covered in class (both in the lectures and in the exercise classes) and after having worked out all homework problems (with the possible exception of those labeled as challenging). Please check the above files for more specific information about how to/not to prepare for this exam.
  Solutions of the Probeprüfung: Again, we strongly suggest to look at the solutions only after having completed the study of the course material and tried the probeprüfung.

We organized three additional office hours sessions during the semester break (end of July / beginning of August, see precise schedule in the section Office hours) as substitution to Ferienpräsenz. Please feel free to exploit this resource (as well as the Forum) to clarify your doubts at due course, possibly well in advance with respect to the exam date.

Forum

In order to easily interact, we set up a forum for our course at the link Topology (Spring 2020) - forum. You have to sign up with your ETH credentials. There you find several topics where you can ask questions and discuss about the lectures, the problem sets, the exam, etc. Use it!

Exercise classes

Exercise classes will be recorded on Mondays 10-12 (only one session per week) on ETH Zoom. You can follow them live with the link and credentials you have received by email. Video recordings and notes of the exercise classes will also be available here on the website. During the exercise classes we will give examples and further explanations about the theory covered during the lectures and we will discuss the questions you asked on the forum; hence it is important that you use this platform to communicate with us.

Please register and enroll for a teaching assistant in myStudies. The enrolment is not needed to attend the exercise class (since there is only one unified exercise class per week), but to hand in your homework.