Topics in Topology ("Topologie III"), Sommersemester 2024

Lecture summaries / reading suggestions / exercises (updated every week; current version now up to date for weeks 1-2) Announcements 19.04.2024: Starting in the second week, all Monday lectures will start at 11:00 instead of 11:15, and end at 12:30. General information Instructor: Prof. Chris Wendl (for contact information and office hours see my homepage) Moodle: https://moodle.hu-berlin.de/course/view.php?id=126989 The enrollment key is: obstruction Important: You must join the moodle for the course in order to receive occasional time-sensitive announcements, e.g. if a lecture has been cancelled or rescheduled. HU students can access moodle using their HU username and password. Non-HU users can access it by following the above link and then clicking on "Login" followed by "Externen Zugang anlegen" to set up a HU Moodle Account with an external e-mail address as a username. You will need to enter the enrollment key printed above. Time and place: Lectures on Mondays 11:00-12:30 in room 1.013 and Thursdays 11:15-12:45 in room 3.008 (RUD25) Problem Class (Übung) Thursdays 15:15-16:45 in room 3.007 (RUD25) Language: The course will be taught in English.

image by Christian Lawson-Perfect from cp's mathem-o-blog

Prerequisites:
The main prerequisites are a solid foundation in point-set topology, the fundamental group and covering spaces, singular and cellular (co-)homology (including computations based on the axioms and some homological algebra, e.g. the universal coefficient theorems), and some willingness to put up with the language of categories, functors and universal properties. If you took last semester's Topologie II course, then you definitely have the essential prerequisites. Some knowledge of smooth manifolds will occasionally be useful, but if you do not have this, you will just need to be willing to accept a small set of facts about tangent spaces, tubular neighborhoods and transversality as black boxes.

Contents:
This is a course on intermediate-level algebraic topology, and is conceived in part as a sequel to last semester's Topology 2 course, though it will also repeat some material on homology and cohomology that was covered in that course, sometimes from a wider perspective. We also aim to prove some useful results from elementary homotopy theory (some of which were mentioned briefly last semester but were not proved), and introduce the essentials of obstruction theory, classifying spaces, characteristic classes, and bordism theory. These are all topics that have wide-ranging applications to other areas of mathematics, especially to problems in differential geometry and topology, such as the existence and classification of exotic smooth structures on manifolds. We will not attempt any deep exploration of modern homotopy theory, as that would far exceed my expertise.

Here is a more detailed plan of topics, provided with the caveat that it is preliminary and subject to change. I cannot yet say more precisely when each topic will be covered, nor can I promise that all of it will be covered, or that the order indicated below will remain unchanged. This is my first time teaching this course, so it will be an adventure.

• Notions from category theory: limits and colimits, (co-)products, fiber products / pullbacks and pushouts, (co-)equalizers
• Elementary homotopy theory: mapping cylinders, reduced suspensions, loop spaces and adjunction, mapping cones, fibrations and cofibrations
• Homotopy groups: definitions and group structure, homotopy exact sequence, Serre fibrations, Freudenthal suspension theorem
• Homotopy theory of CW-complexes: weak homotopy equivalences and Whitehead's theorem, n-connectedness, cellular and CW-approximation, Eilenberg-MacLane spaces, homotopical characterization of cellular cohomology, Hurewicz theorem
• Topology of fiber bundles: vector/principal/fiber bundles and structure groups, induced bundles and homotopy invariance, universal bundles and classifying spaces, reduction of structure groups, stable equivalence and K-groups
• Homological algebra: mapping cones, chain contractions and chain homotopy equivalences
• Singular (co-)homology: excisive couples and Mayer-Vietoris sequences, acyclic models, cross/cup/cap products and their relations to each other
• Axiomatic (co-)homology: generalized (co-)homology theories, compact support axiom, mapping telescope and homotopy colimit, the lim¹ functor, multiplicative cohomology theories, Leray-Hirsch theorem, Thom isomorphism
• Axiomatic (co-)homology of manifolds: orientation bundle, fundamental class, Poincaré and Alexander duality, transversality and intersection product, Euler number
• Obstruction theory: singular and cellular (co-)homology with local coefficients, obstruction cocycles for lifting/extension problems
• Characteristic classes: Chern, Stiefel-Whitney, Pontryagin, Euler, application to exotic smooth structures (only a sketch)
• Bordism theory: definitions as generalized homology, stable homotopy groups of spheres, stable Pontryagin-Thom theorem, tangential structures, computation of the rational oriented bordism ring, Hirzebruch signature theorem

Literature:
I will not be writing detailed lecture notes for this course, but the link at the top of this page marked "Lecture summaries / reading suggestions / exercises" will be updated every week to contain a brief summary of what was covered, plus (as you might guess) reading suggestions and exercises. The bulk of what we plan to cover is contained in the union of the following two books, both of which are available electronically through the HU library:

• Tammo tom Dieck: Algebraic Topology (EMS, 2010)
• James F. Davis and Paul Kirk: Lecture Notes in Algebraic Topology (AMS, 2001)

• George W. Whitehead: Elements of Homotopy Theory (Springer GTM, 1978)
• Norman Steenrod: The Topology of Fibre Bundles (Princeton, 1951). A bit antiquated, but a timeless classic... this is the original source for obstruction theory.
• Daniel S. Freed: Bordism, Old and New (lecture notes, 2012). An excellent source both for bordism theory and several other topics, making somewhat more liberal use of smoothness (i.e. methods from differential topology) than the typical book on "algebraic" topology does.

• Allen Hatcher: Algebraic Topology (Cambridge, 2002); you'll never be free of it. This recommendation comes with the caveat that if you want to learn about products in singular (co-)homology but get all the signs right, then you should look at a different book.
• Raoul Bott and Loring W. Tu: Differential Forms in Algebraic Topology (Springer GTM, 1982). If days were 36 hours long instead of 24, I would teach a whole parallel course following this book.
• Edwin H. Spanier: Algebraic Topology (Springer, 1966). Every time I think I am finished learning new things from this book, I turn out to be wrong.
• J. P. May: A Concise Course in Algebraic Topology (U. Chicago Press, 1999); also downloadable from Peter May's homepage

Homework:
The link near the top of this page will be updated each week to include exercises on current material. Sometimes I will discuss them in the problem class. They will not be graded. I may sometimes also use the problem class to fill in gaps on details that did not fit into the regular lectures.

Grades:
Since this is an advanced course, I have a fairly relaxed attitude about grades. If you come to the course with adequate prerequisites and stay with it for the whole semester, you can come to my office at the end for a conversation (let's pretend that's the English translation of “mündliche Prüfung”). The format is as follows: you pick one particular coherent topic from the course to focus on, typically the contents of four to six lectures, and we will talk about that. If you demonstrate that you learned something interesting from the course, you'll get a good grade.