Lectures on Algebraic Topology II

These notes represent my lectures in 18.906, the second half of a two-semester course on Algebraic Topology offered at MIT in spring 2020. I led both 18.905 and 18.906 in 2016–2017, and the record of lectures in the first course is available on MIT OpenCourseWare. The lectures of both the earlier courses were live TEXed by Sanath Devalapurkar, and his notes formed the invaluable starting point, especially for 18.905. I made many changes in the 2020 version of 18.906 and less remains of Sanath’s notes in the present text. This second course was about homotopy theory and its applications. The course divides into five units. As in the first semester, I emphasize basic categorical structures – here, limits, colimits, and adjunctions. These suggest the “convenient” refinement of the category of topological spaces provided by the cartesian closed category of compactly generated spaces. I discuss fibrations, cofibrations, and homotopy groups. Much of the second unit deals with the conveniences afforded by working with CW complexes. I state the Hurewicz and Whitehead theorems, but delay proofs till we have Serre’s machinery set up, at which point their generalization modulo a Serre class can be proven simultaneously. I gave some hints about obstruction theory, but not complete proofs. In the third unit I discuss vector bundles and principal bundles. I briefly develop the theory of G-CW-complexes to give a proof that the BunG is a homotopy functor. In these notes I develop the theory of classifying spaces from Graeme Segal’s simplicial perspective, since these method have proven so valuable and provide a very concrete construction of these spaces. Unfortunately, this was the spring of COVID-19, and this topic fell prey to four cancelled classes. The remaining two units were conducted online. Next is the theory of spectral sequences, with special reference to the Serre spectral sequence. I set up the theory of Serre classes, and carry out Serre’s proof of the Hurewicz theorem. In lieu of a detailed treatment of multiplicative structure, I describe Dress’s bisimplicial construction of the Serre (or more generally Leray) spectral sequence. This section ends with applications to the Gysin sequence and Thom isomorphism. The final unit applies this machinery to the study of characteristic classes, and sketches some of their applications. I start with Grothendieck’s construction of Chern (and Stiefel-Whitney) classes, verify the splitting principle, and discuss the Thom isomorphism and the Thom class. I introduce the Pontryagin classes and sketch a computation of the cohomology of BSO(n) and BO(n) away from 2. Then I construct Steenrod operations, talk about cobordism, the Pontryagin-Thom collapse, and transversality; then introduce Hopf algebras as a useful tool and sketch Thom’s work, including a brief account of his counterexample to Steenrod’s question; and end with a discussion of the rational oriented bordism ring and the proof of the Hirzebruch signature theorem. These notes reflect approximately what actually happened in the classroom. Many details are consequently omitted, in favor of more discussion of examples and how the theory fits together.