Algebraic Topology

Math 427/527

Winter 2025

This course focusses on a fundamental, and rather ubiquitous, object in algebraic topology. As such this course might better be titled Chain complexes and where to find them. It has, very broadly, two parts that are split by the winter break:

Part 1: homology and cohomology of cell complexes

Part 2: chain complexes and categorification

Everything you need for the course is here. To see what this course looked like the last time I taught it, go here; the course is a little different this time around.

Please note that Math 426 (or similar) is an essential prerequisite. In particular, familiarity with the fundamental group will be assumed.

The image on the right was borrowed from the cover of the proceedings Gauge theory and low-dimensional topology: progress and interactions.

• Instructor
  
  Liam Watson
  liam(at)math(dot)ubc(dot)ca
  Office hours: Wednesdays 10:00 to 11:00 or by appointment.

• Time
  
  Class is on Wednesdays and Fridays from 8:30 to 9:45.

• Homework
  
  This course includes independant reading and exercises (from the reading and/or suggested during lectures). Neither will be graded, but both will be essential to your success in the class.

• Evaluation
  This course is evaluated based on (1) a midterm worth 40% of your final grade; and (2) a final exam worth 60% of your final grade.
  
  The midterm will be in class on Friday February 28, after reading week, covering the material on Part 1 covered up to the reading week. The final exam will be during the exam period.

• Supporting material (updated as we go)
  
  Algebraic Topology by Allen Hatcher, available here.
  This is the main reference for the first part of the course.
  
  Basic Category Theory by Tom Leinster, available here.
  We will need some basic category theory. There are many great references, but I particularly like this one.
  
  Abstract Algebra by David Dummit and Richard Foote.
  You can ruse whatever your favourite reference is for the basic algebraic objects that show up in this course. This book is the standard one used in our year-three algebra sequence. It is not bad if a little wordy; there are lotes of examples which is the main plus.
  
  Frobenius Algebras and 2D Topological Quantum Field Theories by Joachim Kock, available online through the UBC library.
  Some neat categories here, and some of this shows up in the second part of the course.
  
  Topology from the Differentiable Viewpoint by John Milnor available at the UBC library.
  Additional reading, for people interested in the index used in the definition of cellular homology.
  
  Morse Theory by John Milnor available at the UBC library.
  Where do cell complexes come from? Here is a different route to finding homology of a space that is smooth enough to do calculus on.
  
  Knots & Links by Dale Rolfsen
  This is a home-grown classic; still the reference for knot theory despite being nearly 50 years old.
  
  On Khovanov's categorification of the Jones polynomial by Dror Bar-Natan
  The paper that taught everyone about Khovanov's incredible (and beautifully simple) construction.

              Lecture summaries

• Lecture 1: Chain complexes (January 8)
  
  This lecture focused on the algebraic object that plays a staring role in the course. No topology today; just algebra.
• Reading
  
  There is no lecture on Friday January 10. Instead I am assigning reading. Pick at least two of the following, making use of the free time block on Friday for one of the two.
  
  Modules: Sections 10.1 and 10.2 of Dummit and Foote.
  If you are taking Math 323 concurrently with this course, it is worth looking at this section in the context of the definitions discussed in Lecture 1.
  
  More modules: Sections 10.3. and 10.2 of Dummit and Foote.
  The discusion in 10.4 about tensor products is really quite good, for context and examples, and certainly useful for the course.
  
  Categories: Introduction and Section 1 of Leinster.
  Worthwhile reading for getting up to speed on the basics of categories essential to this course.
  
  Cell complexes: pages 5 to 10 of Hatcher.
  This is material from Math 426 that We will assume in the upcoming lectures. If this is new to you, definitely choose this as one of your readings! While you're here, I would also suggest a review of the fundamental group.
  
  Homology: pages 97 to 113 of Hatcher.
  For anyone in a rush to see why the algebraic objects from Lecture 1 are the right ones; Hatcher's subsection on The Idea of Homology (starting on Page 98) is part of what we will discuss next week.
• Lecture 2: The homology of a graph (January 15)
  
  By considering graphs we attempted to generate some intuition for what homology might measure. Ultimately, we concluded that the one dimensional homology of a space should count cycles (essentially un-based loops) modulo boundaries. This gives a first example of a chain complex, since it is possible to show that the homology we computed does not change when the choices in the construction are varied.
• Lecture 3: Chain groups from cell complexes (January 17)
  
  The material from Lecture 3 motivates the spaces we will consider: those constructed out of cells, each of which is an i-dimensional disk. The set of cells in each dimension then gives a candidate generating set for a free abelian group C_i.
• Lecture 4: Homology as a functor (January 22)
  
  We collected our intuition from last lecture into a suite of definitions, including chain complexes of modules and the homology of such complexes. This allowed us to axiomatically define the functors that we wish to construct (and single out their essential properties), taking the based homotopy category to a category of modules. The image of these functors, which we still need to construct, will ultimately be the homology modules (groups, when the underlying ring is Z) associated with a CW complex.
• Lecture 5: Homology as a functor (January 24)
  
  Continuing the discussion from last time, we constructed and studied the differential on our complex that hinges on the the degree of a map between spheres, which is determined by the attaching maps for our cell complex.
• Lecture 6: Flavours of homology (January 29)
  
  To compute cellular homology, we introduced a notion of local degree that generalizes intuition gained from the case of circles and winding. At this stage, we know how to compute homology, before knowing if the functors we have exist! This was a good time to introduce simplicial and singular homology: Hatcher proves that these theories are equivalent, and deduces cellular homology from them.
• Reading
  
  Homology: Pages 102-113 from Hatcher (if you have not read this already); and
  
  Excision and equivalence: Pages 113-130 from Hatcher
  
  For info, Hatcher discusses degree and cellular homology on pages 134 to 141.
• Lecture 7: Cell complexes via Morse theory (January 31)
  
  We discussed how a cell structures arrise naturally in the context of a Morse theory, sketching a paralel between critical points associated with a (smooth) height function on a compact manifold M and cells for a CW-structure on M, and we saw how a Morse-Smale function on smooth, compact manifold generates a cell structure. Indeed, these considerations allowed us to define the Morse-Witten chain complex, and even see parts of the proof that this complex computes the cellular homology groups we've been studying.
• Lecture 8: Introduction to cohomology, by example (February 5)
  
  We are now shifting gears from homology to cohomology. In order to motivate why this might be an interesting thing to to, today we thought through an essay by Roger Penrose on the cohomology of impossible figures which describes how to "measure" the impossibility (that is, the fact that as a 2-dimensional drawing it cannot represent a 3-dimensional object) of the tribar as a cohomology class.
• Lecture 9: The relation between homology and cohomology (February 7)
  
  Guest Lecture by Sebastien Gant: Given and chain complex of abelian groups, we define a cochain complex by dualizing with Hom(-,G). We looked in detail at the degree to which this functor determines the cohomology from the homology.
• Lecture 10: The Ext functor (February 12)
  
  Continuing the discussion from last time, we analyzed the kernel of a natural map to the dual of the homology, and this led to a new functor: Ext. Computing Ext boils down to the homology of a new co-chain complex obtained as the dual of a free resolution of a module.
• Lecture 11: Group cohomology (February 14)
  
  Towards studying group extensions (restricting to abelian groups), we constructed a co-homology theory by hand, motivated by the desier to turn a product of sets into a group given a group law on each of the factors. We found that trying to do this generates a natural cocycle, and this fits into a general framework called group cohomology.
• Lecture 12: Some background on knots (February 26)
  
  Kauffman's bracket gives a recursive construction of a Laurent polynomial associated with a choice of knot Diagram. We set this up following some conventions due to Khovanov, as described in Bar-Natan's paper On Khovanov's categorification of the Jones polynomial
• Midterm February 28
  
  Covers all of the course material, focussing on exercises suggested in class, up to and including Lecture 11.
• Lecture 13: Khovanov's homology theory (March 5)
  
  Towards building a cochain complex categorifying the Kauffman bracket, we set up a bi-graded vector space and recovered this polynomial as a grade Euler characteristic.
• Lecture 14: Differentials from a TQFT (March 7)