Algebraic Topology

Lecture summaries

Lecture 1: Introduction (January 6)
I like this quote from Tom Leinster: "A category is a system of related objects. The objects do not live in isolation: there is some notion of map between objects, binding them together." Today we reviewed some basics of category theory, particularly functors between categories, and used this to formailze a proof of the Brower fixed point theorem.
Lecture 2: Euler characteristic (January 8)
Our aim is to compute a suite of functors called homology groups. These will be relatively easy to work with, but the cost is a somewhat lengthy definition. To motivate the objects we'll require, we looked at the Euler characteristic of a surface, which suggests a notion of cellular decomposition (to be formalized). Our goal, ultimately, is to find a categorical lift of the Euler characteristic.
Lecture 3: CW complexes (January 10)
In order to define homology groups for X, we need some sort of decomposition of the space X. There are various ways to do this. To start with, we will use CW complexes, which were introduced today.
Lecture 4: Some intuition for homology (January 17)
By considering low-dimensional CW complexes, namely graphs, we attempted to generate some intuition for what homology might count. Ultimately, we concluded that the one dimensional homology of a space should count cycles (essentially un-based loops) modulo boundaries. That is, while the 1-dimensional chain group is generated by the 1-cells, the 1-dimensional homology classes were equivalence classes of those cycles differing by some boundary.
Lecture 5: 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 define the functors that we wish to construct (and single out their essential properties), taking the based homotopy category to the category of modules. The image of these functors will ultimately be the homology groups associated with a CW complex.
Lecture 6: Homology is stable (January 24)
Continuing our discusion from last time, we showend that homology is stable with respect to suspension. From this fact it follows that a homotopy equivalence of spheres S^m and Sⁿ is equivalent to m=n. While this still depends on a class of functors that we have yet to fully construct, it does indicate that the attaching maps we are interested in leveraging into chain maps should be completely determined by their degree, that is, the value of the identity in the ring R in the image of a morphism induced from a map between spheres.
Lecture 7: The degree of a map between spheres (January 31)
We formalized the notion of degree, and checked the basic properties forced on us by functoriality. We saw what the degree needs to boil down to in the case of S^- (it has to be the degree of the associated cover!), which points to the right way to calculate in general: local degree.
Lecture 8: The local degree (February 5)
We'd like to be able to calculate the degree by hand, in fact, what we would really like is for all of this to line up with the attaching maps coming from our cell complexes. To this end, the right definition of local degree follows from removing a point y of Sⁿ in the image of f and removing the preimage of this point x₁,...,x_m in Sⁿ. The fact that this was the right thing to do followed from a discussion about the Eilenberg-Steenrod axioms (in relation to our definition of reduced homology groups), and included a detour into the proof of the five-lemma.
Lecture 9: Degree as a sum of local degrees (February 7)
We proved that the degree of a map can be computed in terms of the local degree. Then, at long last, we assembled all of the pieces together to build the cellular homology chain complex.
Lecture 10: Calculation (February 12)
We checked our work from last time on closed, orientable surfaces. These came with very special CW structures, from which we were able to determine that all differentials in the complex vanished. This seems too good to be true, in general, and begs what I think is a really important question: where to CW structures really come from. By way of a lengthy aside, this led us to consider the gradient flow of a real valued function on a smooth manifold.
Lecture 11: Cell complexes via Morse theory (February 14)
Expanding on the discussion started last time, we saw in detail 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.

Reading week
Please read section 2.1 of Hatcher on Simplicial and singular homology (pages 102 through 131). Important points are homotopy invariance (which you should understand the details of) and excision (which you should at least be aware of).

Lecture 12: Infinite cyclic covers (February 24)
Today focused on a concrete interplay between algebra and topology: we considered CW complexes equipped with a cover having infinite cyclic deck group and concluded that the homology (with field coefficients) of the cover carried a natural and explicit module structure over a ring of Laurent polynomials. This latter being a PID, a natural invariant of the cover is the order of the module in each homological degree, which in this case is a product of laurent polynomials. Along the way, we saw a short exact sequence inducing a long exact sequence: This clip from the film It's my turn gives a really quick proof of the Snake Lemma.
Lecture 13: Abelianization and knot complements (February 26)
After sketching the proof the that the Hurewicz map abelianizes the fundamental group of a path connected space and recovers the first homology group, we set about considering a lengthy example by considering the complement of an explicit knot. Our aim, ultimately, is to show that the module structure on the infinite cyclic cover of the knot complement records structure despite the fact that the homology of the complement itself is rather boring. Here, we're really motivating a deeper dive that will lead to Alexander duality, but in the meantime we opted for an ad hoc approach in order to calculate by hand. To find a homology generator for our knot, we picked a nice presentation for the fundamental group and argued as in George K. Francis' Topological picturebook; a recreational trip to the Barber stacks is recommended.
Lecture 14: The Alexander invariant (March 4)
Today we gave a complete calculation of the Alexander invariant of the trefoil knot (that is, the module structure on the infinite cyclic cover of the knot complement), building on the material developed last time. An important geometric step led us to consider the analog of Seifert-van Kampen for homology: The Mayer-Vietoris exact sequence. Many of the important points are revisited in the most recent homework assignment.
Lecture 15: Fox free calculus (March 6)
To round out our discussion on the homology of an infinite cyclic covers, we calculated the differential on the 2-cells explicitly as a map of Z[t,t^-1]-modules. This can be given as a matrix with entries determined by the Fox free derivative applied to the relators in a group presentation for the fundamental group of the base.
Lecture 16: Introduction to cohomology, by example (March 11)
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 trough 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 17: The Ext functor (March 13)
Given and chain complex of abelian groups, we define a cochain complex by dualizing with Hom(-,G). We looked in detail at the way in wich this functor determines the cohomology from the homology: it does, up to a functor Ext(-,G).

COVID-19 update
As of Monday March 16, we will be using video conferencing tools to complete the course material. We will remove the final homework, add more reading to pair with the lectures, and work towards finding a workable solution for the final exam. Please read Chapter 3 of Hatcher on Cohomology. Our lectures, which I'll continue to summarize as we go, will complement this material. The main topics we will focus on in rounding out the course are: cup products and duality.

Lecture 18: Graded abelian groups and tensor products (March 18)
Today we saw the first glimpse of a potential product structure on cohomology, by considering the isomorphism between H^*(S²) and Z[x]/(x²) as graded abelian groups, if x is in degree 2. In fact, we'd like to capture the entire ring structure, and towards this end, we started with tensor products of chain complexes.
Lecture 19: The cup product (March 20)
Continuing with the material from last time, we built a product in cohomology using the dualized diagonal map. As a result, the cohomology groups H*(X) may be regarded as a graded ring.

Final topic and further adjustments
There will be three more lectures in for the course, focusing on Poincaré duality, which will follow Hatcher's Section 3.3 closely. We will meet on Wednesdays using Zoom, since campus is now closed. I will leave reading for you to do on your own (using our usual Friday time, say) that complements/fills out this material. If you find them useful, here are the notes, mildly annotated post lecture.

Lecture 20: Orientations on manifolds (March 25)
We used the infinite cyclic local homology H_n(M|x) of an n dimensional manifold M to give a notion of orientability inheriting familiar properties that we are used to from Rⁿ. Of particular use going forward will be the fundamental class of an orientable manifold, which is an element of H_n(M) whose image in H_n(M|x) is a generator, for all points x in M.

Assigned reading: Hatcher's subsection Orientations and Homology, pages 233 through 239.
Lecture 21: The cap product (April 1)
The cap product uses the evaluation map together with the diagonal map in order to pair a cohomology class with a homology class. In particular, when M is an orientable manifold and [M] is a fundamental class, then capping with [M] gives rise to the isomorphism in Poincaré duality.

Assigned reading: first part of Hatcher's subsection The Duality Theorem, pages 239 through 242, together with Hatcher's subsection Cohomology of Space, pages 197 through 204.
Lecture 22: Cohomology with compact support and duality (April 8)
Using directed systems of cohomology groups associated with compact subspaces we defined homology with compact support via limits of abelian groups. This is a key to the proof of Poincaré duality, but also gives rise to other duality theorems such as Alexander duality. This latter highlights some properties that have come up earlier in the course when we were studying infinite cyclic covers and computing the modules associated with them.

Assigned reading: Hatcher's subsection The Duality Theorem, pages 239 through 249, together with the subsections that follow: Connections with Cup Product and Other forms of Duality.

Algebraic Topology

Math 427/527

Winter 2020

Lecture summaries