This page was the home of the Xtraordinary cohomology theory & K-theory Collective Discussion group* for the fall 2010 semester.fall 2010 archive spring 2011 archive fall 2011 archive spring 2012 archive fall 2012 archive spring 2013 archive fall 2013 archive spring 2014 archive fall 2014 archive spring 2015 archive
This page was the home of the Xtraordinary cohomology theory & K-theory Collective Discussion group* for the fall 2010 semester.
All talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).
11:00-12:30 -- Vector fields on spheres, part 1
The problem of finding the maximal number of linearly independent vector fields on a sphere is fundamental to algebraic topology. Without much background, we'll be able to construct explicit solutions in a pretty neat way, although we won't know that these are really the best possible without using K-theory. A teaser: many of you know and love the hairy ball theorem, which says that for Sn-1 if n is odd then the number is zero; it turns out that if we write n=(2a+1)2b, the number only depends on b!
Aaron Mazel-Gee
references: This talk was developed from the VFOS notes. They are still under construction, but the early parts have been edited.
1:00-3:30 -- K-theory
K-theory is an expansive subject, with lots of subfields; I'll begin by sketching a broad picture, then specialize to the case of topological complex K-theory. We'll construct some relevant classifying spaces and study them using tools from algebraic topology to get a sense of what they look like.
Eric Peterson
references: Lecture notes.
11:00-12:00 -- The Steenrod algebra and its applications, part 1 (and Vector fields on spheres, part 2)
Cohomology is a more powerful invariant than homology. One major reason is that the cohomology groups of a space can be made into a graded ring using the cup product, and induced maps on cohomology must then be ring homomorphisms. However, there is even more structure around. In a series of talks, we'll explore one very important such structure, the action of cohomology operations known as "Steenrod squares". In this first talk, we'll introduce the Steenrod squares, discuss their basic properties, and deduce as a nice little consequence that if the sphere Sn-1 is parallelizable then n=2k. In the longer run, we'll use the Steenrod algebra to compute some homotopy groups of spheres; this is a central and notoriously difficult problem in algebraic topology. The end goal will be to present the Adams spectral sequence, a vital tool which (roughly) computes the homotopy classes of maps from a space X to a space Y.
Aaron Mazel-Gee
references: Mosher & Tangora, Cohomology Operations and Applications to Homotopy Theory. Lectures notes. A short note outlining a geometric interpretation of the squares.
1:00-2:00 -- Cobordism and the Thom-Pontrjagin construction, part 1
How can we prove that every closed 3-manifold is the boundary of a compact 4-manifold? One way is to compute the unoriented cobordism ring, whose elements are equivalence classes of compact manifolds, and two closed manifolds are equivalent if together they form the boundary of a higher-dimensional compact manifold. I'll talk about the Thom-Pontrjagin construction and how it forms an isomorphism between this ring and the stable homotopy ring of a specific spectrum. The actual calculation of the ring is something I'm still working through, but it involves cohomology of BO(r) and the Steenrod algebra.
Cary Malkiewich
references: Lecture notes for the entire series. H.M. Stong, Notes on Cobordism Theory. Rene Thom's thesis, Some properties of differentiable varieties (in French). Expository paper by Tom Weston.
11:00-12:00 -- The Steenrod algebra and its applications, part 2
We'll see a number of important concepts (Bockstein homomorphisms, fibrations, Serre's spectral sequence, the transgression, cohomology of K(π,n)'s) that will be lemmatic in our calculation of the homotopy groups of spheres.
Aaron Mazel-Gee
references: Lecture notes.
1:00-2:00 -- Cobordism and the Thom-Pontrjagin construction, part 2
We'll dive into the Pontrjagin-Thom construction in more detail, showing how a cobordism on one end results in a homotopy on the other end. If we have time, we'll start computing the unoriented cobordism ring.
Cary Malkiewich
references: Jump to the first lecture in this series.
11:00-12:00 -- The Steenrod algebra and its applications, part 3
We'll calculate the stable homotopy groups πn+k(Sn) for 0≤k≤7, n»0, and then we'll calculate either π3+k(S3) or π4+k(S4) for 0≤k≤3 and leave the other as homework.
Aaron Mazel-Gee
references: The previous lecture notes will get you up to speed, but there's no point in me just copying Chapter 12 in Mosher & Tangora. However, to help you avoid the tedious parts of the calculations, here's an Adem relations calculator and here's a table of the first few monomials in the Steenrod algebra organized by degree and excess.
12:30-1:30ish -- The geometry of formal varieties in algebraic topology, part 1
Algebraic topology is full of computations with rings, and where we find rings we should seek geometry through methods of algebraic geometry. The geometry of formal varieties turn out to organize many interesting computations in topology, and certain formal varieties called commutative, one-dimensional formal groups give the best global picture of stable homotopy theory currently available. I will give as friendly an introduction to these ideas as can be managed; in particular, I will not assume the audience knows any algebraic geometry.
Eric Peterson
references: Lecture notes.
11:00-12:00 -- Cobordism and the Thom-Pontrjagin construction, part 3
We'll piggyback on Aaron's results about Steenrod squares to get the cohomology of the Thom spectrum MO.
Cary Malkiewich
references: Jump to the first lecture in this series.
12:30-1:30ish -- The geometry of formal varieties in algebraic topology, part 2
This time, we'll discuss complex oriented cohomology theories in general, including the various pieces of Quillen's theorem describing complex bordism and its relationship to the moduli stack of formal groups. Investigating specific features of this stack will give rise to descriptions of cooperations in familiar (and unfamiliar) homology theories. We'll still try to keep things down at an introductory level for those not well-versed in such scary algebro-geometric objects as stacks.
Eric Peterson
references: Lecture notes.
11:00-12:00 -- Cobordism and the Thom-Pontrjagin construction, part 4
We'll talk a bit about Hopf algebras, and we'll show that the Steenrod algebra acts freely on H*(MO), the stable cohomology co-algebra of the unoriented cobordism spectrum MO.
Cary Malkiewich
references: Jump to the first lecture in this series.
11:00-12:00 -- Cobordism and the Thom-Pontrjagin construction, part 5
We'll calculate π*(MO), the unoriented cobordism ring, as a graded group.
Cary Malkiewich
references: Jump to the first lecture in this series.
12:30-1:30ish -- A spectral sequence for the mapping space [K,X]
The first goal of homotopy theory is to understand the homotopy category of topological spaces. For two topological spaces K and X, we denote by [K,X] the set of morphisms in this category, i.e. the homotopy classes of maps from K to X. The obvious question is: what does this set look like? Our search for answers will lead us inevitably to a spectral sequence that computes [K,X] (when K is a CW-complex). Of course, one of the inputs of this spectral sequence is πn(X) for 0≤n≤dim K, which makes it useful as an actual tool primarily when dim K is not too much larger than the connectivity of X. It is nevertheless still of interest, if for no other reason than that it illustrates the close interplay of the homotopy of X and the action of the Steenrod squares on H*(K). It will also lead us to higher cohomology operations (a subject for another talk).
Aaron Mazel-Gee
11:00-12:00 -- Cobordism and the Thom-Pontrjagin construction, part 6
We'll show (finally!) how Stiefel-Whitney numbers tie into our calculations, deducing that two manifolds are cobordant iff they have the same Stiefel-Whitney numbers. Then, we'll examine the coaction of the dual Steenrod algebra A* on the homology of MO.
Cary Malkiewich
references: Jump to the first lecture in this series.
12:30-1:30 -- Higher cohomology operations
Higher cohomology operations are an important refinement of the usual "primary" cohomology operations. Roughly speaking, they encode relations between primary cohomology operations. One neat example is the Massey triple product, which (via Poincare duality) can detect the Borromean rings: three circles which are all linked even though pairwise they are unlinked. Higher cohomology operations determine the higher differentials in the spectral sequence for [K,X] from last week's talk, they control composition in the stable homotopy groups of spheres, and they play a crucial role in the Adams spectral sequence.
Aaron Mazel-Gee
references: Lecture notes.
12:30-1:30 -- Cobordism and the Thom-Pontrjagin construction, part 7
We'll take a look at the co-action of the Steenrod squares on the homology of MO, and use this to finish the calculation that we have been working towards.
Cary Malkiewich
references: Jump to the first lecture in this series.
2:00-3:00 -- Introduction to computations with Hopf rings
In the late 70s and early 80s, Ravenel and Wilson systematically computed E*Xq for various homology theories E and various families of infinite loopspaces {Xq}. I will introduce all of their methods in the simple case of H*(K(Z/2,q);Z/2), which we will compute using the immense amounts of algebraic structure available with hardly any reference to topological arguments. I will also indicate how the same methods can be modified to substitute other Eilenberg-Mac Lane spaces for X_q and Morava K-theory for E; these are the ingredients used to prove the Conner-Floyd conjecture in bordism theory.
Eric Peterson
* the xkcd group -- making xkcd not stand for nothing since 2010!