Algebraic topology from a geometric perspective.

  The materials below are recordings of remote lectures, along with the associated whiteboards and other supporting materials. These lectures started on March 30, 2020. There are also office hours and perhaps other opportunties to learn together. If you are interested in joining send an e-mail to dps **at*** uoregon ++DOT+++ edu.    

• Welcome lecture and corresponding PDF of the whiteboard.  
  • An overview of the mathematics, and a brief personal introduction.   
 
• Catch-up lecture and  clips from the whiteboard.
  • An introduction to Thom cochains, which will play a role throughout, and application to the cohomology of loop spaces and cohomology of configuration spaces.  Friedman, Medina and I plan to post a paper on the more elementary aspects of Thom cochains soon.  The material on homology and cohomology of configuration spaces is treated in this expository paper. A much more extensive introduction to configuration spaces is here.
 
• First ten minutes and   last forty five minutes of the first lecture, along with clips from the whiteboard. 
  • A treatment of Hopf invariants, which are explicit linking invariants preimages of submanifolds under maps from spheres.  Remarkably, the geometry is perfectly governed by a Quillen functor which we'll discuss in the next lecture.  The material in this lecture is the subject of this paper.
 
• Second lecture and clips from the whiteboard. 
  • We continue to develop Hopf invariants, this time giving a full definition as well as defining the Lie coalgebraic bar construction. We state the main theorem, that Hopf invariants are sharp, and share a brief outline of proof. As was the case for the last lecture, the material in this lecture is the subject of this paper.
   
• Third lecture and clips from the whiteboard. 
  • We finish our story of Hopf invariants (except for the fundamental group) by connecting them with previous work, starting with Hopf, Freudenthal, Whitehead, Hilton and Boardman-Steer and then especially the approaches to rational homotopy theory of Chen, Quillen and Sullivan. We end with a (vague) conjecture about rational homotopy classes of maps in general. The above linked paper is still the primary source, but there is a large literature which introduces rational homotopy theory more broadly. One paper that an audience member shared which I hadn't seen but is quite relevant to the perspective from commutative algebra is "Through the looking glass: a dictionary between rational homotopy theory and local algebra." by Avramov and Halperin.
   
• Fourth lecture and whiteboards. 
  • New topic: classifying spaces. These are of central importance in algebraic topology - associating a homotopy type canonically to a group (algebraic topology!). In this introduction we try to bring together key definitions/ perspectives: the simplicial BG, the homotoptical characterization, and natural geometric models. We use the simplicial construction to give concrete models for Eilenberg-MacLane spaces.
   
• Fifth lecture and whiteboards. 
  • We develop characteristic classes through Thom cochains, establishing cases of the Whitney sum formula as well as all other axioms geometrically. We also develop both homology and cohomology of infinite Grassmannians and understand the pairing between them. Along the way we touch on related topics such as the Thom isomorphism theorem, a proof by Bruno Harris of Bott Periodicity, and restriction maps in group cohomology.
   
• Sixth lecture and whiteboards. 
  • After addressing a technical issue about the Thom cochains proposed in lectur five, we discuss the Wu formula, wich relates characteristic classes to Steenrod operations. We then turn to the homology of Eilenberg-MacLane spaces, developing a Hopf ring structure and giving Wilson's concise description.
   
• Additional lecture, on linking of letters and whiteboards. 
  • We develop Hopf invariants for the fundamental group of wedges of circles (that is, free groups), which combinatorially correspond to letter linking. These give a new approach to determining the representative of a word in the lower central series subquotients for free groups (which constitute a free Lie algebra). The results are newly written up in this paper.
   
• Seventh lecture and whiteboards. 
  • We start iterated loop space theory. The central theorems of this theory are
    1. the little disks operad acts on iterated loop spaces
    2. thus the homology of an iterated loop space is a graded Poisson algebra and Dyer-Lashof operations act on their homology
    3. (recognition) an H-space with inverses up to homotopy on which the Fulton MacPherson operad in dimension n acts is an n-fold loop space
    4. (free models) the group completion of the free little n-disks algebra on X provides a model for n-fold loops on the n-fold suspension of X.
    Today we discussed the n=1 case of these theorems, defined the little disks operad and developed the Poisson algebra structure on homology of an n-fold loop space. Next time: the Fulton-MacPherson operad, the recognition theorem, factorization homology and Dyer-Lashof operations.
   
• Eigth lecture and whiteboards. 
  • We develop the Fulton-MacPherson operad and use it to sketch the proof of the recognition theorem, namely that spaces on which it acts whose components form a group are iterated loop spaces. Note that we need to modify what was in the lecture using configurations on the sphere, which when one passes to configurations with labels have points "vanish" at the point at infinity. Extending the construction from configurations in the sphere to those in framed manifolds gives a definition of factorization homology, and we discuss non-abelian Poincare duality. For the material on the Fulton-MacPherson operad, one source is these two papers. This paper uses the FM operad for construcing classifying spaces and in non-abelian Poincare duality.
   
• Ninth lecture and whiteboards. 
  • We finish, for the moment at least, our discussion of iterated loop spaces through development of Kudo-Araki(-Dyer-Lashof) operations on their homology. We then turn to mod-two homology and cohomology of symmetric groups, both of which we take together as a single object. Both end up being free over suitable structure, and we see those structures as overlapping but distinct. A main focus is and will be cup product structure for the cohomology of symmetric groups, which is necessarily complicated (many generators and relations in minimal presentations) for individual symmetric groups but when incorporated into a Hopf ring stucture is determined by little data. If you want to "skip ahead", you can see more about cohomology of symmetric groups in these slides from a talk or this paper.
   
• Tenth lecture and whiteboards. 
  • We give the definitions needed for the description of the cohomology of symmetric groups as a free primitively generated divided powers component Hopf ring. Then we show how these give rise to additive bases and rules for computing cup products. While the object in total is free, we start to see relations arising from Hopf ring distributivity, relations also present in classical rings of symmetric polynomials.
   
• Eleventh lecture and whiteboards. 
  • We elaborate one line of proofs for the calculation of homology and cohomology of symmetric groups, through subgroups. We focus on the example of S_4. We start with the elementary fact that the mod-p cohomology of a group injects into the cohomology of its p-Sylow subgroup. For symmetric groups, the p-Sylow subgroups are given by wreath products, whose homology is given by "formal Kudo-Araki operations." Switching to cohomology, we show further that the cohomology of the wreath product (and thus S_4) is itself detected on the cohomology of two subgroups, both isomorphic to S_2 x S_2 (but not conjugate to each other). These are the maximal elementary abelian subgroups, and detection on such was proven by Madsen-Milgram for symmetric groups. More generally Quillen proved for general groups that these detect modulo nilpotents. Our proof is more elementary and geometric. We finish by discussing some open questions in this area.
   
• Twelfth lecture and whiteboards. 
  • We develop Fox-Neuwirth cell complexes, which though standardly defined in terms of configuration spaces may be viewed as the cell structure on subquotients of the iterated bar complex for the group of order two. They have been studied in many guises. We show how they illuminate a surprising connection between homology of Eilenberg-MacLane spaces and cohomology of symmetric groups. And we also share some ideas (some established, some speculative) as to how to use these ideas to model mapping spaces. Some of these ideas are developed in this paper, though note that unlike what is claimed without proof there, the cup product structure is not modeled at the cochain level. So, in a sense there are two skyline bases for the cohomology of symmetric groups -- one which is the Hopf monomial basis and one formed by associating representative Fox-Neuwirth cycles to a skyline diagram.
   
• Thirteenth lecture and whiteboards. 
  • We start with the Blakers-Massey theorem, a fundamental theorem about the extent to which homotopy groups have a Mayer-Vietoris sequence (or spectral sequence) in settings where there is such in homology. We apply this to spaces of embeddings, showing how Goodwillie's cutting method allows for decomposition of the homotopy types of spaces of embeddings in terms of homotopy types of simpler such spaces which are in fact configuration spaces (embeddings of zero-manifolds). These ideas are explained in much more detail (five hundred pages vs. just an hour lecture) in this book.
   
• Fourteenth lecture and whiteboards. 
  • In the previous lecture we related embedding spaces to simpler embedding spaces, which are homotopy equivalent to configuration spaces. Here we explain a more direct relationship through induced maps on configuration spaces (also known as evaluation maps or Gauss maps). We discuss the structure of spectral sequences for homotopy and homology of spaces of knots. This material is the focus of this paper, and the collapse results mentioned are in this paper.
   
• Fifteenth lecture and whiteboards. 
  • We discuss the Goodwillie-Weiss tower of approximations to the space of classical knots, and the theory of finite-type knot invariants. We present cases of and progress on the conjecture that the tower serves as a universal such invariant. The "quadrisecant result" is in this paper.
    Universality over the real numbers is established in this paper.
    Recent collapse results at any prime are in this paper.
    Along with results announced here, this establishes the universality conjecture at the prime p in degrees less than p (but does not - yet - give new knot invariants).