# Talbot 2015: Little Disks Operads

## Mentored by Kathryn Hess and Dev Sinha

###
April 5-11, 2015

Mt. Hood, Oregon

**Pre-Talbot seminars**

This page will list "pre-Talbot seminars" organized by departments in the run-up to Talbot - these seminars will discuss prerequistes and other material related to the workshop. If you're organizing a seminar and you'd like it listed here, please email talbotworkshop@gmail.com. Below you'll find some suggestions for pre-Talbot topics from the mentors.

The Northwestern pre-Talbot seminar, organized by Dylan Wilson, with notes.

The Boston-area pre-Talbot seminar, organized by Lukas Brantner.

**From Kathryn Hess:**

Related to the **homotopy-theoretic** side of the workshop, I'd like to suggest the following.

Introduction to (non-algebraic) operads and their homotopy theory: There are a number of distinct approaches to the homotopy theory of operads. The one we will prefer most of the time in this workshop is that in the Berger-Moerdijk paper below (see also the book by Fresse and the articles by Muro). Rezk developed an alternate, interesting approach in his thesis to simplicial operads, which is worth comparing to the Berger-Moerdijk approach. To get a flavor of the dg context (which we will not cover in the workshop), the book of Loday and Vallette is highly recommended.

Possible references:

-- Berger and Moerdijk's paper on axiomatic homotopy theory of operads.

--Muro's papers on model categories of operads: 1, 2.

--Benoît Fresse's book "Modules over operads and functors"

--Loday and Vallette's book on algebraic operads.

Also, there are videos of many lectures from the 2013 Newton Institute emphasis semester Grothendieck-Teichmüller groups, operads and deformation" or from MSRI that could be useful for preparing lectures in this series, as well as for the workshop.

--Intro to operads by Vallette (1, 2), by Dwyer and by me.

--If you want to think about (bi)modules over operads (which we will use extensively during the workshop) as well, here's a lecture I gave on the subject.

--Rational homotopy theory: an introduction to the elements of rational homotopy theory, in particular the notion of formality and examples thereof. Possible references:

--Notes from my mini-course at a 2004 summer school in Chicago.

--Course notes from Alexander Berglund.

--The big book by Félix, Halperin, and Thomas (lots of examples and geometric motivation).

--The older book by Griffiths and Morgan, which is less complete, but easier to read.

--A recent book by Félix, Oprea and Tanré, "Algebraic models in geometry", to get glimpse of the power of rational homotopy theory for capturing geometric structure.

**From Dev Sinha:**

Some possible series of talks related to Talbot seminar on Little Disks as a bridge between Homotopy Theory and Geometry, and in particular the knot theory thread within the seminar. Note: these could use elaboration at points, especially with respect to references. Contact Dev Sinha at dps@uoregon.edu and I can directly help a seminar group which is interested.

Series I - Finite type knot theory with an eye towards the connection with Goodwillie-Weiss calculus. The study of spaces of knots and of finite-type knot theory is one of the main areas the in which homotopy theory and geometry interface, with configuration spaces playing a significant role throughout. The Talbot seminar will not have enough time to thoroughly develop the knot theoretic side, so one possible lead-in would be to do so in a seminar series.

Primary resource: “On the self linking of knots” by Bott and Taubes. Primary suggestion is to carefully go through the paper in multiple talks. Pay particular attention to the “leaking cosimplicial space” which inspired later work.

Possible supplementary background material needed for primary resource:

- a more thorough treatment of the compactifications used is in “Manifold theoretic compactifications of configuration spaces” by Sinha (this will be used in the Talbot seminar itself).

- the relationship between integration and push-forward in de Rham theory. It would be good to see for example that in the case of a fiber bundle where the base and fiber and thus total space are all compact manifolds without boundary then integration along the fiber corresponds to “apply Poincare duality to the total space; map from total space to base in homology; apply Poincare duality in the base.” It should also be noted that for Thom forms concentrated in a tubular neighborhood of a submanifold, integration along the fiber produces a Thom form concentrated around a tubular neighborhood of the image.

Secondary resources:

- the canonical reference for finite-type knot theory is Bar Natan’s seminal paper. Habiro’s clasper surgery approach is also worth being aware of, at least at the level of the statement of the main result that finite type invariants are exactly those invariant under clasper surgery.

- the best construction of finite-type invariants is the Kontsevich integral, which uses complex numbers in an essential way. A good resource for that is Chapter 8 of the openly available book of Chmutov-Duzhin-Mostovoy.

- an intriguing follow up for people who want to see the connections with topological field theories is: http://arxiv.org/pdf/hep-th/9905057v1.pdf

as well as the monograph of Sawon. With TFT’s being a major research thread now (cobordism hypothesis, factorization homology and their applications), this “classical” material might be of particular interest. But this is perhaps better as a post-Talbot topic.

Series II - Development of the Goodwillie-Weiss tower for knots. There are very elementary dimension-counting arguments which show that Goodwillie’s “cutting" approximations converge to the homotopy type of the space of knots. The Blakers-Massey theorem plays a central role (just as it does in the homotopy calculus), and one can actually prove that theorem using dimension counting as well. The Talbot seminar will not have time to develop these ideas, so they make for a great preparatory series.

Primary resource: Munson-Volic’s book on cubical homotopy theory. One also needs a key dimension-counting argument which I don’t know is in the literature. It should be, but doesn’t seem to be, a culminating application in the Munson-Volic book. It might also be in notes by Pascal Lambrechts. Contact Dev if you want to track it down (or learn it from him say over Skype to pass on, once the table has been set).

Outline: Start with the Blakers-Massey theorem, as treated in Chapters 4-6 of the text. The proof in the two-cube case, in section 4.4 (especially the dimension counting of 4.4.1 and if people want the more homotopy-theoretic “coordinate counting” of 4.4.2), and the applications of section 4.3 are the highlights. Outside of those, knowing statements in the general case is the main need.

Then, contact Dev about the dimension-counting argument that the Goodwillie-Weiss tower approximates knot spaces in dimensions five and greater.

And finally, relate cosimplicial and cubical diagrams through Section 9.4 of the Munson-Volic text, and then go through Section 6 of Dev's “The topology of spaces of knots: cosimplicial models” to apply this in the case of the Goodwillie-Weiss tower. See also a recent paper by Budney-Conant-Koytcheff-Sinha for more analysis of the interrelationship between cosimplicial and cubical approaches.

While this lecture series may sound narrow in focus, in fact the material on Blakers-Massey as well as that on cosimplicial and cubical diagrams are fairly basic tools which the broader algebraic topology community ought to be aware of. The former is at the heart of the interrelationship between stable and unstable homotopy theory (and as such is the primary idea animating Goodwillie’s homotopy calculus). The latter shows that while simplicial techniques are ubiquitous and cubical less so, cubical diagrams provide a method to fruitfully analyze (co)simplicial diagrams and their realizations/ totalizations. The only material in the lecture series on top of these topics which “all homotopy theorists should know”, is just one or maybe two lectures on the application to the Goodwillie-Weiss tower for knots.