BBS webpage

### Building Bridges Seminar: Invariants of embedding spaces

(Winter 2020)

This is a kind of a learning seminar, whose goal is to bridge geometric and algebraic approaches to spaces of embeddings. This includes invariants of classical knots, 2-knots in 4-manifolds, homotopy groups of diffeomorphism groups. One of the goals is to learn embedding calculus through its applications.

The seminar is held online on Wednesdays at 15:00 CET. Contact me if you wish to be added to the mailing list and obtain the meeting details.

(see abstracts of talks and slides for more links)

[Weiss] Immersion theory for homotopy theorists (fancy .pdf)

The first block:
embedding calculus

Nov 11

Introduction and motivation

I will talk about why a topologist might care about spaces of embeddings, and sketch why embedding calculus might help understand them.

handwritten slides
Nov 18 Pedro Boavida de Brito

Configuration categories and embedding calculus

An introduction to embedding calculus, and its relation to the little disks operad $$\mathrm{E}_n$$. Partly based on doi/abs/10.1112/topo.12048.

tex notes
Nov 25 Geoffroy Horel

Action of GT on the tower for knots

I will explain how the action of the Grothendieck-Teichmüller group $$GT$$ on braid groups, originally studied by arithmetic geometers, induces an action on the embedding calculus tower for long knots. This can be used to deduce some integral results about the universal finite type invariant for knots. This is joint work with Pedro Boavida de Brito. Based on arxiv.org/abs/2002.01470.

beamer slides
Dec 02 Gregory Arone

Operad formality and rational homology of embedding spaces

The Goodwilile-Weiss tower can be described as a space of maps between modules of $$\mathrm{E}_n$$. The formality of $$\mathrm{E}_n$$ has rather far reaching consequences for the rational homotopy type of embedding spaces. I will focus on rational homology. Based on projecteuclid.org/euclid.gt/1513732795.

beamer slides
Dec 09 Victor Turchin

Rational homotopy type of embedding spaces

I will talk about my joint work with Benoit Fresse and Thomas Willwacher. Using embedding calculus and methods of the rational homotopy theory we construct $$L_\infty$$-algebras of diagrams that encode the rational type of connected components of embedding spaces in $$\mathbb{R}^n$$. This type depends on the component. Different known invariants of embeddings seem to be responsible for the rational homotopy type. Some examples will be discussed. Based on arxiv.org/abs/2008.08146.

handwritten slides (.svg)
Dec 16 Pascal Lambrechts

Bonus Talk: Back to basics

Homotopy limits for a working low-dimensional/differential topologist

beamer slides

~ Winter Break ~

Jan 6 Dev Sinha

Algebraic topology of embedding spaces and its application to knot theory, from a geometric perspective

Geometric algebraic topology makes use of representations of homology and cohomology by manifolds along with explicit maps from spheres for homotopy. My interest in this aspect of algebraic topology has grown in part from my study of spaces of embeddings.
In the first third of the talk I will state and give evidence for conjectures (some precise, some not) about the geometric algebraic topology of embedding spaces. At a high level, these conjectures are:
- homology can be represented by families of embeddings defined through resolutions of singularities.
- cohomology can be represented by counting special configurations in families of embeddings or closely related integrals arising from Chern-Simons perturbation theory.
- some key homotopy representatives can be represented by families of clasper surgeries.
In the middle third of the talk I will develop Hopf invariants, which provide geometry for homotopy periods (informally, “rational cohomotopy”).
In the last third of the talk I will discuss the conjecture that the Goodwillie-Weiss tower serves as a universal additive Vassiliev invariant over the integers, with the additional aim to produce new knot invariants in the process of establishing the conjecture.
I will share plenty of open questions which seem approachable (but some have proven to be difficult, at least to me).
Those who want to get a head start can look at my lectures pages.uoregon.edu/dps/GeometricAlgebraicTopology which culminated in the last two lectures with material which overlaps with this talk (as well as with some of the previous presentations in the seminar)

handwritten slides 1
handwritten slides 2

The second block:
embeddings into 4-manifolds

Jan 13 Slava Krushkal

Embedding obstructions in 4-space from the Goodwillie-Weiss calculus and Whitney disks

Given a 2-complex K, I will explain how to use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embeddings into $$\mathbb{R}^4$$. I will also introduce a geometric analogue, based on intersections of Whitney disks.  Focusing on the first obstruction beyond the classical embedding obstruction of van Kampen, I will show that the two a priori very different approaches in fact give the same result, and also relate it to the Arnold class in the cohomology of configuration spaces. The obstructions are realized in a family of examples. Joint work with Greg Arone.

Jan 20 Ben Ruppik

Unknotting 2-knots with Finger- and Whitney moves

This is joint work arxiv.org/abs/2007.13244 with Jason Joseph, Michael Klug, Hannah Schwartz. Any smoothly knotted 2-sphere in the 4-sphere is regularly homotopic to the unknot. This means that every 2-knot K in $$\mathbb{S}^4$$ can be obtained by first performing a number of trivial finger moves on the unknot, and then removing the resulting intersection points in pairs via Whitney moves along possibly complicated Whitney discs. We define the Casson-Whitney unknotting number of the 2-knot K as the minimal number of finger moves needed in such a process to arrive at K.
In this talk, I would like to show examples of families of 2-knots (ribbon 2-knots, twist-spun 2-knots) and tell you why they are interesting. We can study algebraic lower bounds for the Casson-Whitney number coming from the fundamental group of the knot complement. Finally, we compare it with the 1-handle stabilization number, another notion of “unknotting number” that has been in use for 2-knots.

Jan 27 Rob Schneiderman

Whitney towers, capped gropes and the higher-order Arf invariant conjecture

This self-contained talk will introduce a theory of Whitney towers which `measures’ the failure of the Whitney move in dimension four and is closely related to certain 2-complexes called capped gropes which are geometric embodiments of commutators. The main goal of the talk is to describe a naturally arising family of link concordance invariants which are conjectured to be non-trivial finite type invariants generalizing the Arf invariant of a knot. Accompanying material for this talk can be found in the first two sections of the expository paper arxiv.org/abs/2012.01475.

Feb 3 Maggie Miller

TBA (Smooth surfaces in 4-manifolds)

...

Feb 10 Anthony Conway

TBA (On topologically isotopic surfaces in 4-manifolds)

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Feb 17 Aru Ray