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.

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.

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.

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.

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.

Bonus Talk: Back to basics

Homotopy limits for a working low-dimensional/differential topologist

~ Winter Break ~

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)

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.

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.

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.

Exotically knotted surfaces in 4-manifolds

I will discuss some open questions (and interesting related constructions) about exotic knotting of surfaces in 4-manifolds. The most general version of this question is, "Given two smooth surfaces in a smooth 4-manifold that are topologically isotopic, when are they also smoothly isotopic?" If such surfaces are not smoothly isotopic, then we call them an exotic pair. If a surface is in an exotic pair with the unknot, then we say it is exotically unknotted.

Whether or not orientable exotic unknots (or even orientable exotic pairs) exist in $\mathbb{S}^4$ is a long-standing open question. (Interestingly, nonorientable examples were constructed by Finashin—Kreck—Viro more than 30 years ago.) However, Sunukjian—Hoffman have constructed exotic unknots in other simply-connected closed 4-manifolds, and Juhasz—M—Zemke and Hayden have constructed exotic pairs in $\mathbb{B^4}$. Relatedly, there are notions of stabilization which can eventually smoothly relate exotic pairs — in particular, Auckly—Kim—Melvin—Ruberman—Schwartz have given conditions that ensure specific exotic surfaces become smoothly isotopic if a single $\mathbb{S^2}\times \mathbb{S^2}$ summand is added to the 4-manifold. In contrast, Baykur—Sunukjian have shown that many examples of exotic pairs become smoothly isotopic after trivially increasing the genus of each surface by one (while leaving the ambient 4-manifold fixed).

In short, I will discuss these various constructions (and maybe other related things but this abstract is already pretty long) and the big ideas behind their proofs (without many details), as a survey(ish) of this specific topic in knotted surface theory.

Knotted surfaces with infinite cyclic knot group

This talk will concern locally flat, embedded surfaces in 4-manifolds. We discuss criteria for two such surfaces to be topologically isotopic, with a focus on the case where the fundamental group of the complement is infinite cyclic. This is based on joint work with Mark Powell.

Filtrations of the knot concordance group

String links and knots, modulo an equivalence relation called concordance, form a group under the connected sum operation. We'll discuss the solvable and bipolar filtrations of these groups, due to Cochran-Orr-Teichner and Cochran-Harvey-Horn respectively. These filtrations provide a systematic framework to study concordance, and the lower order terms encapsulate classical concordance invariants. Moreover, they provide an infinite sequence of obstructions to sliceness and can be rephrased in terms of gropes or Whitney towers. The bipolar filtration can also be used to distinguish between smooth concordance classes of topologically slice knots. The goal of the talk is to motivate the definition of the filtrations, describe the sliceness obstructions and geometric reinterpretations, and discuss some open problems.

A survey of open problems in 4-manifold theory

Embedding calculus and smooth structures

I will explain joint work with Ben Knudsen about the extent to which embedding calculus is sensitive to the smooth structures of the domain and target. In particular, we will prove that in dimension 4 the approximations provided by embedding calculus depends only on the underlying topological manifolds and the vector bundle reductions of their tangent microbundles provided by the smooth structures. If time permits, I will also give some examples of high-dimensional exotic spheres distinguished by embedding calculus and ask some questions.

Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins

I'm interested in the smooth mapping class group of $\mathbb{S}^4$, i.e. $\pi_0(\mathrm{Diff}^+(\mathbb{S}^4))$; we know very little about this group beyond the fact that it is abelian (proving that is a fun warm up exercise). We also know that every orientation preserving diffeomorphism of $\mathbb{S}^4$ is pseudoisotopic to the identity (another fun exercise, starting with the fact that there are no exotic 5-spheres). Cerf theory studies the problem of turning pseudoisotopies into isotopies using parametrized Morse theory. Most of what works in Cerf theory works in dimension 5 and higher, but with a little digging one discovers statements that work in dimension 4 as well.

Putting all this stuff together we can show that there is a surjective homomorphism from (a certain direct limit of) fundamental groups of spaces of embeddings of 2-spheres in connected sums of $\mathbb{S}^2\times\mathbb{S}^2$ onto this smooth mapping class group of $\mathbb{S}^4$. Furthermore, we can identify two natural, and in some sense complementary, subgroups of this fundamental group, one in the kernel of this homomorphism and one whose image we can understand explicitly in terms of Dehn twist-like diffeomorphisms supported near pairs of embedded $\mathbb{S}^2$'s in $\mathbb{S}^4$ (Montesinos twins).

In this talk I'll give an overview of this story and, depending on the audience's interest, give some details here and there.

Introduction: characteristic classes of manifold bundles

In this introductory talk, I will introduce some of the most important objects and objectives of modern algebraic-geometric topology: diffeomorphism groups and their classifying spaces, manifold bundles and their characteristic classes, and moduli spaces of manifolds.

I will first explain how, at the beginning of the 21th century, the Madsen—Weiss theorem opened a new pathway to think about these topics in the case of surface bundles. Perhaps this can be understood more easily through the alternative proof that uses cobordism categories. After dwelling on this, I will assemble some of the more recent developments in the field, including a better understanding of the behavior of generalized Miller—Morita—Mumford classes in higher dimensions. I will end by reviewing some classical and new results on diffeomorphisms of discs.

Self-embedding calculus and tautological classes

The space of diffeomorphisms of a closed manifold coincides with the space of self-embeddings and can thus be studied via the homotopy theoretic approximations from embedding calculus. This perspective has led to much recent progress in understanding the space of diffeomorphisms and it is believed that the approximation is quite close. In this talk, I will discuss how one might detect the difference between the approximation and the space of diffeomorphisms using classical invariants of fibre bundles.

The Weiss fiber sequence and applications

In this talk I will explain how to obtain a fiber sequence due to Weiss and Kupers which compares diffeomorphisms of a manifold-with-boundary $M$ which fix the boundary pointwise, with self-embeddings of $M$ which are allowed to "move" a codimension $0$ submanifold $N$ of the boundary. The fiber of this fibration can be identified with the space of diffeomorphisms of the "movable part" $Nx[0,1]$.

I will indicate how, by choosing $M$ and $N$ wisely, one can derive qualitative information about (classifying spaces of) diffeomorphism groups like finiteness and infiniteness results for their homotopy groups.

Diffeomorphisms of even-dimensional discs

I will explain joint work with Oscar Randal-Williams on the rational homotopy type of the classifying space of the topological group of diffeomorphisms of even-dimensional discs of dimension at least 6. This is done by combining the work of Galatius and Randal-Williams on homological stability for moduli spaces of high-dimensional manifolds with embedding calculus through the Weiss fiber sequence.

Diffeomorphisms of odd-dimensional discs

I will explain ongoing work with M. Krannich in which we describe the rational homotopy groups of $BDiff_\partial(D^{2n+1})$ in degrees $* < 3n - const$, for $2n+1 \geq 7$. In this range of degrees there are three kinds of classes. Firstly, classes in degrees $4,8,12,...$ coming from algebraic $K$-theory of the integers; these were discovered by Farrell$-$Hsiang in the pseudoisotopy stable range. Secondly, classes in degrees $2n-2, 2n+2, 2n+6, ...$ coming from the nontriviality of higher Pontrjagin classes in the cohomology of $BTop(2n+1)$; these were originally discovered by Weiss. Thirdly, there are two sporadic classes in degrees $2n-2$ and $2n-1$; these both seem to be new.

In his earlier work, Watanabe proved the nontriviality of various rational homotopy groups of $BDiff_\partial(D^{2n+1})$ by constructing families of diffeomorphisms associated to trivalent graphs, and detecting them using configuration-space integrals. The simplest such class is associated to the theta-graph, and has degree $2n-2$. Somewhat surprisingly, this is not the sporadic class in this degree, but rather is the lowest of the Pontrjagin$-$Weiss classes.

In this talk I will first explain the above in detail, and then say something about a crucial step in the argument which is related to Watanabe's clasper construction and uses his result for the theta-graph.

Embedding calculus in codimension zero

Embedding calculus provides an approximation $Emb(M,N)\rightarrow T_\infty Emb(M,N)$ to the homotopy type of the space of smooth embeddings $Emb(M,N)$ by a space $T_\infty Emb(M,N)$ of more homotopy-theoretical nature. Results of Goodwillie, Klein, and Weiss ensure that this approximation is in fact an equivalence as long as the (handle) codimension of $M$ and $N$ is at least $3$, which has led to a variety of applications, some of which discussed in this seminar.

In handle codimension $\le 2$ however, little is known about the (potential) difference between $Emb(M,N)$ and $T_\infty Emb(M,N)$. In this talk, I will report on aspects of ongoing joint work with A. Kupers in which we study this (potential) difference, especially in handle codimension $0$. For instance, our results imply that if $M=N$ is high-dimensional and spin, then the homotopy fibre of the map $Emb(M,N)\rightarrow T_\infty Emb(M,N)$ is a nontrivial infinite loop space that depends only on the fundamental group and the dimension of $M$.

Scanning diffeomorphisms

Cerf gave a beautiful homotopy-equivalence, a map we call scanning, between the space of diffeomorphisms of the n-disc (fixing boundary) $Diff(D^n)$ and the loop space of embeddings of a co-dimension one disc $\Omega Emb(D^{n-1}, D^n)$. The "barbell manifold" is our term for the boundary connect-sum of two copies of $S^{n-1} \times D^2$. We will use a weak form of Cerf's scanning map to show a family of diffeomorphisms of the barbell manifold is non-trivial. We then proceed to embed the barbell manifold in $S^1 \times D^n$ and check the extensions of the barbell diffeomorphism families to $Diff(S^1 \times D^n)$ are homotopically non-trivial, using another scanning map. This allows us to show $\pi_{n-3} Diff(S^1 \times D^n)$ is not finitely generated for all $n\geq3$. One way of restating this result is that the component of the unknot in the space of smooth embeddings $Emb(S^{n-1}, S^{n+1})$ has a not-finitely-generated $(n-2)$-nd homotopy group, for all $n\geq3$.

Knotted $3$-Balls in $S^4$ and knotted $3$-spheres in $S^1\times S^3$

This is the second of two talks on joint work with Ryan Budney. We will show that $\pi_0( Diff_0(S^1\times S^3)/Diff(B^4~fix~\partial) )$ has an explicit infinite set of linearly independent elements.

The diffeomorphism group of a 4-manifold

Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor’s discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group.  The lowest dimension for such classical phenomena is 5.

I will discuss recent joint work with Dave Auckly about these groups in dimension 4. For each n, we construct a simply connected 4-manifold Z and an infinite subgroup of the nth homotopy group of Diff(Z) that lies in the kernel of the natural map to the corresponding homotopy group of Homeo(Z). These elements are detected by (n+1)—parameter gauge theory. I will give a brief sketch of the detection results and concentrate on the topological construction.

Hopf invariants - from embedding questions to homotopy questions and back to linking

We have seen in the talks of Krushkal and Budney, among others, that when one applies embedding calculus one is often led to study induced maps on configuration spaces, with restrictions, up to homotopy. This invites the development of explicit homotopy invariants of maps, an area which has been neglected.

Motivated by this, in work with Walter we found that for simply connected spaces the vector spaces Hom $(\pi_*(X), Q)$ are spanned by numerical invariants defined geometrically through “higher linking with correction.” For example, if $f : S^4 \to S^2 \vee S^2 \vee S^2$ is a map, assuming transversality we consider the preimages of three points on the wedge factors, which are three framed surfaces $P$, $Q$, $R$. Cobounding two of those surfaces and intersecting with the third gives a discrete intersection which can be counted. Such invariants determine the rational homotopy class of $f$ in this case, and similar invariants do so in general. Moreover, the full formalism of rational homotopy theory can be understood through such a linking perspective.

We are starting to apply these ideas to $\pi_1$, giving an approach to the lower central series filtration through (higher) linking of letters in words. Jeff Monroe has applied these ideas to give a simple way to make computations at any level in the Johnson filtration of mapping class groups of surfaces with boundary, for example constructing explicit simple examples of products of Dehn twists which are non-trivial in the fourth subquotient.

~ Seminar Break / End ~