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Timetable (HHHW04)

Manifolds

Monday 3rd December 2018 to Friday 7th December 2018

Monday 3rd December 2018
09:00 to 09:50 Registration
09:50 to 10:00 Welcome from Christie Marr (INI Deputy Director) INI 1
10:00 to 11:00 Soren Galatius (University of Copenhagen)
H_{4g-6}(M_g)
The set of isomorphism classes of genus g Riemann surfaces carries a natural topology in which it may be locally parametrized by 3g-3 complex parameters. The resulting space is denoted M_g, the moduli space of Riemann surfaces, and is more precisely a complex orbifold of that dimension. The study of this space has a very long history involving many areas of mathematics, including algebraic geometry, group theory, and stable homotopy theory. The space M_g is not compact, essentially because a family of Riemann surface may degenerate into a non-smooth object, and may be compactified in several interesting ways. I will discuss a compactification due to Harvey, which looks like a compact real (6g-6)-dimensional manifold with corners, except for orbifold singularities. The combinatorics of the corner strata in this compactification may be encoded using graphs. Using this compactification, I will explain how to define a chain map from Kontsevich's graph complex to a chain complex calculating the rational homology of M_g. The construction is particularly interesting in degree 4g-6, where our methods give rise to many non-zero classes in H_{4g-6}(M_g), contradicting some predictions. This is joint work with Chan and Payne (arXiv:1805.10186).
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Alexander Kupers (Harvard University)
Cellular techniques in homological stability 1: general theory
This is the first of two talks about joint work with S. Galatius and O Randal-Williams on applications higher-algebraic structures to homological stability. The main tool is cellular approximation of E_k-algebras, and we start with a discussion of the general theory of such cellular approximations. This culminates in a generic homological stability result.
INI 1
12:30 to 13:30 Lunch at Churchill College
14:30 to 15:30 George Raptis (Universität Regensburg)
The h-cobordism category and A-theory
A fundamental link between Waldhausen's algebraic K-theory of spaces (A-theory) and manifold topology is given by an identification of A-theory in terms of stable homotopy and the stable smooth h-cobordism space. This important result has had many applications in the study of diffeomorphisms of manifolds. In more recent years, the theory of cobordism categories has provided a different approach to the study of diffeomorphism groups with spectacular applications. In collaboration with W. Steimle , we revisit the classical Waldhausen K-theory in light of these developments and investigate new connections and applications. In this talk, I will first discuss a cobordism-type model for A-theory, and then I will focus on the h-cobordism category, the cobordism category of h-cobordisms between smooth manifolds with boundary, and its relationship to the classical h-cobordism space of a compact smooth manifold. This is joint work with W. Steimle.
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Christopher Schommer-Pries (University of Notre Dame)
The Relative Tangle Hypothesis
I will describe recent progress on a non-local variant of the cobordism hypothesis for higher categories of bordisms embedded into finite dimensional Euclidean space.
INI 1
17:00 to 18:00 Welcome Wine Reception at INI
Tuesday 4th December 2018
09:00 to 10:00 Wolfgang Lueck (Universität Bonn)
On the stable Cannon Conjecture
The Cannon Conjecture for a torsionfree hyperbolic group $G$ with boundary homeomorphic to $S^2$ says that $G$ is the fundamental group of an aspherical closed $3$-manifold $M$.  It is known that then $M$ is a hyperbolic $3$-manifold.  We prove the stable version that for any closed manifold $N$ of dimension greater or equal to $2$  there exists a closed manifold $M$ together with a simple homotopy equivalence $M \to N \times BG$. If $N$ is aspherical and $\pi_1(N)$ satisfies the Farrell-Jones Conjecture, then $M$ is unique up to homeomorphism.
This is joint work with Ferry and Weinberger.
INI 1
10:00 to 11:00 Thomas Willwacher (ETH Zürich)
Configuration spaces of points and real Goodwillie-Weiss calculus
The manifold calculus of Goodwillie and Weiss proposes to reduce questions about embedding spaces of manifolds to questions about mapping spaces of the (little-disks modules of) configuration spaces of points on those manifolds. We will discuss real models for these configuration spaces. Furthermore, we will see that a real version of the aforementioned mapping spaces is computable in terms of graph complexes. In particular, this yields a new tool to study diffeomorphism groups and moduli spaces.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Alexander Kupers (Harvard University)
Cellular techniques in homological stability 2: mapping class groups
This is the second of two talks about joint work with S. Galatius and O Randal-Williams on applications higher-algebraic structures to homological stability. In it we apply the general theory to the example of mapping class groups of surfaces. After reproving Harer's stability result, I will explain how to prove the novel phenomenon of secondary homological stability; there are maps comparing the relative homology groups of the stabilization map for different genus and there are isomorphisms in a range tending to infinity with the genus.
INI 1
12:30 to 13:30 Lunch at Churchill College
14:30 to 15:30 Victor Turchin (Kansas State University)
Embeddings, operads, graph-complexes
I will talk about the connection between the following concepts: manifold calculus, little discs operads, embedding spaces, problem of delooping, relative rational formality of the little discs, and graph-complexes. I will review main results on this connection by Boavida de Brito and Weiss, my coauthors and myself. At the end I will briefly go over the current joint work in progress of Fresse, Willwacher, and myself on the rational homotopy type of embedding spaces.

Co-authors: Gregory Arone (Stockholm University), Julien Ducoulombier (ETH, Zurich), Benoit Fresse (University of Lille), Pascal Lambrechts (University of Louvain), Paul Arnaud Songhafouo Tsopméné (University of Regina), Thomas Willwacher (ETH, Zurich).
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 17:00 Nathalie Wahl (University of Copenhagen)
Homotopy invariance in string topology
In joint work with Nancy Hingston, we show that the Goresky-Hingston coproduct, just like the Chas-Sullivan product, is homotopy invariant. Unlike the Chas-Sullivan product, this coproduct is a "compactified operation", coming from a certain compactification of the moduli space of Riemann surfaces. I'll give an idea of the ingredients used in the proof.
INI 1
Wednesday 5th December 2018
09:00 to 10:00 Cary Malkiewich (Binghamton University)
Periodic points and topological restriction homology
I will talk about a project to import trace methods, usually reserved for algebraic K-theory computations, into the study of periodic orbits of continuous dynamical systems (and vice-versa). Our main result so far is that a certain fixed-point invariant built using equivariant spectra can be "unwound" into a more classical invariant that detects periodic orbits. As a simple consequence, periodic-point problems (i.e. finding a homotopy of a continuous map that removes its n-periodic orbits) can be reduced to equivariant fixed-point problems. This answers a conjecture of Klein and Williams, and allows us to interpret their invariant as a class in topological restriction homology (TR), coinciding with a class defined earlier in the thesis of Iwashita and separately by Luck. This is joint work with Kate Ponto.
INI 1
10:00 to 11:00 Christine Vespa (University of Strasbourg)
Higher Hochschild homology as a functor
Higher Hochschild homology generalizes classical Hochschild homology for rings. Recently, Turchin and Willwacher computed higher Hochschild homology of a finite wedge of circles with coefficients in the Loday functor associated to the ring of dual numbers over the rationals. In particular, they obtained linear representations of the groups Out(F_n) which do not factorize through GL(n,Z).

In this talk I will explain how viewing higher Hochschild homology of a finite wedge of circles as a functor on the category of free groups provides a conceptual framework which allows powerful tools such as exponential functors and polynomial functors to be brought to bear. In particular, this allows the generalization of the results of Turchin and Willwacher; this gives rise to new linear representations of Out(F_n) which do not factorize through GL(n,Z).

(This is joint work with Geoffrey Powell.)

INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Fabian Hebestreit (University of Bonn)
The homotopy type of algebraic cobordism categories
Co-authors: Baptiste Calmès (Université d'Artois), Emanuele Dotto (RFWU Bonn), Yonatan Harpaz (Université Paris 13), Markus Land (Universität Regensburg), Kristian Moi (KTH Stockholm), Denis Nardin (Université Paris 13), Thomas Nikolaus (WWU Münster), Wolfgang Steimle (Universität Augsburg). Abstract: I will introduce cobordism categories of Poincaré chain complexes, or more generally of Poincaré objects in any hermitian quasi-category C. One interest in such algebraic cobordism categories arises as they receive refinements of Ranicki's symmetric signature in the form of functors from geometric cobordism categories à la Galatius-Madsen-Tillmann-Weiss. I will focus, however, on a more algebraic direction. The cobordism category of C can be delooped by an iterated Q-construction, that is compatible with Bökstedt-Madsen's delooping of the geometric cobordism category. The resulting spectrum is a derived version of Grothendieck-Witt theory and I will explain how its homotopy type can be computed in terms of the K- and L-Theory of C.
INI 1
12:30 to 13:30 Lunch at Churchill College
13:30 to 18:00 Free afternoon
Thursday 6th December 2018
09:00 to 10:00 Alexander Berglund (Stockholm University)
Rational homotopy theory of automorphisms of manifolds
I will talk about differential graded Lie algebra models for automorphism groups of simply connected manifolds M. Earlier results by Ib Madsen and myself on models for block diffeomorphisms combined with rational models for Waldhausen's algebraic K-theory of spaces suggest a model for the group of diffeomorphisms homotopic to the identity, valid in the so-called pseudo-isotopy stable range. If time admits, I will also discuss how to express the generalized Miller-Morita-Mumford classes in the cohomology of BDiff(M) in terms of these models.
INI 1
10:00 to 11:00 Johannes Ebert (Westfalische Wilhelms-Universitat Munster)
Cobordism categories, elliptic operators and positive scalar curvature
We prove that a certain collection of path components of the space of metrics of positive scalar curvature on a high-dimensional sphere has the homotopy type of an infinite loop space, generalizing a theorem by Walsh. The proof uses an version of the surgery method by Galatius and Randal--Williams to cobordism categories of manifolds equipped with metrics of positive scalar curvature. Moreover, we prove that the secondary index invariant of the spin Dirac operator is an infinite loop map. The proof of that fact uses a generalization of the Atiyah--Singer index theorem to spaces of manifolds. (Joint work with Randal--Williams)
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Ben Knudsen (Harvard University)
Configuration spaces and Lie algebras away from characteristic zero
There is a close connection between the theory of Lie algebras and the study of additive invariants of configuration spaces of manifolds, which has been exploited in many calculations of rational homology. We begin the computational exploration of this connection away from characteristic zero, exhibiting a spectral sequence converging to the p-complete complex K-theory of configuration spaces---more generally, to their completed Morava E-(co)homology---and we identify its second page in terms of an algebraic homology theory for Lie algebras equipped with certain power operations. We construct a computationally accessible analogue of the classical Chevalley--Eilenberg complex for these Hecke Lie algebras, and we use it to perform a number of computations. This talk is based on joint work in progress with Lukas Brantner and Jeremy Hahn.
INI 1
12:30 to 13:30 Lunch at Churchill College
14:30 to 15:00 Manuel Krannich (University of Cambridge)
Contributed talk - Mapping class groups of highly connected manifolds
The group of isotopy classes of diffeomorphisms of a highly connected almost parallelisable manifold of even dimension 2n>4 has been computed by Kreck in the late 70’s. His answer, however, left open two extension problems, which were later understood in some particular dimensions, but remained unsettled in general. In this talk, I will explain how to resolve these extension problems in the case of n being odd, resulting in a complete description of the mapping class group in question in terms of an arithmetic group and the cokernel of the stable J-homomorphism.
INI 1
15:00 to 15:30 Rachael Boyd (Norwegian University of Science and Technology)
Contributed Talk - The low dimensional homology of Coxeter groups
Coxeter groups were introduced by Tits in the 1960s as abstractions of the finite reflection groups studied by Coxeter. Any Coxeter group acts by reflections on a contractible complex, called the Davis complex. This talk focuses on a computation of the first three integral homology groups of an arbitrary Coxeter group using an isotropy spectral sequence argument: the answer can be phrased purely in terms of the original Coxeter diagram. I will give an introduction to Coxeter groups and the Davis complex before outlining the proof.
INI 1
15:30 to 16:00 Afternoon Tea
16:00 to 16:30 Csaba Nagy (University of Melbourne)
Contributed Talk - The Sullivan-conjecture in complex dimension 4
The Sullivan-conjecture claims that complex projective complete intersections are classified up to diffeomorphism by their total degree, Euler-characteristic and Pontryagin-classes. Kreck and Traving showed that the conjecture holds in complex dimension 4 if the total degree is divisible by 16. In this talk I will present the proof of the remaining cases. It is known that the conjecture holds up to connected sum with the exotic 8-sphere (this is a result of Fang and Klaus), so the essential part of our proof is understanding the effect of this operation on complete intersections. This is joint work with Diarmuid Crowley.
INI 1
16:30 to 17:00 Danica Kosanović (Max-Planck-Institut für Mathematik, Bonn)
Contributed talk - Extended evaluation maps from knots to the embedding tower
The evaluation maps from the space of knots to the associated embedding tower are conjectured to be universal knot invariants of finite type. Currently such invariants are known to exist only over the rationals (using the existence of Drinfeld associators) and the question of torsion remains wide open. On the other hand, grope cobordisms are certain operations in ambient 3-space producing knots that share the same finite type invariants and give a geometric explanation for the appearance of Lie algebras and graph complexes.

I will explain how grope cobordisms and an explicit geometric construction give paths in the various levels of the embedding tower. Taking components recovers the result of Budney-Conant-Koytcheff-Sinha, showing that these invariants are indeed of finite type. This is work in progress joint with Y. Shi and P. Teichner.
INI 1
18:15 to 19:15 Reception at Cambridge University Press Bookshop
19:30 to 20:30 Formal Dinner at Christs College (Hall 110)
Friday 7th December 2018
10:00 to 11:00 Andre Henriques (University of Oxford)
The complex cobordism 2-category and its central extensions
I will introduce a symmetric monoidal 2-category whose objects are 0-manifolds, whose 1-morphisms are 1-dimensional smooth cobordisms, and whose 2-morphisms are Riemann surfaces with boundary and cusps. I will introduce a certain central extension by ℝ₊ and explain its relevance in chiral conformal field theory. Finally, I will explain the state of my understanding on the question of classification of such extensions by ℝ₊.
INI 1
11:00 to 11:30 Morning Coffee
11:30 to 12:30 Sam Nariman (Northwestern University)
Topological and dynamical obstructions to extending group actions.
For any 3-manifold $M$ with torus boundary, we find finitely generated subgroups of $\Diff_0(\partial M)$ whose actions do not extend to actions on $M$; in many cases, there is even no action by homeomorphisms. The obstructions are both dynamical and cohomological in nature. We also show that, if $\partial M = S^2$, there is no section of the map $\Diff_0(M) \to \Diff_0(\partial M)$. This answers a question of Ghys for particular manifolds and gives tools for progress on the general program of bordism of group actions. This is a joint work with Kathryn Mann.
INI 1
12:30 to 13:30 Lunch at Churchill College
13:30 to 14:30 Ryan Budney (University of Victoria)
Some prospects with the splicing operad
Roughly six years ago I described an operad that acts on spaces of `long knots'. This is the space of smooth embeddings of R^j into R^n. The embeddings are required to be standard (linear) outside of a disc, and come equipped with a trivialisation of their normal bundles. This splicing operad gives a remarkably compact description of the homotopy-type of the space of classical long knots (j=1, n=3), that meshes well with the machinery of 3-manifold theory: JSJ-decompositions and geometrization. What remains to be seen is how useful this splicing operad might be when n is larger than 3. I will talk about what is known at present, and natural avenues to explore.
INI 1
14:30 to 15:30 Diarmuid Crowley (University of Melbourne)
Relative kappa-classes
Diff(D^n), the the space of diffeomorphisms of the n-disc fixed near the boundary has rich rational topology. For example, Weiss's discovery of ``surreal'' Pontrjagin classes leads to the existence of rationally non-trivial homotopy classes in BDiff(D^n).

For any smooth n-manifold M, extension by the identity induces a map BDiff(D^n) \to BDiff(M). In this talk I will report on joint work with Wolfgang Steimle and Thomas Schick, where we consider the problem of computing the image of the ``Weiss classes'' under the maps on homotopy and homology induced by extension. This problem naturally leads one to consider relative kappa-classes.

Via relative kappa-classes, we show that the maps induced by extension are rationally non-trivial for a wide class of manifolds M, including aspherical manifolds (homology, hence also homotopy) and stably parallelisable manifolds (homotopy). When M is aspherical, our arguments rely on vanishing results for kappa-classes due to Hebestreit, Land, Lueck and Randal-Williams.

INI 1
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons