Manifolds
Monday 3rd December 2018 to Friday 7th 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 H_{4g6}(M_g)
The set of isomorphism classes of genus g Riemann surfaces carries a natural topology in which it may be locally parametrized by 3g3 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 nonsmooth object, and may be compactified in several interesting ways. I will discuss a compactification due to Harvey, which looks like a compact real (6g6)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 4g6, where our methods give rise to many nonzero classes in H_{4g6}(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 Cellular techniques in homological stability 1: general theory
This is the first of two talks about joint work with S. Galatius and O RandalWilliams on applications higheralgebraic structures to homological stability. The main tool is cellular approximation of E_kalgebras, 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 The hcobordism category and Atheory
A fundamental link between Waldhausen's algebraic Ktheory of spaces (Atheory) and manifold topology is given by an identification of Atheory in terms of stable
homotopy and the stable smooth hcobordism 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 Ktheory in light of these developments and investigate new connections and applications. In this talk, I will first
discuss a cobordismtype model for Atheory, and then I will focus on the hcobordism category, the cobordism category of hcobordisms between smooth manifolds
with boundary, and its relationship to the classical hcobordism 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 SchommerPries The Relative Tangle Hypothesis
I will describe recent progress on a nonlocal 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 
09:00 to 10:00 
Wolfgang Lueck 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 FarrellJones 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 Configuration spaces of points and real GoodwillieWeiss calculus
The manifold calculus of Goodwillie and Weiss proposes to reduce questions about embedding spaces of manifolds to questions about mapping spaces of the (littledisks 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 Cellular techniques in homological stability 2: mapping class groups
This is the second of two talks about joint work with S. Galatius and O RandalWilliams on applications higheralgebraic 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 Embeddings, operads, graphcomplexes
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 graphcomplexes. 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. Coauthors: 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 Homotopy invariance in string topology
In joint work with Nancy Hingston, we show that the GoreskyHingston coproduct, just like the ChasSullivan product, is homotopy invariant. Unlike the ChasSullivan 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 
09:00 to 10:00 
Cary Malkiewich Periodic points and topological restriction homology
I will talk about a project to import trace methods, usually reserved for algebraic Ktheory computations, into the study of periodic orbits of continuous dynamical systems (and viceversa). Our main result so far is that a certain fixedpoint invariant built using equivariant spectra can be "unwound" into a more classical invariant that detects periodic orbits. As a simple consequence, periodicpoint problems (i.e. finding a homotopy of a continuous map that removes its nperiodic orbits) can be reduced to equivariant fixedpoint 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 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 The homotopy type of algebraic cobordism categories
Coauthors: 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 quasicategory 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 GalatiusMadsenTillmannWeiss. I will focus, however, on a more algebraic direction. The cobordism category of C can be delooped by an iterated Qconstruction, that is compatible with BökstedtMadsen's delooping of the geometric cobordism category. The resulting spectrum is a derived version of GrothendieckWitt theory and I will explain how its homotopy type can be computed in terms of the K and LTheory of C.

INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 18:00  Free afternoon 
09:00 to 10:00 
Alexander Berglund 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 Ktheory of spaces suggest a model for the group of diffeomorphisms homotopic to the identity, valid in the socalled pseudoisotopy stable range. If time admits, I will also discuss how to express the generalized MillerMoritaMumford classes in the cohomology of BDiff(M) in terms of these models.

INI 1  
10:00 to 11:00 
Johannes Ebert 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 highdimensional 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 RandalWilliams 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 AtiyahSinger index theorem to spaces of manifolds. (Joint work with RandalWilliams)

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Ben Knudsen 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 pcomplete complex Ktheory of configuration spacesmore generally, to their completed Morava E(co)homologyand 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 ChevalleyEilenberg 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 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 Jhomomorphism.

INI 1  
15:00 to 15:30 
Rachael Boyd 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 Contributed Talk  The Sullivanconjecture in complex dimension 4
The Sullivanconjecture claims that complex projective complete intersections are classified up to diffeomorphism by their total degree, Eulercharacteristic and Pontryaginclasses. 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 8sphere (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 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 3space 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 BudneyConantKoytcheffSinha, 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) 
10:00 to 11:00 
Andre Henriques The complex cobordism 2category and its central extensions
I will introduce a symmetric monoidal 2category whose objects are 0manifolds, whose 1morphisms are 1dimensional smooth cobordisms, and whose 2morphisms 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 Topological and dynamical obstructions to extending group actions.
For any 3manifold $M$ 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 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 homotopytype of the space of classical long knots (j=1, n=3), that meshes well with the machinery of 3manifold theory: JSJdecompositions 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 Relative kappaclasses
Diff(D^n), the the space of diffeomorphisms of the ndisc fixed near the boundary has rich rational topology. For example, Weiss's discovery of ``surreal'' Pontrjagin classes leads to the existence of rationally nontrivial homotopy classes in BDiff(D^n). For any smooth nmanifold 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 kappaclasses. Via relative kappaclasses, we show that the maps induced by extension are rationally nontrivial 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 kappaclasses due to Hebestreit, Land, Lueck and RandalWilliams. 
INI 1 