# Topology of moduli spaces

## University of Copenhagen

Conference: 19-23 August 2024

This is a conference in honour of Ulrike Tillmann's 60th birthday. While celebrating Ulrike's work in, and influence on, the topology of moduli spaces, the event will be a research conference, focused on the latest developments in this subject. The talks will take place in Aud 3. The conference is supported by the Clay Mathematics Institute and the Danish National Research Foundation. |

### Contributed talks

Monday | Tuesday | Wednesday | Thursday | Friday |

8:45 Registration and tea | |
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9:30 Søren Galatius | 9:30 Manuel Krannich | 9:30 Alexis Aumonier | 10:00 Paolo Salvatore |
9:30 Rachael Boyd |

10:30 tea | 10:30 tea | 10:30 tea | 11:00 tea | 10:30 tea |

10:55 Lucianna Basualdo Bonatto | 10:55 Martin Palmer | 10:55 George Raptis | 11:30-12:30 Andreas Stavrou | 10:55 Fabian Hebestreit |

12:00 Sadok Kallel | 12:00 Irakli Patchkoria | 12:00 Oscar Harr | 12:00 Florian Kranhold | |

12:30 lunch | 12:30 lunch | 12:30 lunch | 12:30 lunch | 12:30 lunch |

14:00 Dan Freed | 14:00 John Klein |
14:00 Robin Stoll 14:35 Samuel Munoz-Echaniz |
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15:00 tea | 15:00 tea | 15:05 tea | ||

15:30 Danica Kosanović | 15:30 Alexander Kupers | 15:30 Jan Steinebrunner | ||

16:30 reception | 18:00 conference dinner. Venue: Munkekælderen, Frue Plads 4. Port A, 1168 København K. If you arrive late and the gate is closed then please phone Jan: +45 2887 3685. I will come and open. |

All lectures are held in AUD 3.

### Alexis Aumonier

#### Moduli of smooth hypersurfaces

The space of all smooth hypersurfaces in a complex projective variety is an algebraic moduli space parameterising algebraic fibre bundles of hypersurfaces. I will explain how one can compute parts of its cohomology via homotopical methods. As a consequence, we will observe a cohomological stability phenomenon, and a relation to classifying spaces of diffeomorphism groups of hypersurfaces.

### Luciana Basualdo Bonatto

#### Link Concordance Invariants via Manifold Calculus Techniques

Many knot and link invariants were originally defined in a combinatorial manner, and subsequently demonstrated to be preserved via Reidemeister relations. However, in the past decades, there has been a shift towards the use of more categorical techniques, such as Manifold Calculus, to derive descriptions of invariants via universal properties. For instance, the knot embedding tower was shown to detect finite-type knot invariants. In particular, this categorical approach can be beneficial in demonstrating the existence of conjectured invariants. Motivated by the conjectured existence of higher-order Arf invariants, we will discuss a new tower which can be used to detect concordance invariants for links and will explore some of its properties. This is work in progress in collaboration with Hyeonhee Jin and Peter Teichner.

### Rachael Boyd

#### Diffeomorphisms of reducible 3-manifolds

I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

### Dan Freed

#### Moduli spaces, orientations, and spin structures

In ongoing joint work with Mike Hopkins and Greg Moore we revisit the orientation question for moduli spaces of self-dual connections on on oriented 4-manifolds, and in addition we consider the question of spin structures. A key step is interpreting the topological index of the self-dual complex as an invertible topological field theory. This unleashes the use of techniques from stable homotopy theory.

### Søren Galatius

#### Hopf algebra structures in the cohomology of moduli spaces.

In joint work with Brown, Chan, and Payne, we describe a bigraded cocommutative Hopf algebra structure on the weight zero compactly supported rational cohomology of the moduli space of principally polarized abelian varieties, and use it to give lower bounds on the dimensions of these cohomology groups. One step is to construct a coproduct on Quillen's spectral sequence, abutting to the rational homology of the one-fold delooping of the algebraic K-theory space of the integers, making it a spectral sequence of Hopf algebras. We also relate this spectral sequence to one involving Kontsevich's graph complexes.

### Oscar Harr

#### E_2-cells and handlebody mapping class groups

A satisfying feature of Galatius, Kupers, and Randal-Williams' multiplicative approach to homological stability is its sensitivity to computations in low degree and genus. We use this to improve upon Hatcher and Wahl's stability result for mapping class groups of 3-dimensional handlebodies. The key input is a simple geometric computation which shows that a certain homology class which does not destabilize nevertheless has the property that its stabilization destabilizes twice. The same argument gives an improved stability range for the integral homology of Aut(F_n), and recovers a result of Kupers--Miller--Patzt for GL_n(Z).

### Fabian Hebestreit

#### On the Weiss-Williams index

I will explain a construction of the Weiss-Williams index classes for manifold bundles in the language of Poincaré categories. The particular view this affords allows one to coalesce these classes into a topological field theory with values in cobordism categories of visibly self-dual parametrised spectra. The underlying homotopy types of these categories are Weiss-Williams' LA-spectra, originally custom made as the home for their indices. Our construction refines and unifies their work with earlier maps by Bökstedt and Madsen with target Waldhausen's A-spectra and Basterra, Bobkova, Ponto, Tillmann and Yaekel with target hermitian K-spectra, provides an index theorem for the Weiss-Williams indices and offers a somewhat fresh perspective on Waldhausen's map connecting block homeomorphisms and Whitehead spectra. Joint with M.Land, T.Nikolaus, W.Steimle

### Sadok Kallel

#### The topology of spaces of maps from a Riemann surface to complex projective space

The space of continuous maps Map(M,N) between two Riemannian manifolds M and N is a fundamental object of study in algebraic topology, more particularly when the source space M is a sphere. We will address the case when M=C is a Riemann surface of positive genus and N is a complex projective n-space. This mapping space has received considerable attention in the literature,by physicists and mathematicians alike. It breaks down into connected components indexed by an integer (the "charge"). We give an overview of the relevant results, and then compute the homology of these components. This is ongoing work with Paolo Salvatore (Rome).

### John Klein

#### On embeddings and acyclic maps

Given an acyclic map X\to Y of Poincaré duality spaces of dimension d, we study the relationship between the existence of Poincaré embeddings of Y in S^n with those of X in S^n when n-d \ge 3. We also consider the analogous problem in the smooth and piecewise linear cases. We then focus on the case when X is a smooth homology sphere and deduce results about the homotopy type of the space of block embeddings of X in a sphere.

### Danica Kosanović

#### Diffeomorphisms from degree one graspers

Not only is it still unknown if the 4-sphere has a unique smooth structure, but we also do not know if its smooth oriented mapping class group is nontrivial. Recently it was shown that any potential class can be realised from a 1-parameter family of embedded spheres. In this talk I will explain how all candidates that have been constructed in fact come from a 1-parameter family of embedded circles. Analogues of such families were shown by Budney—Gabai and Watanabe to be nontrivial in some non-simply connected 4-manifolds.

### Florian Kranhold

#### A stable splitting of factorisation homology of generalised surfaces

For a manifold W and an E_d-algebra A, the factorisation homology of W with coefficients in A admits an action by the diffeomorphism group of W and we consider its homotopy quotient W[A]. For W_{g,1} a generalised surface of genus g with a single boundary sphere, the collection of all W_{g,1}[A] is a monoid by taking boundary-connected sums. We discuss its homological stability and describe its group-completion in terms of a tangential Thom spectrum and the iterated bar construction of A. We do so by identifying the above collection with an algebra over the generalised surface operad, establishing a splitting result for such algebras, and studying the free infinite loop space over a given (framed) E_d-algebra.

### Manuel Krannich

#### Decomposing high-dimensional manifold theory

I will explain a partial pullback decomposition of the category of d-manifolds and spaces of embeddings between them. It is valid in dimensions d > 4 and involves the derived automorphism group of the little d-discs operad. Applications include a rational section of the stabilisation map for the space of homeomorphisms of d-dimensional Euclidean space for d > 5. This is joint work with Alexander Kupers.

### Alexander Kupers

#### Automorphisms of E_d-operads

In this talk we explain what we know and do not know about the (derived) automorphisms of the little d-discs operad. This operad plays a central role in higher algebra and Goodwillie-Weiss’ embedding calculus approach to the study of manifolds. An important role is played by the action on this operad by linear transformations or more generally homeomorphisms of Euclidean space, yielding maps BO(d) -> BTop(d) -> BAut(E_d). We will explain why the fibres of these maps can be studied through embedding calculus and the consequences that this has. This is joint work with Manuel Krannich and Geoffroy Horel.

### Samuel Munoz-Echaniz

#### Mapping class groups of h-cobordant manifolds

Given two h-cobordant manifolds M and M', how different can the homotopy types of the diffeomorphism groups Diff(M) and Diff(M') be? The homotopy groups of these two spaces are the same “up to extensions” in a range of strictly positive degrees (depending on the dimension of M). Contrasting this fact, I will present examples of h-cobordant manifolds in high-dimensions with different mapping class groups. In doing so, I will introduce a moduli space of “h-block” bundles and analyse its difference with the moduli space of ordinary block bundles from surgery theory.

### Martin Palmer

#### Homology of big mapping class groups supported on compact subsurfaces

If S is a surface of infinite type (i.e. its fundamental group is infinitely generated), then the mapping class group Mod(S) is uncountable and its group homology is in many cases uncountably generated in every degree. A natural question is whether any of these homology classes are supported on compact subsurfaces of S. In the infinite-genus setting and with rational coefficients, for example, this equivalently asks whether the dual Miller-Morita-Mumford classes vanish on Mod(S). We will discuss this question and the analogous question about support on finite-type subsurfaces, giving an almost-complete answer when S has positive (for example infinite) genus and a partial answer when S has genus zero, in which case it depends very subtly on the topology of S. This represents joint work with Xiaolei Wu.

### Irakli Patchkoria

#### Chromatic congruences and Bernoulli numbers

For every n and a fixed prime p, we construct a new congruence for the orbifold Euler characteristic of a group which we call the chromatic congruence at the height n. Here the word “chromatic” refers to the chromatic stable homotopy theory, though to understand this talk no background in stable homotopy theory is required. The p-adic limit of these congruences when n tends to infinity recovers the well-known Brown-Quillen congruence. We apply these results to mapping class groups and using Harer-Zagier we get an infinite family of congruences for Bernoulli numbers. At the end we will see that these congruences in particular recover classical congruences for Bernoulli numbers due to Kummer, Voronoi, Carlitz and Cohen.

### George Raptis

#### Topological and homotopical aspects of bounded cohomology

Bounded cohomology is a variant of singular (and group) cohomology with deep connections with differential geometry and group theory. I will review and discuss aspects of bounded cohomology, focusing on the interactions with homotopy theory and the comparison with usual cohomology. I will also review the properties of the simplicial volume, a closely related homotopy invariant of closed oriented manifolds, also in connection with the cobordism category, and discuss an intriguing open question of Gromov about the vanishing of the Euler characteristic of aspherical manifolds with vanishing simplicial volume. (This is partially based on joint works with Li, Löh, and Moraschini.)

### Paolo Salvatore

#### Disc embeddings, deloopings and operad actions

A classical theorem by Morlet identifies the (n+1)-fold delooping of the group of diffeomorphisms of the n-disc relative to the boundary. We extend the result to spaces of embeddings of m-discs into n-discs, together with their versions modulo immersions and with frames. In the framed case we construct an action of the framed (m+1)-disc operad that combines an O(m+1)-action defined by Hatcher and an E_{m+1} action defined by Budney. This is joint work with Victor Turchin.

### Andreas Stavrou

#### Configuration spaces of surfaces and the Johnson filtration

The mapping class group of a surface acts naturally on the homology of the configuration spaces of the surface. The kernels of the representations which arise this way vary with the number of configuration points and with the flavour of the configuration space. In this talk, we will discuss recent results and work in progress comparing these kernels with the Johnson filtration of the mapping class group.

### Jan Steinebrunner

#### 2-dimensional topological field theories via the genus filtration

By a folk theorem (non-extended) 2-dimensional TFTs valued in the category of vector spaces are equivalent to commutative Frobenius algebras. Upgrading the bordism category to an (infinity, 1)-category whose 2-morphism are diffeomorphisms, one can study 2D TFTs valued in higher categories, leading for example to modular functors and cohomological field theories.

I will explain how to describe such more general (non-extended) 2D TFTs as algebras over the modular infinity-operad of surfaces. In genus 0 this yields an E2SO-Frobenius algebra and I will outline an obstruction theory for inductively extending such algebras to higher genus. Specialising to invertible TFTs, this amounts to a genus filtration of the classifying space of the bordism category and hence the Madsen--Tillmann spectrum MTSO2. The aforementioned obstruction theory identifies the associated graded in terms of curve complexes and thereby yields a spectral sequence starting with the unstable and converging to the stable cohomology of mapping class groups.

### Robin Stoll

#### An equivariant rational model for automorphisms of bundles

I will report on joint work with Alexander Berglund, in which we use ideas of Sullivan and Berglund--Zeman to incorporate the action of (a quotient of) the fundamental group into a rational model for the classifying space of the topological monoid of automorphisms of certain types of bundles. This allows to access the whole classifying space, rather than just its universal covering. Moreover, this yields a rational model for the classifying space of block diffeomorphisms of high dimensional manifolds. Lastly, I will comment on an application of this result to the cohomology of the classifying space of diffeomorphisms of connected sums of products of spheres.

The conference/masterclass will take place at the Department of Mathematical Sciences, University of Copenhagen, in Aud 3 (HC Ørsted building). See detailed instructions on how to reach Copenhagen and the conference venue.

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB ticket office on your right-hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation.

A journey planner in English is available.

More information on the "find us" webpage.

We kindly ask the participants to arrange their own accommodation.

We recommend Hotel 9 Små Hjem, which is pleasant and inexpensive and offers rooms with a kitchen. Other inexpensive alternatives are CabInn, which has several locations in Copenhagen: the Hotel City (close to Tivoli), Hotel Scandinavia (Frederiksberg, close to the lakes), and Hotel Express (Frederiksberg) are the most convenient locations; the latter two are 2.5-3 km from the math department. Somewhat more expensive – and still recommended – options are Hotel Nora and Ibsen's Hotel.

An additional option is to combine a stay at the CabInn Metro Hotel with a pass for Copenhagen public transportation (efficient and reliable). See information about tickets & prices.

Registration has closed.

Nathalie Wahl: wahl@math.ku.dk

Oscar Randal-Williams: o.randal-williams@dpmms.cam.ac.uk

Administrative matters: Jan Tapdrup jt@math.ku.dk