We prove that every rational homology cobordism class in the subgroup generated

by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in

any other element in the same class. As a consequence we show that several natural maps to

the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility

condition between the determinants of certain 2-bridge knots and other knots in the same

concordance class. This is joint work with Daniele Celoria and JungHwan Park.

# Past Topology Seminar

In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined

topological operads with homological stability (OHS) and proved that the

group completion of an algebra over an OHS is weakly equivalent to an

infinite loop space.

In this talk, I shall outline a construction which to an algebra A over

an OHS associates a new infinite loop space. Under mild conditions on

the operad, this space is equivalent as an infinite loop space to the

group completion of A. This generalises a result of Wahl on the

equivalence of the two infinite loop space structures constructed by

Tillmann on the classifying space of the stable mapping class group. I

shall also talk about an application of this construction to stable

moduli spaces of high-dimensional manifolds in thesense of Galatius and

Randal-Williams.

We will discuss a variant of Taubes’s Seiberg-Witten to Gromov theorem in the context of a 4-manifold with cylindrical ends, equipped with a nontrivial harmonic 2-form. This harmonic 2-form is allowed to be asymptotic to 0 on some (but not all) of its ends, and may have nondegenerate zeros along 1-submanifolds. Corollaries include various positivity results; some simple special cases of these constitute a key ingredient in Kutluhan-Lee-Taubes’s proof of HM = HF (Monopole Floer homology equals Heegaard Floer homology). The aforementioned general theorem is motivated by (potential) extensions of the HM = HF and Lee-Taubes’s HM = PFH (Periodic Floer homology) theorems.

X.S. Lin defined an invariant of knots in S^3 by counting represenations

of the knot group into SU(2) with fixed meridinal holonomy. Lin's

invariant was subsequently shown to coincide with the Levine-Tristam

signature. I'll define an analogous total Lin invariant which counts

repesentations into both SU(2) and SL_2(R). Unlike the SU(2) version, this

invariant does not (as far as I know) coincide with other known

invariants. I'll describe some applications to left-orderability of Dehn

fillings and branched covers, as well as a curious connection with the

Alexander polynomial. This is joint work with Nathan Dunfield.

I will explain how Lurie‘s approach to L-theory via Poincaré categories can be extended to yield cobordism categories of Poincaré objects à la Ranicki. These categories can be delooped by an iterated Q-construction and the resulting spectrum is a derived version of Grothendieck-Witt-theory. Its homotopy type can be described in terms of K- and L-theory as conjectured by Hesselholt-Madsen. Furthermore, it has a clean universal property analogous to that of K-theory, localisation sequences in much greater generality than classical Grothendieck-Witt theory, gives a cycle description of Weiss-Williams‘ LA-theory and allows for maps from the geometric cobordism category, refining and unifying various known invariants.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.

The large-scale features of groups and spaces are recorded by asymptotic invariants. Examples of asymptotic invariants are the asymptotic cone and, for hyperbolic groups, the Gromov boundary.

In his study of asymptotic cones of connected Lie groups, Yves Cornulier introduced a class of maps called sublinearly biLipschitz equivalences. Like the more traditionnal quasiisometries, sublinearly biLipschitz equivalences are biLipschitz on the large-scale, but unlike quasiisometries, they are generally not coarse. Sublinearly biLipschitz equivalences still induce biLipschitz homeomorphisms between asymptotic cones. In this talk, I will focus on Gromov-hyperbolic groups and show how the Gromov boundary can be used to produce invariants distinguishing them up to sublinearly biLipschitz equivalences when the asymptotic cones do not. I will especially give applications to the large-scale sublinear geometry of hyperbolic Lie groups.

The concept of an acylindrically hyperbolic group, introduced by D. Osin, generalizes hyperbolic and relatively hyperbolic groups, and includes many other groups of interest: Out(F_n), n>1, most mapping class groups, directly indecomposable non-cyclic right angled Artin groups, most graph products, groups of deficiency at least 2, etc. Roughly speaking, a group G is acylindrically hyperbolic if there is a (possibly infinite) generating set X of G such that the Cayley graph \Gamma(G,X) is hyperbolic and the action of G on it is "sufficiently nice". Many global properties of hyperbolic/relatively hyperbolic groups have been also proved for acylindrically hyperbolic groups.

In the talk I will discuss a method which allows to construct a common acylindrically hyperbolic quotient for any countable family of countable acylindrically hyperbolic groups. This allows us to produce acylindrically hyperbolic groups with many unexpected properties.(The talk will be based on joint work with Denis Osin.)

In the late seventies, Casson and Gordon developed several knot invariants that obstruct a knot from being slice, i.e. from bounding a disc in the 4-ball. In this talk, we use twisted Blanchfield pairings to define twisted generalisations of the Levine-Tristram signature function, and describe their relation to the Casson-Gordon invariants. If time permits, we will present some obstructions to algebraic knots being slice. This is joint work with Maciej Borodzik and Wojciech Politarczyk.

I will discuss Agol's proof of the Virtually Fibred Conjecture of

Thurston, focusing on the role played by the `RFRS' property. I will

then show how one can modify parts of Agol's proof by replacing some

topological considerations with a group theoretic statement about

virtual fibring of RFRS groups.

In a joint work with Matt Tointon, we study the fine structure of approximate groups. We deduce various applications on growth, isoperimetry and quantitative estimates for the the simple random walk on finite vertex transitive graphs.