We show that the mutant 2-component pretzel links P(p,q,-q,-p) and P(p,q,-p,-q) are not concordant for any distinct odd integers p and q greater than 1. As a corollary, we obtain a proof of the slice-ribbon conjecture for 4-stranded 2-component pretzel links. In order to distinguish mutant links up to concordance we consider 3-fold branched covers and use an obstruction based on Donaldson's diagonalization theorem. This is joint work with Min Hoon Kim, JungHwan Park and Arunima Ray.
SEMINARI DI GEOMETRIA
Si consideri la famiglia dei gruppi finitamente generati che ammettono
una scomposizione k-acilindrica, non-elementare (l'acilindricità è da
intendersi nel senso di Sela). Mostreremo l'esistenza di una funzione
(esplicita) f( - ;k):N--->N, dipendente esclusivamente da k e divergente
all'infinito tale che per ogni gruppo G ed ogni sistema finito di
generatori S di G l'entropia Ent(G,S) di G rispetto ad S sia maggiore o
uguale a f(#(S);k). Spiegheremo come tale disuguaglianza sia la chiave
We study links in 3-manifolds which have alternating diagrams onto orientable surfaces of positive genus. When the diagram is sufficiently complicated, we are able to obtain topological and geometrical information about the link exterior. In particular, we can tell if the link is hyperbolic and obtain bounds on volume, know whether the checkerboard surfaces are essential or quasi-fuchsian, and rule out exceptional Dehn fillings. Joint work with Jessica Purcell.
The simplicial volume is a homotopy invariant of compact manifolds introduced in 1982 by Gromov
in his pioneering paper "Volume and Bounded Cohomology". Roughly speaking, the simplicial volume
measures how difficult is to describe a manifold in terms of real singular chains.
In this talk, we will define the ideal simplicial volume, a variation of the ordinary simplicial volume for
compact manifolds with boundary. The main difference between ideal simplicial volume and the ordinary
(Collaborazione con Thang Le) Le algebre skein delle superfici chiuse sono oggetti estremamente ricchi di struttura e studiati. Recentemente, partendo da lavori di Bonahon-Wong e di Muller, Thang Le ha definito una versione delle algebre skein per superfici a bordo. In questo seminario, dopo aver richiamato le definizioni di base, cercherò di spiegare perché questa versione delle algebre skein è particolarmente interessante e cercherò’ di dare un’idea dei risultati (ancora in corso di stesura) di una collaborazione con Thang Le che puntano ad inserire queste algebre in una costruzione di un
Non-semisimple constructions in quantum topology produce strong invariants and TQFTs with unprecedented properties. The first family of non-semisimple quantum invariants of 3-manifolds was defined by Hennings in 1996. The construction enabled Lyubashenko to build mapping class groups representations out of every finite-dimensional factorizable ribbon Hopf algebra. Further attempts at extending these constructions to TQFTs only produced partial results, as the vanishing of Hennings invariants in many crucial situations made it impossible to treat non-connected surfaces.
Since the volume of a hyperbolic manifold is often regarded as a measure of its complexity, it is interesting to study the examples of minimal volume.
The 2 and 3-dimensional cases are well understood.
In dimension four there is an abundance of examples, but we are far from classifying them. In this talk, we will survey all known examples of minimal volume hyperbolic 4-manifolds and prove that these fall into three commensurability classes. This is joint work with Stefano Riolo.
Given a closed 3-manifold Y with the action of a finite group G, we show how to find a closely related hyperbolic G-manifold Y’. The two manifolds are related by an invertible equivariant homology cobordism; a cobordism from Y to Y’ is invertible if there is a second cobordism from Y’ to Y such that the union of the two along Y’ is a product cobordism. I will give a collection of applications in 3 and 4-dimensional topology. (Joint work with Dave Auckly, Hee Jung Kim, and Paul Melvin.)
Most applications of gauge theory in 4-dimensional topology are concerned with simply-connected manifolds with non-trivial second homology. I will discuss the opposite situation, first describing the classical Rohlin invariant for manifolds with first homology = Z and vanishing second homology. I will give an interpretation in terms of a Seiberg-Witten theory, with an unusual index-theoretic correction term. I will discuss recent work with Jianfeng Lin and Nikolai Saveliev giving a new formula for this invariant in terms of monopole homology.