The triangulation conjecture stated that any n-dimensional topological manifold is homeomorphic to a simplicial complex. It is true in dimensions at most 3, but false in dimension 4 by the work of Casson and Freedman. In this talk I will explain the proof that the conjecture is also false in higher dimensions. This result is based on previous work of Galewski-Stern and Matumoto, who reduced the problem to a question in low dimensions (the existence of elements of order 2 and Rokhlin invariant one in the 3-dimensional homology cobordism group). The low-dimensional question can be answered in the negative using a variant of Floer homology, Pin(2)-equivariant Seiberg-Witten Floer homology. At the end I will also discuss a related version of Heegaard Floer homology, which is more computable.

# Past Topology Seminar

In this talk I will show how given a finitely generated relatively hyperbolic group G, one can construct a finite generating set X of G for which (G,X) has a number of metric properties, provided that the parabolic subgroups have these properties. I will discuss the applications of these properties to the growth series, language of geodesics, biautomatic structures and conjugacy problem. This is joint work with Yago Antolin.

Unordered configuration spaces on (connected) manifolds are basic objects

that appear in connection with many different areas of topology. When the

manifold M is non-compact, a theorem of McDuff and Segal states that these

spaces satisfy a phenomenon known as homological stability: fixing q, the

homology groups H_q(C_k(M)) are eventually independent of k. Here, C_k(M)

denotes the space of k-point configurations and homology is taken with

coefficients in Z. However, this statement is in general false for closed

manifolds M, although some conditional results in this direction are known.

I will explain some recent joint work with Federico Cantero, in which we

extend all the previously known results in this situation. One key idea is

to introduce so-called "replication maps" between configuration spaces,

which in a sense replace the "stabilisation maps" that exist only in the

case of non-compact manifolds. One corollary of our results is to recover a

"homological periodicity" theorem of Nagpal -- taking homology with field

coefficients and fixing q, the sequence of homology groups H_q(C_k(M)) is

eventually periodic in k -- and we obtain a much simpler estimate for the

period. Another result is that homological stability holds with Z[1/2]

coefficients whenever M is odd-dimensional, and in fact we improve this to

stability with Z coefficients for 3- and 7-dimensional manifolds.

If G is a semi-simple Lie group, it is known that all lattices

are arithmetic unless (up to finite index) G=SO(n,1) or SU(n,1).

Non-arithmetic lattices have been constructed in SO(n,1) for

all n and there are infinitely many non-arithmetic lattices in

SU(1,1). Mostow and Deligne-Mostow constructed 9 commensurability

classes of non-arithmetic lattices in SU(2,1) and a single

example in SU(3,1). The problem is open for n at least 4.

I will survey the history of this problem, and then describe

recent joint work with Martin Deraux and Julien Paupert, where

we construct 10 new commensurability classes of non-arithmetic

lattices in SU(2,1). These are the first examples to be constructed

since the work of Deligne and Mostow in 1986.

A permutation group is called sharply n-transitive if it acts freely and transitively on the set of ordered n-tuples of distinct points. The investigation of such permutation groups is a classical branch of group theory; it led Emile Mathieu to the discovery of the smallest finite simple sporadic groups in the 1860's. In this talk I will discuss the case where the permutation group is assumed to be a locally compact transformation group, and explain how this set-up is related to Gromov hyperbolicity and to arithmetic lattices in products of trees.

Let $G$ be a reductive group such as $SL_n$ over the field $k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the affine Weyl group of $G$. The associated affine Deligne-Lusztig varieties $X_x(b)$ were introduced by Rapoport. These are indexed by elements $x$ in $G$ and $b$ in $W$, and are related to many important concepts in algebraic geometry over fields of positive characteristic. Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and representation theory to address these questions in the case that $b$ is a translation. Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns. Since we work only in the standard apartment of the building for $G$, which is just the tessellation of Euclidean space induced by the action of the reflection group $W$, our results also hold over the p-adics. This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).

Building a suitable family of walls in the Cayley complex of a finitely

presented group G leads to a nontrivial action of G on a CAT(0) cube

complex, which shows that G does not have Kazhdan's property (T). I

will discuss how this can be done for certain random groups in Gromov's

density model. Ollivier and Wise (building on earlier work of Wise on

small-cancellation groups) have built suitable walls at densities <1/5,

but their method fails at higher densities. In recent joint work with

Piotr Przytycki we give a new construction which finds walls at densites

<5/24.

I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a

free group $F$ is exactly $Aut(F)$, provided that $F$ has rank at least $3$. I shall begin by sketching the fruitful analogy between automorphism groups of free groups, mapping class groups, and arithmetic lattices, particularly $SL_n({\mathbb{Z}})$. In this analogy, the free factor complex, introduced by Hatcher and Vogtmann, appears as the natural analogue in the $Aut(F)$ setting of the spherical Tits building associated to $SL_n $ and of the curve complex associated to a mapping class group. If $n>2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of the building is $PGL_n({\mathbb{Q}})$. Ivanov proved an analogous theorem for the curve complex, and our theorem complements this. These theorems allow one to identify the abstract commensurators of $GL_n({\mathbb{Z}})$, mapping class groups, and $Out(F)$, as I shall explain.

The study of closed geodesics on a Riemannian manifold is a classical and important part of differential geometry. In 1969 Gromoll and Meyer used Morse - Bott theory to give a topological condition on the loop space of compact manifold M which ensures that any Riemannian metric on M has an infinite number of closed geodesics. This makes a very close connection between closed geodesics and the topology of loop spaces.

Nowadays it is known that there is a rich algebraic structure associated to the topology of loop spaces — this is the theory of string homology initiated by Chas and Sullivan in 1999. In recent work, in collaboration with John McCleary, we have used the ideas of string homology to give new results on the existence of an infinite number of closed geodesics. I will explain some of the key ideas in our approach to what has come to be known as the closed geodesics problem.

By Thurston's geometrisation theorem, the complement of any knot admits a unique hyperbolic structure, provided that the knot is not the unknot, a torus knot or a satellite knot. However, this is purely an existence result, and does not give any information about important geometric quantities, such as volume, cusp volume or the length and location of short geodesics. In my talk, I will explain how some of this information may be computed easily, in the case of alternating knots. The arguments involve a detailed analysis of the geometry of certain subsurfaces.