# Workshop in Geometric Topology

## Schedule and Abstracts

All talks will take place in Room 138 of the Tucker Technology Center on the TCU campus. See the campus map for location of Tucker Technology Center.

The full conference schedule can be found here

## Abstracts of talks:

### Principal lectures by Wolfgang Lück

Introduction to $$L^2$$-invariants
Slides
Abstract: We give an introduction to $$L^2$$-homology and $$L^2$$-Betti numbers which generalizes the well-known classical notions of homology and Betti numbers. They have suprising applications to problems in topology, geometry, and group theory which a priori seem not to be related but whose proofs require $$L^2$$-techniques. We also discuss some open conjectures and will briefly treat $$L^2$$-torsion, which is the analogue of Reidemeister torsion.

Introduction to the Farrell-Jones Conjecture
Slides
Abstract: The Farrell-Jones Conjecture identifies the algebraic K- and L-groups for group rings to certain equivariant homology groups. We will not focus on the technical details of the precise formulation or of the proofs but try to convince the audience about its significance by considering special cases and presenting the surprizing large range of its applications to prominent problems in topology, geometry, and group theory. This talk will be independent of the the first talk.

Some applications to 3-manifolds
Slides
Abstract: Thanks to the recent breakthrough about the Fibering Conjecture due to Agol and others, there are nice applications of L2-invariants to 3-manifolds which also use the Farrell-Jones Conjecture. Essentially we will introduce and relate four invariants which are of different natures: the Thurston norm, the degree of higher order Alexander polynomials, the degree of the L2-torsion function, and a version of the L2-Euler characteristic. We will use some of the results of the first two talks here, but this talk will nevertheless be self-contained.

### Other talks

Ashwini Amarasinghe, University of Florida
On separators of space of non negatively curved planes
Abstract: The spaces of Riemannian metrics with positive scalar curvature are subjects of intensive study. The connectedness properties of such spaces on the plane were studied recently by Belegradek and Hu. We shall prove that the Hilbert cube cannot be separated by a weakly infinite dimensional subset. As a corollary we obtain that the complement of a weakly infinite dimensional subset of the space of complete non negatively curved Riemannian metrics is continuum connected. We can extend this result to the associated moduli space when the set removed is a Hausdorff space with Haver's property C. These results are refinements of theorems proven by Belegradek and Hu.

Michael Andersen,Brigham Young University
The Existence of a Discontinuous Homomorphism Requires a Strong Axiom of Choice
Abstract: Conner and Spencer used ultrafilters to construct homomorphisms between fundamental groups that could not be induced by continuous functions between the underlying spaces. We use methods from Shelah and Pawlikowski to prove that Conner and Spencer could not have constructed these homomorphisms with a weak version of the Axiom of Choice. This led us to define and examine a class of pathological objects that cannot be constructed without a strong version of the Axiom of Choice, which we call the class of inscrutable objects. Objects that do not need a strong version of the Axiom of Choice are scrutable. We show that the scrutable homomorphisms from the fundamental group of a Peano continuum are exactly the homomorphisms induced by a continuous function.

Kyle Austin,University of Tennessee
The Geometry of Scales
Abstract: In the very beginning of the development of topology, topologists were studying uniform spaces which are a kind of global topological space. To put it simply, uniform topology is the study of spaces as one "uniformly looks at the space closer and closer". The dual concept of coarse geometry, which is the study of spaces when you uniformly "view them from greater distances", became popular in the late 20'th century. It has only been recently that mathematicians have been mixing the two concepts and studying coarse geometry and uniform topology alongside each other. J. Dydak, M. Holloway, and I are finishing up three papers which focus of the combination of coarse and uniform concepts and on the duality between the two concepts. I will discuss our results and try and show you how unifying coarse and uniform concepts enriches both simultaneously.

Bryce Chriestenson, University of Heidelberg
Equivariant Moore Approximation and Fiber bundles
Abstract: In this talk we will introduce the notion of an equivariant-Moore approximation to a G-space. We will use such approximations to study fiber bundles whose fibers admit an equivariant-Moore approximations in all degrees. We will show how the existence of such an equivariant Moore approximation implies the collapse of of the Serre sequence at the $$E_2$$ term. Thus, showing how non-zero differentials in the Serre spectral sequence can be viewed as obstructions to the existence of an equivariant Moore decomposition of the fiber. We connect this to work of Banagl, who uses a similar method involving differential forms to give such a collapse, but with very different assumptions.

Eric Freden, Southern Utah University
Growth series for rooted trees
Abstract: A standard approach to computing the growth function for a finitely generated group relies on finding geodesic normal forms for group elements. Usually the set of paths corresponding to these normal forms constitute a rooted spanning tree in the Cayley graph. Such a tree has bounded, but usually not constant, valence. Each Baumslag-Solitar group has Cayley 2-complex homeomorphic to tree cross real line (in fact the Bass-Serre tree). The projection of the 2-complex yields a rooted tree with constant valence (except at the root node). However, for the projection tree to isometrically embed in the 2-complex, the tree edges will carry different (but finitely many and uniformly bounded) weights. The above paragraphs are the motivation for studying growth functions for rooted trees with bounded vertex valences and bounded edge weights. The current inquiry concerns the relationship between recursive schemes for edge weights/vertex valences and recursive algorithms for computing the growth series. If the former takes polynomial time, so does the latter. What about the converse? This presentation gives partial results in that direction and is illustrated with counter-intuitive examples.

Andrew Geng, University of Chicago
Classification and examples of 5-dimensional geometries
Abstract: Thurston's eight homogeneous geometries formed the building blocks of 3-manifolds in the Geometrization Conjecture. Filipkiewicz classified the 4-dimensional geometries in 1983, finding 18 and one countably infinite family. I have recently classified the 5-dimensional geometries. I will review what a geometry in the sense of Thurston is, survey related ideas, and outline the classification in 5 dimensions. Salient features and suspicious patterns will be illustrated using particular geometries from the list. The classification touches a number of topics including foliations, fiber bundles, representations of compact Lie groups, Lie algebra cohomology, Galois theory in algebraic number fields, and conformal transformation groups.

Shijie Gu, University of Wisconsin-Milwaukee
Nested defining sequences and 1 dimensionality of decomposition elements
Abstract: This talk addresses the question of whether a cell-like map from an n-manifold to a finite dimensional space can be approximated by another cell-like map having 1-dimensional point preimages. The normal form for decompositions due to Cannon and Quinn assures this can be done when $$n>4$$ and $$X$$ is an n-manifold except for a singular set $$S(X)$$ of dimension at most $$n-3$$. The same conclusion also holds when S(X) is contained in an $$(n-1)$$-manifold. However, the general case remains unresolved. Here we treat an interesting special case in which the map arises as the decomposition map associated with a nested defining sequence in the domain.This is a joint work with Bob Daverman.

Burns Healy, Tufts University
On groups with the RFRS property
Abstract: In 2008, a large step towards proving Thurston's Virtual Fibering Conjecture was taken by Ian Agol. In a paper entitled "Criteria for Virtual Fibering", he proves that an irreducible 3-manifold with fundamental group satisfying a certain group-theoretic condition called RFRS must be virtually fibered. We explore the relationship the RFRS condition has to certain groups, and look at ways to extend the class of groups which can be shown to be RFRS.

Tyler Hills, Brigham Young University
Warsawanoid - To Be Or Not to Be a Sharkovskii Space
Abstract: Sharkovskii's Theorem is a well-known result in dynamical systems. It provides an interesting, complete ordering of the natural numbers which completely describes the orbits of points under iterations of a continuous function from an interval to itself. If this theorem holds for every continuous function f:M --> M for a space M, then we call M a sharkovskii space. We consider criterion for a space to be sharkovskii, in particular, we look at the Solenoid and Warsawanoid (an analog of the Solenoid, with a Warsaw Circle).

Michael Holloway, University of Tennessee
Duality of Scales
Abstract: The Higson Compactification of a metric space was introduced by Higson in studying the Novikov conjecture. The Higson compactification of a proper metric space X is characterized by the property that a continuous function from X to a compact metric space extends to a continuous function on the Higson compactification if and only if that function is slowly oscillating. In this talk, the notion of a function being slowly oscillating will be extended to maps between coarse spaces and uniform spaces. This provides a means to connect the structures of the domain and codomain of the map and to study properties of both spaces. Using slowly oscillating functions, we also create a Galois connection between the set of coarse structures on the domain and the set of uniform structures on the codomain.

Peter Horn, Syracuse University
Noncommutative knot Floer homology
Abstract: Noncommutative knot Floer homology (ncHFK) is a variant of combinatorial knot Floer homology with an Alexander ‘filtration’ taking values in a nonabelian group. The noncommutative grading lifts the integer-valued Alexander filtration in a certain sense. In this talk I will define the noncommutative Alexander filtration and the chain complex for ncHFK and discuss the difficulties that lie ahead.

Chris Hruska, University of Wisconsin-Milwaukee
Local topology of CAT(0) boundaries
Abstract: It is known, by a theorem of Swarup, that the boundary at infinity of any word hyperbolic group is locally connected. But when is the boundary of a CAT(0) group locally connected? I will give a complete solution to this question in the setting of CAT(0) spaces with isolated flats. In this setting the boundary is an invariant of the group (not depending on which space the group acts on). The solution uses Bowditch's notion of peripheral splittings of a relatively hyperbolic group. This is joint work with Kim Ruane

Qayum Khan, Saint Louis University
Free transformations of $$S^1 \times S^n$$ of prime period
Abstract: Let $$p$$ be an odd prime, and let $$n$$ be a positive integer. We classify the set of equivariant homeomorphism classes of free $$C_p$$-actions on the product $$S^1 \times S^n$$ of spheres, up to indeterminacy bounded in $$p$$. The description is expressed in terms of number theory.
The techniques are various applications of surgery theory and homotopy theory, and we perform a careful study of $$h$$-cobordisms. The $$p=2$$ case was completed by B. Jahren and S. Kwasik (2011). The new issues for the odd $$p$$ case are the presence of nontrivial ideal class groups and a group of equivariant self-equivalences with quadratic growth in $$p$$. The latter is handled by the composition formula for structure groups of A Ranicki (2009).

Sang Rae Lee, Texas A & M University
Twisted conjugacy classes of Houghton's groups
Abstract: Houghton's groups $$H_n$$ consists of translations at infinity of $$n$$ rays of discrete points. We show every automorphism of Houhgton's groups has infinitely many twisted conjugacy classes. This is joint work with JongBum Lee and JangHyun Jo.

Ash Lightfoot, Indiana University
Abstract: To a link map $$f:S^2_+\cup S^2_-\to S^4$$, a map of two 2-spheres in the 4-sphere with disjoint images, Kirk associated a pair of integer polynomials $$\sigma(f)=(\sigma_+(f),\sigma_-(f))$$ that obstructs link homotoping $f$ to an embedding. The obstruction measures whether (up to link homotopy) the self-intersections of each component $$f|S^2_{\pm}:S^2_\pm\to S^4-f(S^2_\mp)$$ occur as pairs of double points equipped with Whitney discs --- the devices for performing the Whitney trick to eliminate self-intersections. A priori these discs may have interior intersections with $$f(S^2_\pm)$$. In this talk I will sketch a proof that if a link map $$f$$ has $$\sigma(f)=(0,0)$$, then after a link homotopy the double points of $$f|S^2_{\pm}$$ can be equipped with Whitney discs whose interiors are, in fact, disjoint from $$f(S^2_\pm)$$. I will then discuss further steps in addressing the open question of whether $$\sigma(f)$$ is a complete obstruction.

Vajira Manathunga, University of Tennessee
A conjecture on amphicheiral knots
Abstract: Detecting chirality is one of main question in knot theory. For this end, various methods have been developed in the past. The polynomial invariants like Jones, HOMFLYPT, Kauffman are the most prominent methods among them. In 2006, James Conant found that for every natural number $$n$$, a certain polynomial in the coefficient of the Conway polynomial is a primitive integer-valued degree $$n$$ Vassiliev invariant, which named as $$pc_n$$. It is conjectured that $$pc_{4n} \mod 2$$ vanishes on all amphicheiral knots. Another way to formulate this conjecture is, if $$K$$ is an amphicheiral knot then there is a polynomial $$F$$ such that $$C(z)C(iz)C(z^2)=F^2$$, where $$F\in Z_4(z^2)$$, $$C(z)$$ is the Conway polynomial of knot $$K$$, and $$i = \sqrt{-1}$$. Using Kawauchi and Hartley's theorems on amphicheiral knots, it can be shown that this is true for all negative and strong positive amphicheiral knots. However the conjecture still remain unsolved for positive amphicheiral knots (not strong). In this talk we summarize our work on this conjecture.

Fedor Manin, University of Toronto
Some maps which are hard to homotope
Abstract: Suppose we have two maps between finite complexes $$X$$ and $$Y$$ which are $$L$$-Lipschitz, or simplicial on a certain subdivision of $$X$$, and which we know are homotopic. To what extent can we control the size of a homotopy between them? In the case where $$Y$$ is simply connected, Gromov (1999) and Ferry-Weinberger (2013) offer a range of conjectures, which I am studying in ongoing work with Chambers, Dotterrer and Weinberger. Rather than attempting to explain our positive results, in this talk I will focus on presenting examples which disprove the most optimistic of the conjectures and demonstrate the richness of the subject.

Kyle Matthews, Texas Christian University
Intersection homology Poincare' duality with coefficient systems
Abstract: We discuss recent advances in Poincare' duality for intersection homology with coefficient systems via cap products. The beginning of the talk will be used to review definitions of pseudomanifolds and intersection homology. If time permits, we will also go over why our result is the best possible using current techniques.

Wiktor Mogilski, University of Wisconsin-Milwaukee
The Weighted Singer Conjecture for Coxeter Groups in Dimensions Three and Four
Abstract: Associated to a Coxeter system (W,S) there is a contractible simplicial complex $$\Sigma$$ called the Davis complex on which W acts properly and cocompactly by reflections. Given a positive real S-tuple Q, one can define the weighted $$L^2$$-(co)homology groups of $$\Sigma$$ and associate to them a nonnegative real number called the weighted $$L^2$$-Betti number. Within the spectrum of weighted $$L^2$$-(co)homology, there is a conjecture of interest called the Weighted Singer Conjecture which was formulated in a 2007 paper of Davis-Dymara-Januszkiewicz-Okun. The conjecture claims that if $$\Sigma$$ is an n-manifold (equivalently, the nerve of the corresponding Coxeter group is an (n-1)-sphere), then the weighted $$L^2$$-(co)homology groups of $$\Sigma$$ vanish above dimension n/2 whenever $$Q\leq 1$$ (that is, all terms of the multiparameter Q are real numbers less than or equal to 1). We present a proof of the conjecture in dimension three that encompasses all but nine Coxeter groups. Then, under some restrictions on the nerve of the Coxeter group, we obtain partial results whenever n=4 (in particular, the conjecture holds for n=4 if the nerve of the corresponding Coxeter group is a flag complex). We then extend our results in dimension four to prove a general version of the conjecture for the case where the nerve of the Coxeter group is assumed to be a flag triangulation of a 3-manifold.

Allison Moore, Rice University
Cosmetic crossing changes in thin knots
Abstract: A classical open problem in knot theory is the Nugatory Crossing Conjecture, which asserts that the only crossing changes which preserve the oriented isotopy class of a knot are nugatory. Knots known to satisfy the conjecture include two-bridge and fibered knots. We will prove that Khovanov-thin knots (a class of knots which includes all alternating and quasi-alternating knots) also satisfy the Nugatory Crossing Conjecture, provided the first homology of the branched double cover decomposes into summands of square-free order. The proof relies on the Dehn surgery characterization of the unknot, a tool coming from Floer homology. This work is joint with Lidman.

Invariants for wild knots
Abstract: It is possible for inequivalent wild knots to have homeomorphic complements; that is unlike the case for tame knots. Some of these examples will be discussed and some invariants for wild knots will be discussed.
Open problems will be presented.

Daniel Ramras, Indiana University-Purdue University Indianapolis
Co-assembly for Bieberbach groups: beyond the Novikov conjecture
Abstract: For a discrete group $$G$$, the (strong) Novikov conjecture states that the assembly map from the K-homology of $$BG$$ to the K-theory of the group $$C^*$$-algebra of G is rationally injective. The deformation K-theory spectrum of $$G$$, built from the category of finite-dimensional unitary representations of $$G$$, admits a co-assembly map to the K-theory of $$BG$$. This map is dual to assembly in the sense that rational surjectivity of co-assembly in high dimensions implies rational injectivity of assembly. These ideas were used in joint work with Willet and Yu to give a proof of the Novikov conjecture for Bieberbach groups (fundamental groups of flat manifolds). In this talk, I'll explain work in progress aimed at proving the co-assembly map is actually a rational isomorphism (in high dimensions) for such groups.

Eric Swenson,Brigham Young University
Recognizing Rank 1 isometries of CAT(0) spaces
Abstract: A rank 1 isometry of a CAT(0) space is an isometry with the classic north south dynamics of a hyperbolic isometry. We give boundary conditions which guarantee an isometry will be rank 1.

Tulsi Srinivasan , University of Florida
The Lusternik-Schnirelmann category of Peano continua
Abstract: We extend the theory of the Lusternik-Schnirelmann category (LS-category) to Peano continua by means of covers by general subsets. We obtain upper bounds for the LS-category of Peano continua by proving analogues to the Grossman-Whitehead theorem and Dranishnikov's theorem, and obtain lower bounds in terms of cup-length, category weight and Bockstein maps. We use these results to calculate the LS-category for some fractal spaces like Menger spaces and Pontryagin surfaces. We compare this definition with Borsuk's shape theoretic LS-category. Finally we discuss some applications of these results to geometric group theory.