Notre Dame Topology Seminar

Spring 2010 Topology Seminar
Graduate Topology/Geometry Seminar
Notre Dame Department of Mathematics

Questions? Contact the organizer, Qayum Khan

Fall 2010 Schedule

Unless otherwise indicated in red, fall talks occur on Thursdays during 3pm--4pm (Eastern Time) in 258 Hurley Hall.

September 23: Justin Thomas (U Notre Dame); The Phantom of the Operad, I
September 30: Justin Thomas (U Notre Dame); The Phantom of the Operad, II
October 7: Justin Thomas (U Notre Dame); The Phantom of the Operad, III
October 14: Justin Thomas (U Notre Dame); The Phantom of the Operad, IV
October 15 at 12PM in H258: Margaret Doig (Indiana U); Heegaard Floer Homology and Knot Surgery
~~October 21~~: NO SEMINAR; Notre Dame Fall Break
October 28: Justin Thomas (U Notre Dame); The Phantom of the Operad, V
November 4: Nathaniel Rounds (Purdue U); Local Poincaré Duality and the Algebraic Structure of Topological Manifolds
November 11: Julia Bergner (U California @ Riverside); Homotopy-theoretic Approaches to Higher Categories
November 18: Rosona Eldred (U Illinois); An Equivalence of Towers
~~November 25~~: NO SEMINAR; Thanksgiving Break
December 2: Dan Lior (U Illinois); A Generalization of Robinson's Bicomplex for the Stable Homotopy of Γ-modules
December 9: Grigori Avramidi (U Chicago); Isometry Groups and Aspherical Manifolds
~~December 16~~: NO SEMINAR; Notre Dame Final Exam Week

Spring 2011 Schedule

Unless otherwise indicated in red, spring talks occur on Thursdays during 12:45pm--1:45pm (Eastern Time) in 258 Hurley Hall.

January 20: Lizhen Qin (Wayne State U); On Moduli Spaces and CW Structures Arising from Morse Theory on Hilbert Manifolds
January 27: Karsten Grove (U Notre Dame); A Knot Characterization and Non-negatively Curved 4-manifolds with Circle Symmetry
February 3: William Dwyer (U Notre Dame); Localization: a Survey, I
February 10: William Dwyer (U Notre Dame); Localization: a Survey, II
February 17: Wolfgang Ziller (U Pennsylvania); Obstructions to Fatness
February 24: Bruce Williams (U Notre Dame); Family Hirzebruch Signature Theorem with Converse
March 3: Shmuel Weinberger (U Chicago); Aspherical Manifolds Whose Fundamental Groups Have Center
March 10: Frank Connolly (U Notre Dame); Involutions on Tori and Topological Rigidity
~~March 17~~: NO SEMINAR; Notre Dame Spring Break
March 24: Daniel Ramras (New Mexico State U); Quillen--Lichtenbaum Phenomena in Stable Representation Theory
March 31 at 10AM in H258: Wolfgang Steimle (U Bonn, GERMANY); Obstructions to Stably Fibering Manifolds
March 31: Bjørn Dundas (U Bergen, NORWAY); Integral Excision in Algebraic K-theory
April 7: Ayelet Lindenstrauss (Indiana U); K-Theory of Formal Power Series
April 14: Kate Ponto (U Kentucky); Coincidence Invariants
April 21: James Davis (Indiana U); Some Remarks on Nil-groups in Algebraic K-theory
April 28: Allan Edmonds (Indiana U); Klein Bottles and Multidimensional Fixed Point Sets
May 3 at 12:15PM in H258: Dmitri Pavlov (U California-Berkeley); Jones Index via a Symmetric Monoidal Bicategory of von Neumann Algebras
May 5: David Rosenthal (St. John's University); On the Asymptotic Dimension of Groups
~~May 12~~: NO SEMINAR; Notre Dame Final Exam Week

Abstracts of Invited Talks

September 23 & 30, October 7 & 14 & 28, 2010: Justin Thomas

The Phantom of the Operad

I will begin with an elementary introduction to operads. It will include the definition of the associative and commutative operads, the E_n operads, and the A_∞ operad of Stasheff. We will move on to the Hochschild cohomology of associative algebras. We will discuss the Gerstenhaber algebra structure on Hochschild cohomology and how it leads naturally to Deligne's conjecture. Finally, we will describe the Swiss cheese operad, the notion of an E_n algebra acting on an E_n+1 algebra, and a proof of Kontsevich's Swiss cheese conjecture.

October 15, 2010: Margaret Doig

Heegaard Floer Homology and Knot Surgery

Heegaard Floer theory has something to say about a wide variety of questions in low-dimensional topology. For example, while it is well known that any 3-manifold can be obtained by Dehn surgery on a link, it is currently unknown which manifolds can be obtained from a knot or which knots can produce them. We will discuss this question for spherical manifolds (other than lens spaces) using the Heegaard Floer correction terms associated to a 3-manifold Y and its torsion Spin^c structures. If H₁(Y) is small, the correction terms completely identify the Y which can be realized as knot surgeries and place restrictions on the knots; for those Y with larger H₁(Y), the invariants still provide useful information.

November 4, 2010: Nathaniel Rounds

Local Poincaré Duality and the Algebraic Structure of Topological Manifolds

Can we associate an algebraic structure to a manifold such that this structure up to equivalence determines the manifold up to homeomorphism? If we replace the word "homeomorphism" with the word "homotopy equivalence", the answer is yes. We will present results due to Mandell, McClure, Wilson, and Chataur showing describing various intricate algebraic structures on a manifold's chains and cochains, all of which prove to be homotopy invariant. We will suggest, however, that the missing idea is that of algebraic locality. The various algebraic structures that we associate to a manifold are all local in an appropriate sense, but the inverse to the Poincaré duality map need not be local. We will show, using Ranicki's algebraic surgery, that considering the inverse to the Poincaré duality map leads to a topological invariant of manifolds. We will end with a conjectural synthesis of all these ideas which gives an affirmative answer to the opening question.

November 11, 2010: Julia Bergner

Homotopy-theoretic Approaches to Higher Categories

Several models for (∞,1)-categories have been defined and shown to be equivalent, and they are all being used in different areas of algebra and topology. More recently, there has been interest in more general (∞,n)-categories, especially with Lurie's recent work on the Cobordism Hypothesis. Comparison of different definitions is still work in progress by several authors. In this talk, we will go over some of the models for (∞,1)-categories and discuss some of the methods for inductively generalizing them to models for (∞, n)-categories.

November 18, 2010: Rosona Eldred

An Equivalence of Towers

By using formal properties of the theory of Calculus of Functors to spectra, in joint work with Bauer, Johnson, and McCarthy, we showed that two constructions suggested by the observations of Waldhausen and Rezk for de Rham cohomology for E_∞ algebras always agree. Recently, Goodwillie suggested a possible equivalence of homotopy limit towers that would imply a strengthening of the results, extending to spaces and not just spectra. I will present a proof of this statement and some of its consequences.

December 2, 2010: Dan Lior

A Generalization of Robinson's Bicomplex for the Stable Homotopy of Γ-modules

Alan Robinson described a bicomplex for the stable homotopy of a Γ-module. Randy McCarthy et. al. developed the discrete Taylor tower of a Γ-module (an analogue of the Goodwillie-Taylor tower). It turns out that the stable homotopy groups of a Γ-module F are the same as the homology groups of the first layer D₁F of F in its discrete Taylor tower. We present a unified method for generating bicomplexes for the higher layers D_kF in the discrete Taylor tower of F.

December 9, 2010: Grigori Avramidi

Isometry Groups and Aspherical Manifolds

Borel showed that for a large class of compact locally symmetric spaces, the standard metric has the most symmetries. Farb and Weinberger gave a criterion for detecting locally symmetric spaces among compact aspherical Riemannian manifolds. These results relate the fundamental group of a compact aspherical manifold to its isometry group (in the first case) and the isometry group of its universal cover (in the second case). I'll describe extensions of these results to some non-compact aspherical manifolds.

January 20, 2011: Lizhen Qin

On Moduli Spaces and CW Structures Arising from Morse Theory on Hilbert Manifolds

This talk will focus on various results concerning the moduli spaces of broken flow lines associated with a Morse function on a Hilbert-Riemannian manifold. We study the compactness of flow lines, manifold structures of certain compactified moduli spaces, orientation formulas, and CW structures on the underlying manifold. I will end the talk with questions concerning how to generalize these results to other more difficult situations.

January 27, 2011: Karsten Grove

A Knot Characterization and Non-negatively Curved 4-manifolds with Circle Symmetry

By work of W.Y. Hsiang and B. Kleiner, utilizing Freedman's classification of simply connected 4-manifolds, it has been known for over 20 years that a closed positively curved simply connected 4-manifold with infinite isometry group, i.e. having circle symmetry, is homeomorphic to either S⁴ or CP². In the case of nonnegative curvature independent work of Kleiner and of Searle--Yang show that in addition S²×S², CP²#CP² and CP²#-CP² occur.

A new construction in, and use of Alexandrov geometry combined with the Poincaré conjecture allows to provide a complete classification of simply connected closed 4-manifolds with nonnegative curvature and circle symmetry up to equivariant diffeomorphism. A new(?) knot characterization plays a crucial role.

This is joint work with Burkhard Wilking.

February 3 & 10, 2011: William Dwyer

Localization: a Survey

This will be a survey of the concept of localization, as it appears in topology and in homological algebra. I'll talk about the meaning of the concept, give examples, describe how localizations are constructed, and maybe discuss how they are used. If time permits, I'll also talk about the dual notion (which isn't called globalization!).

Februrary 17, 2011: Wolfgang Ziller

Obstructions to Fatness

A.Weinstein introduced the concept of fat principle bundles and sphere bundles in an attempt to understand conditions for the existence of Kaluza-Klein metrics (connection metrics) with positive sectional curvature. He also observed that there are some obstructions. We explore the precise conditions that such obstructions put on Chern classes and Pontrjagin classes.

February 24, 2011: Bruce Williams

Family Hirzebruch Signature Theorem with Converse

Let X be a space which satisfies 4k-dimensional Poincaré Duality, and let σ(X) be the signature of X. If X is a manifold, then σ(X) can be "disassembled," i.e. σ(X) is determined by a local invariant, the Hirzebruch L-polynomial. Ranicki has given an enriched version of σ(X) which is defined in all dimensions. If dim(X) > 4, Ranicki's enriched version can be disassembled if and only if X admits topological manifold structure. There is a further enrichment which yields an analogous result for families of spaces, i.e. fibrations.

This is joint work with Michael Weiss.

March 3, 2011: Shmuel Weinberger

Aspherical Manifolds Whose Fundamental Groups Have Center

Aspherical manifolds are manifolds whose universal covers are contractible. The first goal of this talk is to explain the central role that these play in topology, at least at the level of conjecture. A key role is played by analogies to the rigidity theory of locally symmetric spaces, but this mirror has several cracks (and more are believed -- at least by me -- to exist). The second main goal of the talk is to explain some constructions of non-classical aspherical manifolds and how these crack the mirror (i.e. lead to counterexamples to some of these problems), and change one's expectations about others.

The talk will be based on work of A.Borel, M.Gromov, I.Piatetski-Shapiro, C.T.C.Wall, S.Cappell, M.Davis, T.Januszkiewicz, P.Conner, F.Raymond, J.Block, J. Fowler, M.Yan, T.Farrell, L.Jones, A.Bartels, W.Lück, and of mine, and will more aim to describe the landscape than give detailed constructions.

March 10, 2011: Frank Connolly

Involutions on Tori and Topological Rigidity

How many topological involutions on the n-dimensional torus have an isolated fixed point?

We prove that there is only one involution on the n-torus Tⁿ, up to conjugacy, for which the fixed set contains an isolated point. But here, n must be of the form 4k or 4k+1, or n must be less than 6. In the other dimensions, we classify all such involutions, using surgery theory and the calculation of the groups UNil_n+1(Z;Z,Z). We also discuss a Topological Rigidity Conjecture, and we show that the above result is a consequence of it.

This is joint work with Jim Davis and Qayum Khan.

March 24, 2011: Daniel Ramras

Quillen--Lichtenbaum Phenomena in Stable Representation Theory

In the early 1960's, Atiyah, Hirzebruch, and Segal constructed and studied a mapping from the representation ring of a compact Lie group G to the K-theory of the classifying space BG. For infinite discrete groups, an analogous map exists on the level of representation spaces and their associated deformation K-theory spectra.

Computations have shown that in certain interesting cases, this map is an equivalence on highly connected covers. This situation is closely analogous to the Quillen--Lichtenbaum conjectures in algebraic K-theory, which are known to fail in low dimensions. In this case, the low-dimensional failure admits a concrete geometric explanation, relying on methods from differential and algebraic geometry.

This is joint work with Tom Baird.

March 31, 2011: Wolfgang Steimle

Obstructions to Stably Fibering Manifolds

Given a map f : M → B between compact topological manifolds, is it homotopic to the projection map of a fiber bundle whose fibers are compact manifolds? Obstructions in higher algebraic K-theory to fibering the given map f will be defined. The vanishing of these obstructions has a concrete geometrical meaning: the obstructions are zero if and only if f fibers stably, i.e. after crossing M with a high-dimensional disk. The methods also provide a classification of the different ways of stably fibering f in terms of algebraic K-theory.

March 31, 2011: Bjørn Dundas

Integral Excision in Algebraic K-theory

Algebraic K-theory has good localization properties, but in the presence of singularities there is a highly nontrivial obstruction to excision. For instance, if a scheme is obtained by gluing two closed subschemes, there is in general no exact Mayer-Vietoris sequence.

However, the necessary correction term is available through variants of cyclic homology, at least in the affine case. This was proved after rationalization by Cortinaz and after profinite completion by Geisser and Hesselholt for the discrete case.

We will discuss the integral case for ring spectra: If a square of ring spectra satisfies hypotheses essentially saying that it is opposite to a gluing of closed embeddings of schemes, then the integral cyclotomic trace K → TC induces an equivalence of correction terms. In other words, the homotopy fiber of the cyclotomic trace satisfies "excision for closed embeddings".

So, if one is able to calculate TC in a given situation, this means that its K-theory is accessible through assembling the K-theories of simpler closed subspaces. The theorem works equally well for the non-commutative case.

This is joint work with Harald Kittang.

April 7, 2011: Ayelet Lindenstrauss

K-Theory of Formal Power Series

We study the algebraic K-theory of parametrized endomorphisms of a unital ring R with coefficients in a simplicial R-bimodule M, and compare it with the algebraic K-theory of the ring of formal power series in M over R.

Waldhausen defined an equivalence from the suspension of the reduced Nil K-theory of R with coefficients in M to the reduced algebraic K-theory of the tensor algebra T_R(M). Extending Waldhausen's map from nilpotent endomorphisms to all endomorphisms, our map has to land in the ring of formal power series rather than in the tensor algebra, and is no longer in general an equivalence (it is an equivalence when the bimodule M is connected). Nevertheless, the map shows a close connection between its source and its target: it induces an equivalence on the Goodwillie Taylor towers of the two (as functors of M, with R fixed), and allows us to give a formula for the suspension of the invariant W(R;M) (which can be thought of as Witt vectors with coefficients in M, and is what the Goodwillie Taylor tower of the source functor converges to) as the inverse limit, as n goes to infinity, of the reduced algebraic K-theory of T_R(M)/Mⁿ.

This is joint work with Randy McCarthy.

April 14, 2011: Kate Ponto

Coincidence Invariants

A fixed point of a continuous endomorphism f of a topological space X is a point x in X so that f(x)=x. A coincidence point of a pair of continuous maps f and g from X to Y is a point x in X so that f(x)=g(x). Coincidence points are a natural generalization of fixed points. I will explain how the formal structure that describes the Lefschetz fixed point theorem also describes a corresponding theorem for (some) coincidence invariants.

April 21, 2011: James Davis

Some Remarks on Nil-groups in Algebraic K-theory

We show how the Farrell--Jones Conjecture in K-theory for Zⁿ strengthens the Fundamental Theorem of Algebraic K-theory and allows one to express the iterated Nil-groups in terms of the non-iterated Nil-groups. A sample corollary is:
If K_qR[x] = K_qR and K_q-1R[x] = K_q-1R then K_qR[x,y] = K_qR.

April 28, 2011: Allan Edmonds

Klein Bottles and Multidimensional Fixed Point Sets

Standard group actions on familiar aspherical manifolds such as tori have the property that all components of the fixed point set have the same dimension. We explore the possibilities for finite group actions on other aspherical manifolds with multi-dimensional fixed point sets, especially in low dimensions.

May 3, 2011: Dmitri Pavlov

Jones Index via a Symmetric Monoidal Bicategory of von Neumann Algebras

I will describe a new symmetric monoidal structure on the bicategory of von Neumann algebras, bimodules and intertwiners, which is motivated by conformal and Euclidean field theories. I will then demonstrate how the bicategorical formalism of shadows of 1-morphisms and traces of 2-morphisms developed by Ponto and Shulman yields the Jones index in a purely categorical way.

May 5, 2011: David Rosenthal

On the Asymptotic Dimension of Groups

The interest in Gromov's notion of finite asymptotic dimension for a finitely generated group stems from its role in establishing the Novikov Conjecture for such groups. The purpose of this talk is to give a brief overview of asymptotic dimension and to use Roe's generalization of this concept to define the asymptotic dimension of a topological group.

This is joint work with Andrew Nicas.