Topology Seminar

Seminars During Academic Year 2008-09

Fall 2008

Wednesday September 17, 2008 in Rawles Hall 104

Matvei Libine, Equivariant cohomology and Berline-Vergne localization formula

Wednesday September 17, 2008 in Rawles Hall 104

Matvei Libine, Equivariant cohomology and Berline-Vergne localization formula

The notion of equivariant cohomology plays an important role in symplectic geometry, algebraic geometry, representation theory and other areas of mathematics. Let G be a compact Lie group acting on a topological space M. For the topologists, the equivariant cohomology of M is defined to be the ordinary cohomology of the space (M x E)/G, where E is any contractible topological space on which G acts freely. (This definition does not depend on the choice of E.) If M is a smooth manifold there is an alternative way of defining the equivariant cohomology groups of M involving de Rham theory. My goal is to explain the Berline-Vergne localization formula (which is a generalization of the Duistermaat-Heckman Theorem) expressing integrals of equivariant forms as sums over fixed points. In a future talk I will describe my own generalization of this formula.

Wednesday September 24, 2008 in Rawles Hall 104

Bert Guillou, Massey products and the motivic fundamental group of the punctured projective line

We will discuss, following a suggestion of S. Bloch, the construction of certain mixed Tate motives associated to the projective line minus the points 0, 1, and infinity. In the end, this will reduce to the question of producing a cellular approximation of a certain dg-module over a dga, and we will see that the construction of this cell module is controlled by Massey products in the motivic cohomology of the ground field.

Wednesday October 8, 2008 in Rawles Hall 104

Allan Edmonds, A survey of group actions on 4-manifolds

A survey of almost 50 years of locally linear, compact and finite group actions on topological 4-manifolds, with an emphasis on history, guiding questions, and open problems, and highlighting some challenges for the existence of smooth actions. A version of a talk presented at the conference on Transformation Groups in Topology and Geometry, University of Massachusetts at Amherst, July 2008.

Wednesday October 15, 2008 in Rawles Hall 104

Allan Edmonds, Survey of group actions on 4-manifolds, Part 2

A continuation of last week's talk, focussing mainly on existence and classification of finite group actions on 4-manifolds.

Wednesday October 22, 2008 in Rawles Hall 104

Steve Mitchell, The Topology of Birkhoff Varieties

In his study of singularities of systems of ODE's (c.1910), Birkhoff introduced the decomposition $$SL_n {\Bbb C} [z,z^{-1}] =\coprod _{\lambda} P^- \lambda P ,$$ where $P=SL_n {\Bbb C} [z]$, $P^-=SL_n {\Bbb C} [z^{-1}]$, and $\lambda$ ranges over the diagonal matrices $diag (z^{a_1},...z^{a_n})$ with $a_1 \geq a_1 \geq ...\geq a_n$ and $\sum a_i =0$. The Birkhoff decomposition can be reinterpreted as a classification of holomorphic vector bundles over ${\Bbb P} ^1$ (due to Grothendieck, 1958). Yet another interpretation involves passing to the {\it affine Grassmannian} ${\cal L} =SL_n {\Bbb C} [z,z^{-1}] /P$, which has the structure of an infinite-dimensional projective variety. Then the $P^-$-orbits define the {\it Birkhoff stratification} of ${\cal L}$. The closure $Z_\lambda$ of an orbit is a {\it Birkhoff variety}. These are analogous to the “opposite” or “dual” Schubert varieties in an ordinary Grassmannian, except that instead of having finite dimension, $Z_\lambda$ has finite codimension. In this talk I will describe recent joint work with Luke Gutzwiller on the topology of Birkhoff varieties. Our main result is that the inclusion of a Birkhoff variety in the affine Grassmannian is a homotopy equivalence. This result holds in all Lie types, and also for a finer version of the Birkhoff decomposition that I will define.

Wednesday November 12, 2008 in Rawles Hall 104

Yazinski, Fundamental Groups of Symplectic 4-Manifolds

Wednesday December 3, 2008 in Rawles Hall 104

Kate Ponto, Traces and fixed points

We can associate an integer called the Lefschetz number to each endomorphism of a closed manifold. The Lefschetz fixed point theorem states that the Lefschetz number of an endomorphism with no fixed points is zero. This theorem is a consequence of the identification of the Lefschetz number, a global invariant, with a local invariant. There are many proofs of this identification, one uses the trace in a symmetric monoidal category. For closed smooth manifolds of dimension at least three, the Lefschetz number of an endomorphism can be refined to an invariant that is zero if and only if the endomorphism has no fixed points. A major step in the proof of this result is the identification of a local invariant with a global invariant. I will explain how to prove this identification using the trace in a bicategory. I will also explain how to use this perspective to define equivariant and fiberwise fixed point invariants.

Wednesday December 10, 2008 in Rawles Hall 104

Daniel Ramras, Gauge theory and homotopical representation theory

To every representation rho: Gamma –> G of a discrete group Gamma into a Lie group G, there is an associated map of classifying spaces Brho: BGamma –> BG, which classifies the G-bundle associated to rho. When Gamma admits a compact manifold as its classifying space, this correspondence can be studied using gauge theory. The resulting computations show interesting relationships between representation spaces of discrete groups, cohomology, and K-theory, and suggest analogies with the Atiyah-Segal Theorem and the Quillen-Lichtenbaum conjectures.

Tuesday December 16, 2008 in Rawles Hall 104

Victor Ostrik, Graded fusion categories

This is a joint work with P.Etingof and D.Nikshych. An additive tensor category C is called graded by a group G if there is a direct sum decomposition C=oplus_{g in G}C_g with C_g otimes C_h subset C_{gh}. This definition implies that a component _e corresponding to the unit element e of G is a tensor subcategory of C and we say that C is a G-graded extension of C_e. In this talk I will describe a theory which aims to describe all graded extensions of a given tensor category in a class of fusion categories (which are in a sense simplest additive tensor categories).

Spring 2009

Wednesday January 14, 2009 in Rawles Hall 104

John Klein, Higher torsion invariants.

will outline the current state of the theory of higher Reidemeister torsion, and discuss examples and open problems.

Wednesday January 21, 2009 in Rawles Hall 104

Chuck Livingston, The Stable 4-Genus of Knots

The 4-genus g(K) of a knot K is the minimal genus of a surface bounded by K in the 4-ball. The limit g(nK)/n as n goes to infinity exists and is called the stable 4-genus. I will discuss examples and properties of this invariant.

Wednesday February 4, 2009 in Rawles Hall 104

John Bryden, The linking form conjecture for 3-manifolds

The linking form conjecture states that any non-degenerate, symmetric bilinear form on a finite abelian group (into Q/Z) is isomorphic to the linking form of a Seifert manifold that is a rational homology sphere. As an immediate consequence it would show that any closed, connected, oriented 3-manifold has linking form isomorphic to the linking form of a Seifert manifold. This may be somewhat surprising. This conjecture arose when Florian Deloup and I were investigating a certain class of quantum invariants. What is more surprising is that the proof arises from homotopy theory.

Wednesday February 18, 2009 in Rawles Hall 104

John Bryden, The linking form conjecture for 3-manifolds

The linking form conjecture states that any non-degenerate, symmetric bilinear form on a finite abelian group (into Q/Z) is isomorphic to the linking form of a Seifert manifold that is a rational homology sphere. As an immediate consequence it would show that any closed, connected, oriented 3-manifold has linking form isomorphic to the linking form of a Seifert manifold. This may be somewhat surprising. This conjecture arose when Florian Deloup and I were investigating a certain class of quantum invariants. What is more surprising is that the proof arises from homotopy theory.

Wednesday February 25, 2009 in Rawles Hall 104

Mike Mandell, Localization Sequences in THH and TC

I will outline the relationship of THH and TC to algebraic K-theory and describe the localization results referred to in Hesselholt's abstract for his lecture next week.

Wednesday March 4, 2009 in Rawles Hall 104

Lars Hesselholt, Vanishing of negative K-groups

A conjecture of Weibel predicts that the Bass negative K-groups of a noetherian scheme X of dimension d vanish in degrees i < - d. A few years ago, the conjecture was verified for schemes separated and essentially of finite type over a field of characteristic 0 by Cortinas-Haesemeyer-Schlichting-Weibel. I will discuss recent work with Thomas Geisser which establishes the conjecture for schemes separated and essentially of finite type over a perfect field of characteristic p > 0 under the assumption of resolution of singularities. The proof in characteristic 0 uses a comparison of K-theory to Connes' cyclic homology. Similarly, the proof in characteristic p > 0 uses a comparison of K-theory to the topological cyclic homology of Bokstedt-Hsiang-Madsen. In both cases, the starting point of the proof is a theorem of Haesemeyer which allows one to show that the difference between K-theory and the respective cyclic theories satisfies descent with respect to Voevodsky's cdh-topology. The proofs are completed by careful analysis of the cyclic theories. A central technical component in the proof is the recent extension from rings to exact categories of the definition of the respective cyclic theories by Keller and Blumberg-Mandell.

Thursday March 5, 2009 in Rawles Hall 104

Tim D. Cochran, New Characteristic Series for Groups and the Fractal Nature of Knot Concordance

For each sequence P={p_1(t),...} of polynomials we define a characteristic series of groups, called the derived series localized at P. Given a knot K, such a sequence of polynomials can arise naturally as the orders of certain submodules of the sequence of higher-order Alexander modules of K. These also yield filtrations of the knot concordance group that refine the n-solvable filtration. We show that the quotients of successive terms of these refined filtrations have infinite rank. These give higher-order analogues of the p(t)-primary decomposition of the algebraic concordance group. We also use these to give evidence that the set of smooth concordance classes of knots is a fractal set.

Wednesday March 11, 2009 in Rawles Hall 104

Vladimir Touraev, Bundles over surfaces from quantum viewpoint

A classical problem in the theory of fiber bundles is to detect whether or not the given fiber bundle has a section. When the base of the bundle is a closed orientable surface and the fundamental group G of the fiber is finite, a complete answer is formulated in terms of characteristic classes of the bundle derived from linear representations of G. The proofs use the techniques of 2-dimensional topological quantum field theory.

Wednesday April 1, 2009 in Rawles Hall 104

Jim Davis, Mapping tori of self-homotopy equivalences of lens spaces

Alternate titles include: 1) There are no exotic beasts in Hillman's zoo 2) Homotopy fibrations and a lemma of Gauss

Wednesday April 15, 2009 in Rawles Hall 104

Teena Gerhardt, Algebraic K-theory of the Dual Numbers

Nearly 30 years ago, Soule showed that the abelian group K_n(Z[x]/x^2, (x)) is finitely generated with rank 0 if n is even and 1 if n is odd. We compute this group for n odd and find its order for n even. Further, we generalize these results to the study of the algebraic K-theory of truncated polynomial algebras, K_n(Z[x]/x^e, (x)). This is joint work with Vigleik Angeltveit and Lars Hesselholt.

Wednesday April 22, 2009 in Rawles Hall 104

John Bryden, Group cohomological methods in low dimensional topology

Wednesday April 29, 2009 in Rawles Hall 104

Allan Edmonds, Groups that act pseudofreely on spheres

Thursday April 30, 2009 in Rawles Hall 104

Matt Hedden, Using Heegaard Floer homology to show that Khovanov homology of satellites detects the unknot

Wednesday May 6, 2009 in Rawles Hall 104

Chuck Livingston, Slice knots not concordant to polynomial 1 knots

Wednesday May 20, 2009 in Rawles Hall 104

Qingxiang Zhang, Dehn surgery on two bridge knots and two generator Klienian groups

Let < f, g > be a two generator Klienian group. We define B(f)=tr^2(f)-4, B(g)=tr^2(g)-4,G(f, g)=tr[f, g]-2 where tr(f) and tr(g) denote the traces of representive matrices of f and g respectively and [f, g] denotes the multiplicative commutator fgf^{-1}g^{-1}. These three complex numbers are called the parameters of the two generator group < f, g>, we write (< f, g>)=(G(f, g),B(f), B(g)). We discuss the G(f, g) plane for g is elliptic of order 2 and f is elliptic of order p for p=3, 4, 5, 6, 7.