Stanford University Topology Progress Seminar 2006-7

Unless otherwise noted, all seminars are on Wednesdays at 4:00 pm in Room 381-T (First floor of Math Building, Bldg 380).
Not to be confused with the stanford topology seminar, the topology progress seminar is intended as a forum for students and postdocs to speak about their work in an informal style.

Winter 2007 Schedule


January 17, 2007

Speaker: Kate Gruher

Title: String topology, fusion, and stacks.

 


January 24, 2007

Speaker: Kate Gruher

Title: String topology, fusion, and stacks (cont.)

Abstract: I will begin with a brief introduction to topological stacks and describe the recent work of Behrend-Ginot-Noohi-Xu constructing string topology operations for inertia stacks. I will then discuss how the Frobenius algebra arising from their work relates to the Frobenius algebras I discussed last week, and to the fusion product on the Verlinde algebra via the Freed-Hopkins-Teleman theorem.


January 31, 2007

Speaker: Kate Gruher

Title: Positive Energy Representations and Fusion


February 7, 2007

Speaker: Ralph Cohen

Title: The Floer homotopy type of the cotangent bundle and string topology

Abstract: I will begin by discussing the general question of when can one realize a chain complex by the cells of a CW complex or spectrum. I will give necessary and sufficient conditions, and will discuss how, when these conditions are satisfied, the Pontrjagin-Thom construction implies that the attaching maps of an underlying stable homotopy type are represented by framed manifolds with corners. In the case of a Floer chain complex, the conditions are basically that the moduli spaces of J-holomorphic cylinders connecting periodic orbits, can have framings (stable trivializations of their tangent bundles) compatible with gluing. I will show why this condition is satisfied in the case of the Floer theory of the cotangent bundle and show that the underlying "Floer homotopy type" is homotopy equivalent suspension spectrum of the free loop space, LM. This extends results of Viterbo. I will then discuss how string topology operations in LM correspond to operations on the Floer theory of the cotangent bundle obtained by counting J-holomorphic curves.


February 14, 2007

Speaker: Ralph Cohen

Title: The Floer homotopy type of the cotangent bundle and string topology (cont.)


February 28, 2007

Speaker: Anders Angel

Title: Orbifold Cobordism

Abstract: Orbifolds, which are like manifolds, except that they locally look like R^n/G, for G a finite group, have all the desirable properties to be able to talk about cobordism between them. The study of the cobordism groups of orbifolds was started by K.Druschel in her thesis during the 90s, where she studied the torsion-free part, by introducing generalized Pontrjagin numbers that determine when a multiple of an orbifold bounds.

Generalizing basic constructions from equivariant cobordism I will explain how I am trying to study the torsion. It is a simple observation that in the unoriented context every orbifold bounds, therefore, it is natural to restrict the type of orbifolds that we study. For orbifolds with groups of odd order we have a decomposition of the cobordism groups in term of bordism groups of classifying spaces of linear groups, this decomposition allows us to define characteristic numbers that determine the cobordism class of an orbifold with groups of odd order. As a corollary we have that if an orbifold bounds another orbifold with groups of odd order, then it actually bounds a manifold.

For oriented cobordism of orbifolds, I will introduce a spectral sequence that allows us to prove that every two and three dimensional oriented orbifold bounds, a result previously obtained by K.Druschel.

In the last part of this lecture I will introduce orbibordism groups of an orbifold and make some conjectures about the relation between these groups and equivariant cobordism.


March 7, 2007

Speaker: Sverre Lunoe-Nielson

Title: The homotopy limit property of Z/p acting on B^p

Abstract: Let B be a bounded below spectrum of finite type over F_p. Then we can form a C_p-equivariant spectrum B^p by smashing B with itself p times and let C_p act by permuting the smash factors.

For any group G and for any G-equivariant spectrum X, there is a map from the strict fixed points of X to the homotopy fixed points. It is an interesting problem to study when this map is an equivalence.

We will see that for many non-trivial spectra B and the action of C_p on B^p, these two notions of fixed points spectra are homotopy equivalent after p-completion.

I hope to spend some time convincing the audience that this is a "surprising result", at least from the point of view of the homotopy fixed point spectral sequence.

The result relies on the cohomological calculations done by Lin and Gunawardena in connection to the Segal conjecture for cyclic groups of prime order, which considers the case of the sphere spectrum B=S.


March 14, 2007

Speaker: Sverre Lunoe-Nielson

Title: The homotopy limit property of Z/p acting on B^p (cont.)


April 4, 2007

SPECIAL TOPOLOGY SEMINAR

Speaker: Ernesto Lupercio (CINVESTAV, Mexico City)

Title: Chen-Ruan cohomology of cotangent orbifolds as orbifold string topology

Abstract: Abstract: In this talk we will report on the paper

math.AT/0610899 Chen-Ruan Cohomology of cotangent orbifolds and Chas-Sullivan String Topology. Ana Gonzalez, Ernesto Lupercio, Carlos Segovia, Bernardo Uribe, Miguel A. Xicotencatl. (to appear Math Res Lett)

In this paper we prove that for an almost complex orbifold, its virtual orbifold cohomology [math.AT/0606573] (related to string topology) is isomorphic as algebras to the Chen-Ruan orbifold cohomology (related to Floer cohomology) of its cotangent orbifold.


April 11, 2007

Speaker: Soren Galatius

Title: Point set topology of spaces of graphs

Abstract: There is a sheaf Phi on R^N such that Phi(U) is the set of subsets of U which are graphs (i.e. closed subsets that locally look like R or like a wedge of half-open intervals). There is a topology on Phi(U) which makes it into a sheaf of topological spaces, and as such Phi is an important object in determining the stable homology of Aut(F_n). In this talk I will define the topology on Phi(U) and prove various point-set topological properties.


April 18, 2007

Speaker: Soren Galatius

Title: Point set topology of spaces of graphs (cont.)


April 25, 2007

AREA EXAM

Speaker: Isidora Milin

Title: Orderability and (non)squeezing for standard contact lens spaces

Abstract: We say that a contact isotopy of a compact contact manifold is "positive" if during the isotopy each point of the manifold moves in a positively transverse direction to the contact structure. The question of whether this natural notion induces a partial order on the universal cover of the contactomorphism group turns out to be sensitive to the topology of the contact manifold, and is related to nonsqueezing phenomena in contact geometry, as studied by Eliashberg, Kim and Polterovich.

I will begin by describing this relation, and continue by introducing an equivariant version of cylindrical contact homology for domains. This will be used to detect contact nonsqueezing leading to orderability in the case of standard contact lens spaces. This result should be contrasted with the case of the standard contact sphere, where the answer to orderability question is negative.

Towards the end, I will mention a possible generalization of this result to arbitrary prequantization spaces, explain what the analogue of the classical notion of rotation number for a circle diffeomorphism is in this context and how one might try to study this analogue further.


May 2, 2007

SPECIAL TOPOLOGY SEMINAR

Speaker: Boris Chorny, Australian National University, Canberra

Title: Brown representability for space-valued functors

Abstract: In this talk we will discuss two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors.

In more detail, we will show that every small contravariant functor from spaces to spaces converting coproducts to products, up to homotopy, and taking homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. This theorem may be considered as a contravariant analog of Goodwillie's classification of linear functors, see his [Calculus II, III] papers. The interpretation of the current result in terms of Homotopy Calculus is still a challenge.

Homological representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets is equivalent to a restriction of a representable functor. This theorem is essentially equivalent to Goodwillie's classification of linear functors.


May 9, 2007

Bergman Lecture: Dennis Sullivan


May 30, 2007

Speaker: Andres Angel

Title: When is a manifold the boundary of an orbifold?

Abstract: The question When is a manifold the boundary of another manifold? was settled by R.Thom in the fifties, the necessary and sufficient condition is the vanishing of characteristic numbers (Stiefel-Whitney number if we do not consider orientations and Pontrjagin and Stiefel-Whitney when take them into account). His construction involves the identification of the cobordism groups with the homotopy groups of certain Thom spectra.

It is a simple observation that every manifold, and even every orbifold, is the unoriented boundary of another orbifold, therefore it is natural to restrict the type of orbifolds that we allow. In this talk I will present a decomposition of the cobordism ring of orbifolds with isotropy groups of odd order in terms of bordism groups of classifying spaces of linear groups. This decomposition will allow us to prove that if a manifold bounds and orbifold with isotropy groups of odd order then it actually bounds a manifold.

For the corresponding question with orientations, I will present a similar decomposition of the rational cobordism ring of oriented orbifolds, the conclusion in this case is that if an oriented manifold bounds an oriented orbifold, then some multiple of the manifold bounds an oriented manifold.