Stanford
University Topology Progress Seminar 20067 Unless otherwise noted,
all seminars are on Wednesdays at 4:00 pm in Room 381T
(First floor of Math Building, Bldg 380). 

Winter 2007 Schedule
Speaker:
Kate Gruher
Title:
String topology, fusion, and stacks.
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 BehrendGinotNoohiXu 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 FreedHopkinsTeleman theorem.
Speaker: Kate Gruher
Title: Positive Energy Representations and Fusion
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 PontrjaginThom 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 Jholomorphic 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 Jholomorphic curves.
Speaker: Ralph Cohen
Title: The Floer homotopy type of the cotangent bundle and string topology (cont.)
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 torsionfree 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.
Speaker: Sverre LunoeNielson
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_pequivariant 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 Gequivariant 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 nontrivial spectra B and the action of C_p on B^p, these two notions of fixed points spectra are homotopy equivalent after pcompletion.
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.
Speaker: Sverre LunoeNielson
Title: The homotopy limit property of Z/p acting on B^p (cont.)
SPECIAL TOPOLOGY SEMINAR
Speaker: Ernesto Lupercio (CINVESTAV, Mexico City)
Title: ChenRuan cohomology of cotangent orbifolds as orbifold string topology
Abstract: Abstract: In this talk we will report on the paper
math.AT/0610899 ChenRuan Cohomology of cotangent orbifolds and ChasSullivan 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 ChenRuan orbifold cohomology (related to Floer cohomology) of its cotangent orbifold.
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 halfopen 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 pointset topological properties.
Speaker: Soren Galatius
Title: Point set topology of spaces of graphs (cont.)
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.
SPECIAL TOPOLOGY SEMINAR
Speaker: Boris Chorny, Australian National University, Canberra
Title: Brown representability for spacevalued 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.
Bergman Lecture: Dennis Sullivan
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 (StiefelWhitney number if we do not consider orientations and Pontrjagin and StiefelWhitney 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.