In topology moduli spaces of Riemann surfaces with boundaries form an operad. In algebraic geometry it is hard to talk about boundary components. Still the compactified moduli spaces of curves form an operad. We will show that categories of Tate sheaves on the open moduli spaces form a cooperad, giving rise to motivic versions of the chains of the above topological operads. We give applications to the motivic action on the Grothendieck-Teichmueller group.
See also the Brave New Rings seminar page.
There are many different notions of a stratified space. One such notion is that of a homotopically stratified space. These were introduced by Frank Quinn and include most other stratified space definitions. Here the strata are related by "homotopy rather than geometric conditions". This makes them ideal for studying the topology of stratified spaces. Two such tools for studying that topology are the holink and popath spaces. We will define these and state a theorem showing how the space of popaths can be "deformed" into the space of holinks in a useful way. We will define what it means for maps to be stratified homotopic and spaces to be stratified homotopy equivalent. Then we will sketch some theorems and give two categorical viewpoints on such a homotopy theory.
We will construct certain stratified spaces inside the category of Frölicher spaces. Various ways of generalizing the notion of tangent space for a manifold will be discussed. For different purposes all of the following seem relevant: Flow spaces, curve spaces, ray spaces and continuous derivations. The differences between these will be illuminated by basic examples. We will introduce type A and type B embeddings and prove that the image of a locally compact sharp stratified space under a type B embedding into an Euclidean space naturally gets the structure of a Whitney stratified set. Furthermore, I will indicate an approach to Thom's first isotopy lemma.
I will explain how equivariant monopole Floer homology for rational homology 3-spheres can be constructed by a limiting process from the (metric and perturbation dependent) irreducible Floer groups. These equivariant groups are related by a long exact sequence similar to that appearing in Kronheimer and Mrowka's monopole Floer homology.
Abstract: Aut(F_n) denote the automorphism group of a free group on n letters. The group homology H_k(Aut(F_n)) is known to be independent of n, as long as n>2k+1, and it is natural to ask what this "stable homology" is. There is a homomorphism from the symmetric group S_n to Aut(F_n) where a permutation acts by permuting the generators of F_n. Taking the limit as n goes to infinity gives a map S_\infty -> Aut_\infty, and in '95 Hatcher proved that H_k(S_\infty) is a direct summand of H_k(Aut_\infty) and conjectured that the complement might vanish. I will explain parts of the proof (arxiv:math/0610216) of Hatcher's conjecture, and how it relates to Madsen-Weiss' generalized Mumford conjecture.
Abstract: (work of Stacy Hoehn) Suppose M is a non-compact manifold. Then a completion of M is a compact manifold (N, ∂N) such that M is homeomorphic to N-∂N. Siebenmann showed that the obstruction to completing a manifold is a certain element in a K_{0} algebraic K-group.
Similarly, suppose p_{1}: E_{1} → B is a fiber bundle with fibers homeomorphic to the non-compact manifold M. Then a fiberwise completion of p_{1} is a fiber bundle p_{2}: E_{2} → B such that the fibers of p_{2} are compact manifolds with boundary such that E_{1} is fiberwise homeomorphic to E_{2}-∂_{B} E_{2}. (Here ∂_{B} E_{2} is the fiberwise boundary of E_{2}.) The goal of this talk is to express the obstruction to such a fiberwise completion in terms of algebraic K- theory of spaces.
Conjecture: For any map f: E → S^{1} from a closed 4-manifold to a circle whose homotopy fiber has the homotopy type of a 3-manifold, there exists a fiber bundle \bar f : \bar E → S^{1} where \bar E is a 4-manifold homotopy equivalent to E.
Theorem (joint with Shmuel Weinberger) The conjecture is true when the 3-manifold is a lens space with odd order fundamental group.
The proof involves a surgery theoretic argument which involves a lemma of Gauss used in his third proof of the law of quadratic reciprocity.
This theorem answers a question of Jonathan Hillman, asked in the context of 4-dimensional geometries:
Theorem: Any 4-manifold with Euler characteristic zero and fundamental group a semidirect product where Z acts on Z/odd is homotopy equivalent to the mapping torus of a self isometry of a lens space.
We start by briefly defining the category of ex-spectra over a base N. Then we define the notion of these being Serre-fibrant, and see how this gives rise to a Serre type spectral sequence. We then discuss a way to define a Chas-Sullivan type product on the "total space", when each fiber is an S-algebra and the base N is a smooth manifold. If time permits we will briefly discuss how these notions occur naturally in spectrum models for Floer-homology in cotangent bundles, but we postpone a more thorough discussion of this for a later seminar.
In the talk we introduce the motivic slice filtration as defined by Voevodsky. We present some of the conjectures made by Voevodsky on the slice filtration of particular spectra and indicate the relationship to the motivic Atiyah-Hirzebruch spectral sequence. Finally we outline how to get insight into the slice filtration of MGL and other motivic Landweber spectra.
This is joint work with Bökstedt, Bruner and Lunøe-Nielsen from 2007, based on work of Ravenel from 1981. For a finite p-group G and a bounded below G-spectrum X of finite type mod p, the G-equivariant Segal conjecture asserts that the canonical map X^G --> X^{hG} is a p-adic equivalence. We show that if the C_p-equivariant Segal conjecture holds for a C_{p^n}-spectrum X, as well as for each of its geometric fixed point spectra Phi^{C_{p^e}}(X) for 0<e<n, then the C_{p^n}-equivariant Segal conjecture holds for X.
First I will introduce the notion of Conley indices and parametrized Conley indices, then I will introduce a concept of parallel transport in parametrized Conley indices. I will then describe how finite reduction of Floer homology in the cotangent bundle of a manifold N describes a fibrant ex-spectrum using these notions.