MATHGEOM

**Séminaire de topologie 2011/12**

le mardi à 14h15

CM 012

(Voir aussi le programme du groupe de travail en topologie et le programme du séminaire de topologie en 2010/11, en 2009/10, en 2008/09, en 2007/08, en 2006/07, et en 2005/06.)

**Berglund:** I will describe a new approach to the rational homotopy theory of
mapping spaces which is inspired by Getzler's Lie theory for nilpotent
L-infinity algebras. Combined with a recent characterization of spaces
that are simultaneously formal and coformal, this yields small chain
complexes for calculating the rational homotopy groups of the space
aut(X) of homotopy automorphisms of such spaces. In joint work with
Madsen, we use this to calculate explicitly the rational homotopy groups
of the homotopy automorphisms of highly connected manifolds. If time
admits, I will also discuss work in progress towards the calculation of
the rational cohomology ring of the classifying space Baut(X).

**Gepner:** The space of units GL_{1}(R) of a commutative ring spectrum R
is the infinite loop space of a spectrum gl_{1}(R). Typically, this spectrum is taken to be connective, meaning that it has
no nonzero negative homotopy groups. However, there are other interesting deloopings of GL_{1}(R) which carry important
algebraic information about R. One in particular has π_{-1} gl_{1}(R) = π_{0}Pic(R), the Picard group of R, and
π_{-2} gl_{1}(R) = π_{0} Br(R), the Brauer group of R. If R is connective, there is a spectral sequence
for computing the homotopy groups of Pic(R) and Br(R) which reveals a close relationship between π_{0} Br(R) and
Br(π_{0} R).

**Scherer:** This is joint work with Boris Chorny. I will explain a very basic construction
in homotopy theory, namely that of Whitehead products, and present the
Lawvere algebraic theory defining nilpotency of class less than n. I will show
that loop spaces which are "nilpotent up to homotopy" enjoy a vanishing
property for iterated Whitehead products and move then to a more
contemporary notion of homotopy nilpotency due to Biedermann and
Dwyer. It is strongly related to Goodwillie calculus and we will see that
the understanding of the stages in the Goodwillie tower yields a new
vanishing result for iterated Whitehead products.

**Müller:** These talks give an overview of my thesis.
They cover an introduction to classical monadic descent and an important example, Grothendieck descent.
Then we examine the question, how this setting can be generalized to the case of categories enriched over
model categories and give the general theory of homotopic descent. This includes cosimplicial structures and
their totalizations, derived completion and the descent spectral sequence associated to homotopic descent.
Finally we give a definition of a generalized Adams spectral sequence
as an example of the theory. In particular, we will examine its relationship to the descent spectral sequence.

**Turner:** Khovanov homology is an abelian-group-valued knot invariant
related to the Jones polynomial. Like many other modern knot
invariants, the underlying topological or geometrical properties that
are being measured remain rather obscure. To better understand the
situation it is natural to ask: is it possible to associate to each
knot a topological space whose classical invariants (for example
cohomology, homotopy groups etc) give Khovanov homology? In this talk
I will discuss one approach to this question using homotopy theory.

**Seal:** Joint work with Dirk Hofmann and Frédéric Mynard. Exponentiable topological spaces showcase a remarkable link between topological and ordered structures, a feature that can be reinterpreted in terms of underlying monadic structures. Indeed, on one hand, exponentiable objects in TOP are those topological spaces whose set of opens forms a continuous lattice. On the other, topological spaces are monoids in the Kleisli category of the filter monad, while continuous lattices form its Eilenberg-Moore algebras. In this
talk, I will present this categorical interpretation of topological structures, and illustrate it with well-known and new results.

**Finster:** Recently a surprising connection has been discovered between
homotopy theory and a class of well studied formal languages used in
logic and computer science know as dependent type theories. This
observation, made independently by Vladimir Voevodsky and the logician
Steve Awodey, is the basis for Voevodsky's "Univalent Foundations"
program, whose goal is to provide an alternative to set theory in
which all homotopy types are regarded as basic entities. In this
talk, I will describe some of the ideas that motivate dependent type
theories, as well as show how to use them to express homotopy
theoretic ideas.

**Hess:** Let **M** be a monoidal category endowed with a
distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids.
We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids
in **M**. These notions generalize both principal bundles and crossed modules and are preserved by nice enough
monoidal functors, such as the normaliized chain complex functor.
We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when **M** is the
category of simplicial sets or the category of chain complexes over a commutative ring.

**Berglund (2012):** I will talk about joint work in progress with Ib Madsen concerning the
problem of calculating the rational cohomology ring of the diffeomorphism
group of a (d-1)-connected 2d-dimensional manifold M. Roughly speaking,
our approach is to use rational homotopy theory to obtain information
about the monoid of homotopy self-equivalences aut(M), and then use
surgery theory to measure the difference between homotopy
self-equivalences and diffeomorphisms. One can obtain very precise
information about the classifying space Baut(M). A surprising result is
that its rational homotopy groups only depend on d and the rank of the
middle dimensional homology group H_{d}(M). To approach the cohomology of
the diffeomorphism group, we prove a stability theorem analogous to the
classical Harer stability theorem for diffeomorphism groups of surfaces.
The cohomology in the stable range was recently calculated by Galatius and Randal-Williams.

**Chataur:** Intersection homology and cohomology
were introduced by Goresky and MacPherson in order to extend the L-genus to singular spaces.
These theories provide a generalization of Poincaré duality and are useful in the study of algebraic
varieties.
In this talk, we will explain how to develop algebraic models for intersection cohomology.

**Dessai:** We report on joint work with
Michael Wiemeler on the classification of low dimensional complete intersections with S^{1}-symmetry.

**Dwyer:** This talk, which should be entirely accessible to doctoral students
and advanced masters students, will consist of an introduction to localization in model
categories, including a number of interesting and important examples.

**Scherer (2012):** This is joint work with Emmanuel Dror Farjoun. We usually think
that the pullback of a fibration along any map is another fibration which cannot become more complicated than the fibration we started
with. This is often true and I will recall a few classical examples. However, when we look at "flatness properties" this philosophical
principle is not the rule. By an L-flat fibration sequence we mean one which remains a fibration sequence after applying the functor L.
The plan is to explain which functors L behave
well with pullbacks in the category of spaces and continue with the analogous question for group extensions which started this project.

**Finster (2012):** The Opetopes are a family of
polytopes which are used in several
definitions of higher category, and in this sense, constitute a kind
of alternative to the more familiar notion of simplices. They arise
very naturally from considering pasting diagrams of "many-in one-out"
operations, and can be regarded geometrically as the higher
dimensional analog of trees. A very succinct definition due to Joyal,
Kock and Batanin constructs the Opetopes using the language of
polynomial functors on Set, giving them a very close connection to the
notion of inductive datatype found in many modern programming
languages. In this talk, I will sketch their definition and explain a
convenient graphical notation for depicting and manipulating them.

**Rovelli:** In this talk, we will introduce the definition of
triangulated category, with a
particular care to similarities and differences between the triangulated
context and the abelian one.
The approach is that followed by Holm and Jorgensen's article: given an abelian
category, we consider both the complexes category and the homotopic category,
and we will easily see that the first one is abelian, while for the second one
we need to change the environment.
In the second part of the talk, we will give an idea of how to build the derived
category out of an abelian one, we will explain why it is more useful (for the
purpose of studying homology) than the homotopic category, and finally we will
discuss whether it is triangulated.

**Sabatini:** (Joint work with Prof. L. Godinho, IST Portugal)
In 2009 Tolman formulated the ``symplectic generalization of
Petrie's conjecture": given a compact symplectic manifold M
with a Hamiltonian S^{1}-action and minimal number of fixed
points, is it possible to characterize all the possible
cohomology rings and Chern classes that can arise?
We turn this question into a computational problem, in the following way.
We derive a simple algebraic identity involving the ûrst Chern class.
This enables us to construct an algorithm to obtain linear relations
among the isotropy weights at the ûxed points.
Since determining the weights at the fixed points determines the
(equivariant) cohomology ring and Chern classes, this allows us to give
a (positive) answer to the question.
In particular, we give a complete list of cohomology rings and Chern
classes when dim(M) is less than or equal to 8.

**Hausel:** We will be looking at the graph of Betti numbers,
depicted as a discrete function of cohomological degree,
of various non-compact complete hyperkahler manifolds: hyperkahler toric, quiver and character varieties.
We find that if they are of large dimension, then the graphs will converge to various distributions, including
Gauss, Gumbel and Airy. This is a joint experiment with Fernando Rodriguez Villegas.

**Seal (2012):** One of the simplest non-cartesian monoidal
structure on a category is given by the tensor product of abelian groups. This tensor has the desirable property of
representing bilinear maps as linear ones. However, the corresponding general notion of a ``bimorphism'' (that is, of a ``morphism in each variable'') is a priori awkward to express in categorical terms. A further analysis reveals that the ``free abelian group'' monad is monoidal, and that this additional structure facilitates a categorical definition of bimorphisms - and of the corresponding tensor. Hence, under appropriate hypotheses a monoidal monad on a monoidal
category induces a monoidal structure on the Eilenberg-Moore category - a fact that invites the study of actions therein.

**Bergner:** With many definitions being given for (∞, n)-categories, one criterion to check
is whether they can be thought of as categories enriched in (∞ , n-1)-categories.
In joint work with Charles Rezk, we are establishing a chain of Quillen equivalences from the
model structure for Θ_{n}-spaces and the model structure for categories enriched
in Θ_{n-1}-spaces. This comparison also gives insight into how to compare to
other known models.

Dernière mise à jour: 13.08.2012