MATHGEOM

Séminaire de topologie 2010/11

le mardi à 14h15

## Programme

Date Titre Orateur
26.08.10
Categorical aspects of the Hausdorff and Gromov metrics
Walter Tholen
York University
02.09.10
Normal maps of groups and loop spaces
Emmanuel Farjoun
Hebrew University
14.09.10
A stable filtration for homotopy limits
Eric Finster
EPFL
21.09.10
Rational homotopy of the space of self-fibre-homotopy equivalences
Greg Lupton
Cleveland State/EPFL
vendredi
01.10.10
13h15
Notions of size in integral geometry and category theory
Tom Leinster
Glasgow
05.10.10
Fibrewise rational H-spaces
Greg Lupton
Cleveland State/EPFL
12.10.10
Elmendorf's Theorem for cofibrantly generated model categories
(Dans le cadre du groupe de travail)
Marc Stephan
EPFL
02.11.10
Le théorème de Blakers-Massey
Jérôme Scherer
EPFL
09.11.10
10h15
CHB330

Catégories simpliciales enrichies et K-théorie de Waldhausen I
Ilias Amrani
EPFL
16.11.10
10h15
CHB330

Catégories simpliciales enrichies et K-théorie de Waldhausen II
Ilias Amrani
EPFL
23.11.10
10h15
MA 10
Global properties of finite simple groups
Wojciech Chachólski
KTH
23.11.10
14h15
Categorical foundations for K-theory I
Nicolas Michel
EPFL
30.11.10
Categorical foundations for K-theory II
Nicolas Michel
EPFL
07.12.10
CHB330

Homotopic descent and codescent
Kathryn Hess
EPFL
14.12.10
CHB330

An introduction to homotopic Hopf-Galois extensions
(Dans le cadre du groupe de travail)
Varvara Karpova
EPFL
17.02.11
13h15
MA 10
Categorical approach to Gödel's first incompleteness theorem in the initial arithmetic pretopos
(Soutenance de PDM)
Clovis Galiez
Ottawa/EPFL
17.02.11
14h15
MA 10
Cobordism, cohomology theories and formal group laws
(Soutenance de PDM)
Lev Kiwi
Northwestern/EPFL
17.02.11
15h15
MA 10
Two invariants of spatial graphs
(Soutenance de PDM)
Elias Weber
EPFL
22.02.11
An operadic model for the space of long knots
Kathryn Hess
EPFL
08.03.11
Le morphisme d'Hurewicz pour les algèbres et les catégories sur un corps
Claude Cibils
Montpellier
29.03.11
13h15
MA 30
Automorphisms of right-angled Artin groups
(Séminaire Topologie/Théorie des groupes)
Ruth Charney
Brandeis
12.04.11
Une histoire de la bar construction
Muriel Livernet
Paris XIII
19.04.11
Crossed complex resolutions of group extensions
Andy Tonks
London Metropolitan
03.05.11
Paolo Masulli
Copenhagen
11-13.05.11
André Memorial Conference
17.05.11
A classification of Taylor towers of functors of spaces and spectra
Michael Ching
Georgia
24.05.11
Small models for functors from spaces to spaces
Bill Dwyer
Notre Dame
14-18.06.11
Young Topologists Meeting 2011
28.06.11
CM 104

Chaînes géométriques bivariantes et produit d'intersection
David Chataur
Lille
17.08.11
MA 31
Classification of small linear functors
Boris Chorny
Haifa

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

## Abstracts

Tholen: (Joint work with A. Akhvlediani and M.M. Clementino) Hausdorff's metric for (non-empty) subsets of a compact metric space has gained renewed interest following Gromov's introduction of a metric that measures the distance between independent (non-empty) compact metric spaces. In this talk we would like to shed some light on the categorical background for both the Hausdorff and the Gromov metric, adopting Lawvere's view of an individual metric space as a small category enriched over the non-negative real line, so that d(x,y) is interpreted as hom(x,y). While this view places the Hausdorff and Gromov metrics into the general context of enriched category theory, in this talk we would like to emphasize more the clarifying role of category theory rather than its generality.

Farjoun: We explain what it means for a map of groups, discrete or not, to be normal, in the sense that the homotopy quotient of the map has a good, compatible group structure. Certain constructions are seen to preserve normality of maps. We examine a number of old and new questions in this light.

Finster: We study the stable homotopy type of spaces X which arise as the homotopy limit of a diagram of pointed spaces F : C → S*. My goal will be to describe a filtration of the corresponding suspension spectrum ∑ X inspired by Goodwillie's Calculus of Functors, and generalizing the work of G. Arone on the the Goodwillie tower of stable mapping spaces. With some connectivity hypotheses imposed on the functor F, we can the recover the spectrum in question as the homotopy inverse limit of its associated filtration.

Lupton (1): (Joint work with Yves Félix and Sam Smith.) Let p: E → B be a fibration and write Aut(p) for the space of the title: the space of self-maps over B of E, each of which is a fibre-homotopy equivalence. Under mild conditions on B, Aut(p) is a group-like space, and hence all components are of the same homotopy type---that of the identity component, say. When E is finite, we identify the rational homotopy type of this component, in terms of derivations of the minimal model of p: E → B. Examples, and some extensions of and variations on the basic result, will be presented.

Leinster: The subject of integral geometry began with problems like that of Buffon's needle: what is the probability that a needle dropped randomly on the floor will land across a crack between the floorboards? More recently, it has been used in tomography, the science of medical scanning: what is the probability that a randomly-fired scanning beam will meet a tumour? I will give a gentle introduction to integral geometry. Notions of size, such as volume and surface area, play an important role. On the other hand, there is an abstract categorical method that produces notions of size in different areas of mathematics. I will explain some emerging connections between the notions of size arising from these two subjects, integral geometry and category theory.

Lupton (2): (Joint work with Sam Smith.) We prove the fibrewise H-triviality of certain fibrewise H-spaces after rationalization, extending a result of Crabb-Sutherland for the universal adjoint bundle. Our result is a consequence of fibrewise versions of the Hopf and Leray-Samelson theorems. We also give a Sullivan model formula for the Samelson Lie algebra of the space of lifts into a fibrewise group-like space.

Stephan: Elmendorf's Theorem in equivariant homotopy theory states that for any topological group G, the model category of G-spaces is Quillen equivalent to the model category of continuous diagrams of spaces indexed by the opposite of the orbit category of G. For discrete G, Bert Guillou explored equivariant homotopy theory for any cofibrantly generated model category C and proved an analogue of Elmendorf's Theorem assuming that C has "cellular" fixed point functors. We will generalize Guillou's approach and study equivariant homotopy theory for topological, cofibrantly generated model categories. Elmendorf's Theorem will be recovered for G a compact Lie group or a discrete group.

Scherer: Ceci est un projet avec W. Chachólski. Considérons le push-out homotopique D d'un diagramme C ←A → B. Construisons ensuite le pull-back homotopique P du diagramme obtenu en oubliant A. On a alors une application A → P. Le théorème classique de Blakers-Massey s'intéresse à la différence entre A et P en termes de connexité. Nous donnerons une interprétation en termes de classes fermées (i.e. de foncteurs de nullification).

Amrani: Le point de départ est la question de la représentabilité (dans le sens homotopique) de la K-théorie de Waldhausen. Dans la première partie on considère la catégorie des petites catégories. La deuxième partie est une généralisation au cas enrichi.

Chachólski: We think about endofunctors of groups as operations. We are interested only in those endofunctors F which are idempotent (F2 and F are isomorphic). To understand how complicated the action of idempotent functors on groups is we study the orbits of this action: the collection of values F(G) for a fixed group G and all idempotent functors F. It turns out that the collection of finite groups is invariant under this action (any idempotent functor takes a finite group into a finite group). It turns out that it is also possible to describe explicitly the orbit of this action for all finite simple groups. The aim of this talk is to present this classification and interpret the answer in terms of homological properties of the groups. This is joint work with E. Dror Farjoun, Y. Segev, and M. Blomgren.

Michel: La définition des différentes K-théories suit le schéma suivant. Etant donné un objet C, on lui associe d'abord une catégorie AC qui est « structurée » (symétrique monoïdale, exacte, Waldhausen, & ). On applique ensuite la « machine » de K-théorie sur AC pour obtenir finalement le spectre de K-théorie de l'objet C. Par exemple, on associe à un anneau R sa catégorie de modules de type fini projectifs pour obtenir la K-théorie usuelle de R.
Dans ma thèse, je me suis intéressé à la première étape de ce processus. Plus précisément, je me suis posé les questions suivantes. A quel type d'objets devrait s'appliquer la K-théorie ? Quelles catégories structurées devrait-on considérer « au-dessus » de ces objets afin d'obtenir une bonne notion de K-théorie ? Comment cette correspondance prend-elle en compte les morphismes de ces objets ? Dans ces exposés, je vais décrire un cadre catégorique qui réponde à ces questions et donner quelques perspectives d'avenir.

Hess (descent): In this talk I'll describe a general homotopy-theoretic framework in which to study problems of (co)descent and (co)completion. Since the framework is constructed in the universe of simplicially enriched categories, this approach to homotopic (co)descent and to derived (co)completion can be viewed as ∞-category-theoretic.
I'll present general criteria, reminiscent of Mandell's theorem on E-algebra models of p-complete spaces, under which homotopic (co)descent is satisfied. I'll also construct general descent and codescent spectral sequences and explain how to interpret them in terms of derived (co)completion and homotopic (co)descent.
To conclude I'll sketch a few applications.

Karpova: After a quick reminder on the Galois correspondence for fields, we will explain how the classical theory of Galois extensions was generalized to the case of commutative rings (Chase, Harrison and Rosenberg, 1965). Essentially, this was done by reinterpreting the classical elements of the Galois correspondence for fields in terms of certain isomorphisms. This approach led to the notion of Hopf-Galois extensions for algebras, introduced by Chase and Swedler (1969), and Kreimer and Takeuchi (1980), among others, which we will define. In her article (2009) Hess laid the foundations of a theory of homotopic Hopf-Galois extensions in monoidal model categories. Our current motivation is to obtain a result similar to a theorem of Schneider, characterizing faithful flatness of commutative rings in terms of the associated descent data, within the homotopic Hopf-Galois context. This should also give us a better understanding of homotopic Hopf-Galois extensions in themselves. After specifying the original statement of Schneider, we will present the context we are working in and our strategy, which uses the tools of homotopic descent theory, recently developed by Hess.

Galiez: Nous parlerons des univers arithmétiques : un concept catégorique permettant de capturer la manipulation de nombres naturels dans un catégorie. On explicitera les concepts catégoriques sous-jacents afin de comprendre la définition d'un univers arithmétique. Nous esquisserons ensuite une démonstration d'une version ad hoc du premier théorème d'incomplétude de Gödel pour cette structure.

Kiwi: I will talk about the link between complex oriented cohomology theories and formal group laws. In particular, I will state the Quillen Theorem and the Landweber exact functor theorem. This will show us that complex cobordism is (in some sense) universal among complex oriented cohomology theories.

Weber: In the first part of this talk I will introduce the notion of spatial graphs. They are a generalisation of knots and one aim of current mathematical research is to classify them. Like in knot theory this is done with the help of invariants. In the second part, I will give a short introduction to two such invariants: the Yamada polynomial and the fundamental graph quandle.

Hess (long knots): The space of framed long knots in Rn, denoted Emb(n), is the space of embeddings of the real line R into Rn, which are equal to a fixed inclusion of R outside of a compact subspace and which are endowed with a "framing". This space admits a natural A-algebra structure, where the multiplication is given by reparametrization and concatenation. Since the space Emb(n) is connected if n>3, it has the homotopy type of a loop space. Furthermore, its multiplication is homotopy-commutative, since one can pass one of the knots homotopically "through" the other, which suggests that Emb(n) may indeed be a double loop space.
Dev Sinha showed that Emb(n) is indeed a double loop space, but without determining its double-delooping. Bill Dwyer and I have shown that Emb(n) has the homotopy type of the double loops on the derived mapping space of operad morphisms from the associative operad to Kontsevich's variant on the little disks operad, thus providing an explicit double-delooping of Emb(n).
In this talk I will explain how we obtained this identification, the key to which is the existence of a certain fiber sequence of simplicial sets, arising from a monoid morphism in any nice enough (but not necessarily symmetric) monoidal model category. Given two monoids A and B, this fiber sequence enables us to compare the derived mapping spaces of monoid morphisms and of A-bimodule morphisms from A to B.

Cibils: Pour une algèbre ou une catégorie sur un corps, il n'existe pas de théorie d'homotopie de lacets tenant compte de la structure linéaire. A la place, on considère l'ensemble des graduations de ces structures par différents groupes et les revêtements galoisiens qui en résultent (en général il n'existe pas de revêtement ou de graduation universel). Le groupe fondamental est alors le groupe des automorphismes d'un foncteur fibre associé à toutes les graduations. Cet invariant "à la Grothendieck" peut être calculé dans certains cas. Par exemple pour les matrices de taille p un nombre premier sur un corps algébriquement clos de caractéristique zéro, il est le produit du groupe libre à p-1 générateurs et du groupe cyclique d'ordre p. Nous verrons que pour chaque graduation, il existe un morphisme d'Hurewicz linéaire à valeurs dans la cohomologie de Hochschild Michell. Pour une catégorie Schurian, une graduation universelle existe et le morphisme d'Hurewicz associé au groupe fondamental est un isomorphisme. Il s'agit de résultats obtenus en collaboration avec M.J. Redondo et A. Solotar.

Charney: Automorphism groups of free groups have many properties in common with automorphism groups of free abelian groups, i.e., GL(n,Z). Interpolating between these are the automorphism groups of right-angled Artin groups. In joint work with Karen Vogtmann, we have shown that many of these properties hold for all such groups. I will give an overview of our techniques and results.

Livernet: Dans cette exposé, nous nous attacherons à présenter différents aspects historiques de la bar construction, pour les algèbres associatives, pour les monades, pour les DGcatégories et nous montrerons dans quelle mesure on peut unifier ces différentes constructions. Plus récemment, la bar construction a ete employée pour les opérades, et nous verrons comment cette bar construction s'insère dans les modèles historiques. En particulier nous définirons la notion d'opérades et montrerons que différents types de bar construction s'appliquent selon la définition prise. Cependant, de manière assez étonnante, la bar construction originalement définie par Ginzburg et Kapranov pour les opérades, n'en n'est pas réellement une! Nous montrerons en quoi cette "bar construction" est en fait reliée à un complexe de Koszul d'une catégorie d'arbres et expliquerons ainsi la raison d'être de cette définition. Enfin, nous montrerons les équivalences entre ces bar constructions.

Tonks: An old paper of CTC Wall shows, with a simple spectral sequence argument, that given free chain resolutions for groups H and K one may construct a resolution for any group extension G of H by K. More recently Brown, Ellis and others have attempted, with some degree of success, to lift this construction from the category of chain complexes to that of crossed complexes or of spaces. This non-abelian situation is considerably harder; one knows, for example, that there is no homological perturbation theory for crossed complexes. In this talk we will give an overview of the problem and present some new results obtained in collaboration with O Gill.

Masulli: In equivariant homotopy, we have the RO(G)-graded cohomology theory HG*(-;M), associated to a Mackey functor M. We will introduce these notions and use the characterization of this cohomology and the algebraic properties of Mackey functors to compute the cohomology of G and of a point, with as coefficients the constant Mackey functor Z. The result will be useful to get information about equivariant Eilenberg-MacLane spaces.

Ching: The layers of the Taylor tower of a functor F from C to D are fairly easy to describe in terms of certain spectra, the derivatives of F, that are often computable. In this talk, I'll explain that the derivatives of F, viewed as a single object, possess additional structure that allows one to reconstruct the entire Taylor tower, and often, therefore, the functor F. This structure is that of a K-coalgebra for a certain comonad K that depends on the categories C and D. These constructions also determine an equivalence between the homotopy categories of n-excisive functors and n-truncated K-coalgebras. I'll finish by giving more explicit descriptions of what this coalgebra amounts to for various combinations of C and D equal to either based spaces or spectra. (This is all joint work with Greg Arone.)

Dwyer: Sometimes it is possible to describe a functor F from spaces to spaces by giving a functor G from finite sets to spaces; the process of passing from G to F goes back to Kan and is especially transparent from a simplicial point of view. The goal of the talk is to give many examples and a few non-examples, as well as to illustrate how the technique can be applied, for instance, to obtain simple models for Snaith splittings.

Chataur: Le but de cet exposé est de présenter une théorie de chaînes géométriques pour les espaces topologiques. Cette théorie permet de construire une structure algébrique partielle liée au produit d'intersection des cycles d'une variété qui sont en position transverse, et de comparer cette structures à la structure multiplicative des cochaines donnée par le cup produit. On montre que ces deux structures sont équivalentes (en un sens opéradique partiel), ce résultat est un relèvement multiplicatif de la dualité de Poincaré pour les variétés au niveau chaînes-cochaînes.

Chorny: Calculus of functors is a homotopy theory defined on the category of continuous functors from the category of spaces or spectra to the category of spaces or spectra. It carries over many formal constructions and concepts from the classical calculus of functions. In particular, a functor from spectra to spectra is linear if it takes homotopy pushouts to homotopy pullbacks. Finitary linear functors (i.e., linear functors commuting up to homotopy with filtered colimits) were classified by T. Goodwillie. We offer a similar classification of a bigger family of linear functors, the small linear functors (i.e., linear functors commuting, up to homotopy, with $\lambda$-filtered colimits for some cardinal $\lambda$). It was shown by Goodwillie that the homotopy category of finitary linear functors is equivalent to the homotopy category of spectra. We prove that the homotopy category of small linear functors is equivalent to the opposite of the homotopy category of pro-spectra.

Dernière mise à jour: 06.09.2011