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Barcelona Algebraic Topology Group

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Friday's Topology Seminar 2014-2015

Date and place: July 10,  2015. 12:00. CRM Aula A1.
Speaker :
Ramsés Fernandez Valencià (Swansea University)
Sobre teorías topologicas conformes de campos no orientadas.
En la charla se aboradará el concepto de TCFT no orientada (KTCFT). Concretamente, se dará una nueva demostración para la clasificación de KTCFTs abiertas, se introducirá la extensión universal de KTCFTs abiertas a teorías abierto-cerradas y se aboradará la relación de la parte cerrada de una KTCFT abierto-cerrada con el complejo de Hochschild involutivo.

Date and place: June 19,  2015. 12:00. CRM Aula A1.
Speaker : Leandro Lombardi (Universidad de Buenos Aires, Argentina)

Álgebras de cactus compatiblemente bigraduadas
Se mostrará una correspondencia entre la estructura de álgebra sobre el operad (de cadenas celulares del operad topológico) de cactus y el álgebra asociativa libre subyacente, en presencia de una bigraduación compatible.
En toda álgebra de cactus puede considerarse un producto pre-Lie.
En el caso de que este álgebra sea de la forma T V para V un espacio vectorial, este producto, restringido a V, resulta asociativo. Se muestra que una estructura de álgebra de cactus en TV induce una estructura de biálgebra asociativa y coasociativa en  H = V + 1 donde 1 es la unidad formal de dicho producto.  Esto muestra, junto con trabajos previos de Kadeishvili y Menichi que estas estructuras están en correspondencia biunívoca con las estructuras álgebra de cactus en TV (que extienden la de álgebra asociativa) con cierta condición de compatibilidad con la graduación, propiedad motivada por el ejemplo del complejo de Hochschild.

Date and place: June 12,  2015. 12:00. CRM Aula A1.
Speaker : Shizuo Kaji (Yamaguchi Univ. Japan / U. of Southampton UK)

An external product in equivariant homology
Let G be a compact Lie group and M be a compact G-manifold. Denote its Borel construction by M_G. Given a homotopy pullback square with the base space M_G, we will introduce a generalised external product in the homology such that it reduces to the ordinary external product when M is a point and G is trivial. (In this case, the homotopy pullback is just the direct product.) It unifies two constructions in string topology; Chas-Sullivan's string product for the free loop space LM over a manifold and Chataur-Menichi's string product for the free loop space LBG over the classifying space of a compact Lie group. We will also introduce its secondary product, which generalises Tate's cup product in the homology of BG. This is joint work with Haggai Tene.

Date and place: March 27,  2015. 12:00. CRM Aula A1.
Speaker : Jean Michel (Université Denis Diderot - Paris 7)

The K(∏,1) property for complex reflection arrangements and the dual braid monoid
David Bessis, while proving the property that complex reflection arrangements are K(∏,1), has constructed an important monoid, the dual braid monoid, which describes the structure of braid groups of well-generated complex reflection groups. This monoid is already very insteresting for finite Coxeter groups. I will explain these topics assuming no previous knowledge.

Date and place: March 13,  2015. 12:00. CRM Aula A1.
Speaker : Jaume Aguadé (Universitat Autònoma de Barcelona)

Reflections and invariants
This lecture is an elementary survey on finite reflection groups and invariant theory, and the intimate relationship between these two fields. We will also discuss some open problems and recent results on generalized and stable invariants of finite reflection groups, as well as the relevance of these questions in topology.

Date and place: March 6,  2015. 12:00. CRM Aula A1.
Speaker : Jesper M. Møller (
Københavns Universitet)
Enumerative group theory for beginners
Enumerative group theory uses methods from combinatorics, group theory, and topology to find relations between numerical quantities associated to finite groups.  The class relation and Alperin's weight conjecture are examples of such (conjectured) relations.  (By the way, we are all beginners in enumerative group theory since I just made up the term.)

Date and place: February 8, 2015. 15:00. CRM Aula A1.
Speaker : Bob Oliver
(Univ. Paris 13)
Fusion systems over small finite 2-groups

Date and place: January 27, 2015. 12:00. C1/366.
Speaker :
Federico Cantero (Univ. Muenster)
Homological stability for configuration spaces on closed manifolds

Date and place: December 15, 2014. 12:00. CRM.
Speaker : David Gepner (Purdue University)

Localization sequences in the algebraic K-theory of ring spectra

Date and place: December 11, 2014. 16:00. CRM.
Speaker : Anthony Blanc (CRM)

Lattice conjecture : statement and examples.
The lattice conjecture refers to the hypothetical existence of a rational lattice in the periodic cyclic homology of a saturated dg-algebra, given by the topological K-theory. It is a holy grail of noncommutative Hodge theory. We discuss the assumptions of the conjecture and investigate examples like algebraic varieties, finite dimensional algebras, DM-stacks and if time permits matrix factorization categories. This work involves tools from homotopy theory and motives.

Date and place: December 12, 2014. 12:00. CRM room A1.
Speaker : Geoffrey Powell (Université d'Angers)

Classifying spaces and Brown-Gitler spectra through the eyes of ko
The classifying spaces of elementary abelian 2-groups are of fundamental importance in relation to the Sullivan conjecture and the theory of unstable modules over the Steenrod algebra. In particular, knowledge of the cohomology of elementary abelian 2-groups with respect to a cohomology theory E gives information on the associated Omega spectrum. Similarly the Brown-Gitler spectra can be constructed by considering the homology theory which they represent.
This talk reports on part of a project to make explicit the relationship between the two approaches to Omega spectra, in particular when E is taken to be ko. In this case, the results are based on structure results on modules over the subalgebra A(1) of the Steenrod algebra generated by Sq^1, Sq^2.

Date and place: December 8, 2014. 17:00. CRM room A1.
Speaker : Nguyen H.V. Hung (National University of Vietnam, Vietnam)

Singer transfer and a conjecture on the squaring operation
The Singer  transfer
Tr_s:\mathbb F_2 \otimes_{GL_s}PH_d(B\mathbb V_s)\to Ext_{\mathcal A}^{s,s+d}(\mathbb F_2,\mathbb F_2)
is expected to be a useful tool in the study of the mysterious cohomology of the Steenrod algebra, $Ext_{\mathcal A}^{*,*}(\mathbb F_2,\mathbb F_2)$.
In this talk, we first recognize the phenomenon that if we start from any degree $d$ and apply $Sq^0$  on the domain of $Tr_s$ repeatedly at most $(s-2)$ times, then we get into the region in which all the iterated  squaring operations are isomorphisms. As a consequence, every finite $Sq^0$-family in the coinvariants has at most $(s-2)$ nonzero elements. Two applications are exploited.
The first main theorem is that $Tr_s$ is not an isomorphism for $s\geq 5$.  Furthermore, for every $s>5$, there are infinitely many degrees in which  $Tr_s$ is not an isomorphism.  We also show that if $Tr_{\ell}$ detects a nonzero element in certain degrees of  $\text{Ker}(Sq^0)$, then it is not a monomorphism and further, for each $s>\ell$,  $Tr_s$ is not a monomorphism in infinitely many degrees.
The second main theorem is that the elements of any $Sq^0$-family in the cohomology of the Steenrod algebra, except at most its first $(s-2)$ elements, are either all detected or all not detected by $Tr_s$, for every $s$. This study is applied to investigate the behavior of $Tr_s$ for $s=4, 5$.
Based on the above study, we state a conjecture that the squaring operation $Sq^0$ is eventually isomorphic on Ext. More precisely, for every natural number $s$,  there is $t= t(s)$ such that any finite $Sq^0$-family in homological degree $s$ of the cohomology of the Steenrod algebra contains  at most $t$ nonzero elements.

Date and place: December 5, 2014. 12:00. CRM room A1.
Speaker : Frank Neumann (University of Leicester, UK)

Weil Conjectures and Moduli of Vector Bundles
In 1949, Weil conjectured deep connections between the topology, geometry and arithmetic of projective algebraic varieties over a field in characteristic p, including an analogue of the celebrated Riemann Hypothesis. These conjectures led to the development of l-adic etale cohomology in algebraic geometry as an analog of singular cohomology in algebraic topology by Grothendieck and his school and culminated with the proof of the Weil conjectures by Deligne in the 70s. In this talk I will outline how analogs of these Weil conjectures can be formulated and proved for the moduli stack of vector bundles over a projective algebraic curve in characteristic p. This basically comes down to counting correctly vector bundles up to isomorphisms via groupoid cardinalities using Behrend's trace formula for algebraic stacks. On the way I will recall the classical Weil conjectures and Grothendieck's ideas for using stacks in moduli problems.

Date and place: November 28, 2014. 15:00. CRM.
Speaker : Ethan Berkove (Lafayette College, USA)

Calculating the cohomology of Bianchi groups
Let $\mathcal{O}$ be an imaginary quadratic extension of the rational numbers.  The Bianchi groups are the matrix groups $PSL_2(\mathcal{O})$ (or sometimes the corresponding $SL_2$ groups).  In this way, the Bianchi groups can be thought of as generalizations of the modular group $PSL_2(\mathbb{Z})$.  Bianchi groups have been studied for over a century, in fields as varied as group theory, number theory, and topology.

In this talk we will provide an introduction to Bianchi groups and overview of techniques for calculating their (co)homology.  Many earlier calculations where done primarily on a case-by-case basis.  Recently, however, Alexander Rahm introduced a cellular space, the $p$-\textit{torsion subcomplex}, on which the Bianchi groups act.  We will define this complex and describe some recent results of Alexander Rahm and the speaker which use it.   

Date and place: November 20, 2014. 16:00. CRM, POL-1.
Speaker : Matthew Gelvin

Minimal characteristic bisets for saturated fusion system
A \emph{characteristic biset} for the fusion system $\mathcal{F}$ on the $p$-group $S$ is a finite set $\Omega$ with commuting left and right $S$-actions.  $\Omega$ must satisfy certain properties that mimic the left and right multiplications of $S$ on a finite Sylow supergroup inducing $\mathcal{F}$.
The parameterization of $\mathcal{F}$-sets implies the existence of a unique minimal characteristic biset $\Omega_{\mathcal{F}}$, which should be thought of as \emph{the} minimal characteristic biset for $\mathcal{F}$.  In this talk, I will defend this claim by discussing the close connections between $\Omega_{\mathcal{F}}$ and other central notions in $p$-local finite group theory, including $p$-constraint, centric linking systems, and $K$-normalizers.

Date and place: November 21, 2014. 12:00. CRM, A1.
Speaker : Alex González
(Kansas State University)
Irreducible components of p-local compact groups
Irreducibility was introduced in earlier work as a criterion to classify all p-local compact groups of rank 1, and could be thought of as an algebraic analog of connectivity. A p-local compact group is said to be irreducible if its underlying fusion system contains no proper normal subsystems of maximal rank. In this talk I will sketch how to show that every p-local compact group has a unique irreducible component, by using group theoretical techniques of Chermak.

Date and place: November 14, 2014. 12:00. CRM, A1.
Speaker : Paolo Salvatore
(Universitá di Roma "Tor Vergata")
Rational models of configuration spaces
I shall describe the rational homotopy type of some configuration spaces on manifolds and I shall report on the problem of their homotopy invariance.

Date and place: November 13, 2014. 16:00. CRM, POL-1.
Speaker : Matthew Gelvin

The combinatorics of the free monoid of $\mathcal{F}$-sets
Given a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$, say that a finite $S$-set $X$ is $\mathcal{F}$-stable, or an $\mathcal{F}$-set, if the action map $S\to\Sigma_X$ respects the fusion data.  Reeh proved that the monoid of $\mathcal{F}$-sets $A_+(\mathcal{F})$ has a free basis indexed by the $\mathcal{F}$-conjugacy classes of subgroups of $S$, which served as a starting point for our joint investigation of the minimal characteristic bisets of fusion systems.
In this talk I will review the essential point of the freeness result, which will emphasize the inductive nature of the construction of the basis of $A_+(\mathcal{F})$.  I will then describe joint work with Reeh and Yalcin that describes these basis elements in terms of the shape of $\mathcal{F}$.  Along the way we'll introduce a new combinatorial gadget ("broken chains") to compute a generalization of Möbius inversion on a finite poset.

Date and place: November 7, 2014. 12:00. CRM, A1.
Speaker : John Foley (Centre for Symmetry and Deformation, University of Copenhagen)

Recognizing nullhomotopic maps between the classifying spaces of Kac--Moody
Among the first applications of the successful proof of the Sullivan conjecture where characterizations of nullhomotopic maps between the classifying spaces of compact Lie groups.  This talk considers the problem of recognizing when maps between the classifying spaces of Kac--Moody groups---which generalize compact Lie groups---are nullhomotopic.  We show that certain known characterizations of nullhomotopic maps from the Lie setting extend to the Kac-Moody setting after completing at some prime $p$.  However, assembling this $p$-local information with the arithmetic fiber square exposes new subtleties for our integral recognition problem.

Date and place: October 17, 2014. 12:00. CRM, small lecture room.
Speaker : Assaf Libman (University of Aberdeen)

On homotopy groups of the spaces of self equivalences of
equivariant spheres.
It is a classical result that the n-th homotopy group of the space of self equivalences of the spheres $S^k$ where k=1,2,3,... stabilizes (on the n-th stable homotopy group of the spheres). Note that $S^k$ is the $k$-fold join of $S^1$.
An equivariant version of this question is: Suppose a finite group $G$ acts on a sphere $S^d$. What can one say about the n-th homotopy groups of the space of equivariant self equivalences of the k-fold joint of $S^r$ where $r=1,2,3,...$. The case when $S^r$ is a linear sphere will be studied.

Date and place: October 10, 2014. 12:00. CRM, small lecture room.
Speaker : Nguyen The Cuong

On  the algebraic EHP sequence
One of the most basic problems in homological algebra is to construct explicit injective (projective) resolutions of modules. We are interested in finding such construction
for the reduced co-homology $\tilde{H}^*(S^n; \mathbb F_2)$ of the spheres $S^n$ in the category of unstable modules U. The pseudo-hyper resolution is developed as an attempt to deal with this problem. The question is at present far from being solved but we manage somehow to archive some interesting properties of minimal injective resolutions
of $\tilde{H}^*(S^n; \mathbb F_2)$ in U. These results provide a new point of view for the problem of computing the E2 page of the unstable Adams spectral sequence converging to homotopy groups of spheres. One of the first applications of these computations, which is also the main goal of this talk, is an elementary reconstruction of the algebraic EHP sequence and an approximation to the sphere of origin question.

Date and place: September 26, 2014. 12:00. CRM, small lecture room.
Speaker :
Constanze Roitzheim (University of Kent, UK)
Homological Localisation of Model Categories
Bousfield localisation with respect to generalised homology theories of either spaces or spectra has proved to be a powerful tool in topology. We present a feasible version of homological localisation for general model categories and show some examples and applications.

Date and place: September 12, 2014. 10:30am. CRM, small lecture room.
Speaker :
Anthony Blanc (University of Vienna)
Topological K-theory of complex dg-categories.
In noncommutative algebraic geometry à la Kontsevitch, the role of de Rham cohomology is played by periodic cyclic homology. It is then natural to ask : what is the noncommutative extension of Betti cohomology of complex algebraic varieties ? Based on Toën and Bondal's idea, we have defined a topological K-theory of dg-categories defined over the complex numbers, as well as a Chern map into periodic cyclic homology. The definition involves a non trivial topological result, namely a homotopical version of Deligne's proper cohomological descent. This invariant furnishes a candidate for a rational structure on the periodic cyclic homology of a smooth proper dg-algebra, making a step toward the dream of the existence of a noncommutative Hodge structure on the latter. It indeed gives a rational structure in the case of a smooth scheme, and of a finite dimensional associative algebra.

Date and place: September 12, 2014. 12am. CRM, small lecture room.
Speaker : Gustavo Granja (Instituto Politécnico Superior de Lisboa)
Symplectic embeddings in infinite codimension

Let $X$ be a union of a sequence of symplectic manifolds whose dimensions tend to infinity and let $M$ be a manifold with a closed $2$-form. We use Tischler's elementary method for constructing symplectic embeddings to prove a lifting property for the map that pulls back the limiting symplectic form on $X$ along an embedding of $M$. As applications we show that, under suitable assumptions, the inclusion of the space of embeddings of $M$ in $X$ which preserve the symplectic form in the space of maps preserving the cohomology class of the closed $2$-forms is a weak equivalence and that a standard family of closed $2$-forms on a product of $\CP^infty$s is versal among compact families of closed $2$-forms not intersecting the trivial cohomology class. This is joint work with Manuel Araujo.