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Barcelona Algebraic Topology Group

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Projects

  • Teoría de homotopía de estructuras algebraicas MTM2016-80439-P

      Project leaders:
     

    This project belongs to the field of Algebraic Topology, in a broad sense, and with applications going both ways between algebra and topology. While originally, algebraic topology meant the study of homotopy types of topological spaces by the means of algebraic invariants assigned to them functorially, the past decade has seen a proliferation of applications of homotopical methods in various areas of mathematics, such as notably algebraic geometry, representation theory and aspects of mathematical physics. The project is concerned more specifically with applications of methods of homotopy theory to the understanding of fusion systems both finite and compact modeling conjugacies in finite groups and Lie groups. In this context we attack several important open problems, some of which are of interest in group theory.

    We pursue the program for the classification of homotopy classes of maps between classifying spaces, also with the aim of developing complex homotopy representation theory. In a more algebraic setting, we aim to understand systematically exotic examples of fusion systems and construct new ones. As a new line of research we will start the study of stable homotopy theory of p-local compact groups, which is the starting point for extending statements on generalized cohomology theories for Lie groups.

    The techniques of localisation and cellularisation belong properly both to algebraic topology and to algebra. We aim to investigate to which extent homotopy idempotent functors applied to classifying spaces detect algebraic structure. A more geometric aspect of the project goes in the direction of properties of the mapping class group, the group of isotopies of an orientable surface, and its relationship with the invariants of homology spheres.

    We develop the theory of homotopy combinatorics, where the homotopy theory of groupoids and infinity-groupoids is applied to problems in enumerative and algebraic combinatorics, with special emphasis on applications to quantum field theory. In particular we develop a unifying framework for incidence algebras, combinatorial Hopf algebras, inductive data types, and related structures, uncovering common patterns in a wide range of application areas.

     


  • 2014SGR 42 Grup de Topologia Algebraica de Barcelona

      Members:
     

    Recerca en topologia algebraica, principalment en l'àmbit de la teoria d'homotopia moderna i
    les seves relacions amb la teoria de grups i la teoria de categories. Teoria de grups p-locals
    finits i compactes, functors homotòpics idempotents, "mapping class group", TQFT, teoria
    homotòpica de tipus.


  • MTM2013-42293-P-Homotopy theory of classifying spaces and function complexes

      Project leaders:
     

    A central problem in homotopy theory is the study of homotopy classes of maps between topological spaces. A basic
    technique in topology is to reduce a geometric classification problem to homotopy classification of maps into a classifying
    space. Much of the research in homotopy theory in recent decades involves the analysis of the homotopy type of
    classifying spaces, or the study of maps between them.

    The study of the homotopy type of a classifying space involves the so-called local analysis: isolation of the p- local
    structure of the object at each prime p. One of the most important and influential results in the subsequent study of
    homotopy classes of maps is the classification theorem by Jackowski , McCLure and Oliver on the space of selfmaps of
    BG, where G is a compact connected simple Lie group in terms of unstable operations Adams and Out (G). They use
    and develop new techniques that emerged from the proof of the Sullivan conjecture. The highlight may be the existence
    of homological decompositions isolating the homotopy type of BG at a prime p in the structure of subgroups and
    conjugates. In representation theory, local analysis is formalized in the fusion system of a finite group G, whose objects
    are p-subgroups of a fixed p-Sylow subgroup and morphisms are homomorphisms given by conjugation with elements of G.

    Algebraic and topological intuitions came together in the notion of p-local finite group (resp. p-compact) introduced by
    Broto, Levi and Oliver. In particular, in the finite case they describe mapping spaces and selfmaps of classifying spaces
    algebraically. For p-local compact groups many relevant questions in the study of mapping spaces remain open.
    The study of the homotopy theory of a space X through its mapping spaces map(A,X) is the basic idea in the
    development of the A-homotopy. In this context, the study is carried out through the functors of A-nullification and Acellularization.
    Determining the precise values of these functors on classifying spaces is a way of understanding which
    algebraic information is encoded in the corresponding mapping spaces.

    In a more geometric context, given an orientable surface, the group of isotopy classes is an ubiquitous object in
    mathematics. The interest and complexity of this group, mapping class group, lies in the fact that it allows to parametrize
    closed manifolds of dimension 3 by Heegard splittings. The study of its properties and its extensions is a very active
    topic in which a lot of progress is being made and in which we plan to contribute with the analysis of its universal central
    extensions.

    Finally, the idea of a homotopy version of combinatorics (Joyal, Baez-Dolan) goes back to Grothendieck: the symmetries
    of objects prevent the existence of classifying spaces as presheaves of sets, it is necessary to use homotopy types. We
    propose to explore the homotopy theory of homotopically finite infinite-grupoids to conceptualize combinatorial constructions
    in algebraic topology and in quantum field theories.
    In parallel, we pay attention to the development of homotopy type theory, recently stablished (Voevodsky), with the goal
    of investigating classical topological interpretations of syntactic arguments in recent progress.


  • MTM2010-20692 Análisis local en grupos y espacios topológicos

      Project leaders:
    • Natàlia Castellana Vila
     

    The nature of the classifying space of a topological group allows both and homotopic and algebraic
    analysis from a local point of view by isolating the relevant information to a prime p. This duality is present
    in the notion of p-local finite group or p-local compact group introduced by Broto, Levi and Oliver. An
    algebraic object that contains the essential information to describe the homotopy type of theoir p-
    completed classifying spaces. Conversely, given the classifying space one recovers the algebraic object.
    In addition to the p-local structure of finite groups and compact Lie groups, many objects which are purely
    homotopic like p-compact groups and new exotic examples recently found can now be discretely modeled
    by an algebraic structure. All of them must join the finite loop spaces whose relationship with the theory of
    p-local groups has not been fully determined. This theory has seen great development in recent years but
    still remain many unsolved issues that occupy much of this project. 

    The local information of p-local groups is defined in terms of certain categories. An abstract approach to
    algebra of a category would give birth to new technologies and strategies for the study of derived functors
    which is essential in obstruction theory. 

    Finite loop spaces are examples of H-spaces with multiplication given by composition of loops. The
    research which started in the last project has several ways of generalizing concepts such as p-compact
    group by weakening the finiteness properties. In addition, the techniques developed for these spaces
    makes plausible the analysis of certain realizability problems in mod p cohomology in the context of H-
    spaces. 

    Finally, taking as its starting point the study of invariants of homology spheres of dimension 3, this project
    continues with the study of 3-dimensional homology spheres modulo p, where p is a prime number.


  • 2009SGR-1092 - Grup de Topologia Algebraica de Barcelona

      Project leaders:

  • 2005SGR-00606 - Grup de Topologia Algebraica de Barcelona

      Project leaders:

  • MTM2004-06686 Estructura de espacios clasificadores de grupos y acciones de grupos

      Project leaders:

  • 2000SGR-00132 Grup de Topologia Algebraica de Barcelona

      Project leaders:

  • BFM2001-2035 Teoría homotópica de grupos finitos, grupos de Lie y grupos de Kac-Moody

      Project leaders:

  • EEC HPRN-CT-1999-00119 Modern Homotopy Theory

      Project leaders:
    • John Hubbuck

  • PB97-0203 Teoría homotópica de Grupos de Lie

      Project leaders:

  • HB1998-0148 Acción integrada Hispano-Britànica

      Project leaders:

  • INTAS 96 0712 Homotopy theory, low dimensional topology and cohomology of groups Comisión Europea

      Project leaders:

  • PB94-0725 Métodos cohomológicos en homotopía inestable

      Project leaders:

  • ERBCHRXCT 940560 Homotopy theory and its aplications

      Project leaders:
    • Renzo Piccinini

  • ERBCHRXCT 940441 Elliptic Cohomology

      Project leaders:
    • Charles Thomas

  • PB91-0467 Grupos y espacios clasificadores

      Project leaders:

  • MTM2007-61545 - Estructuras multiplicativas en topología y teoría de homotopía

      Project leaders:

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