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Barcelona Algebraic Topology Group

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

University: Universitat Autònoma de Barcelona
Status: Finished  
Project leaders: Collaborators:
Proposed start date: 2014-01-01 Proposed end date: 2016-12-31

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

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.