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.