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.