The loop space on a topological space is a fundamental
construction in topology: it is the space of all continous paths joining
a prescribed point to itself. The concatenation of loops endows the loop space
with a structure of a group (up to homotopy). Conversely, it is known that any
group is in a sense equivalent to a loop space, therefore loop spaces give a
way to study groups by looking to some associated topological space.
In my talk I will explain the so-called ``scanning
method'' which gives a combinatorial approach to the study of loop spaces. It
shows that interesting loop space can be viewed as some space of configurations
of finitely many points in a given manifold. And vice versa, some important
configuration spaces turn out to be equivalent to the loops on certain easily
constructed spaces. This gives in particular the different viewpoint on some
groups like the symmetric group on infinitely many letters or the braid groups.
This approach was also used very recently to understand the group of
diffeomorphisms of surfaces with very large genus (i.e. the mapping class
group) which is a fundamental object in algebraic geometry in the study of
bundles by surfaces.