分野別セミナー

The realizability problem for (finite) groups in a category C consists on deciding whether for a (finite) group G, there exists an object in C such that the automorphisms group of that object is isomorphic to G .
If we consider, for example, the category {\it Groups}, no such an object is found for its automorphisms group to be isomorphic to Zp, p odd. In contrast, the problem of realizability of groups does admit a positive answer in the category {\it Graphs}.
Considered as one of the most relevant questions in the study of homotopical equivalences, i.e., automorphisms groups in the homotopy category of well-pointed spaces HoTop_{*}, it has been appearing in lists of open problems since the 60's.
In this talk, we will explain how to solve the realizability problem for finite groups in HoTop_{*}, by introducing a general technique which happens to be useful in subjects of different nature, such as Differential Geometry or Representation Theory. We will also present further developments for the infinite case.