Realisability problems in Algebraic Topology

Abstract

Realisability problems in Algebraic Topology are very easy to state and extremely hard to solve. Three classic examples of this are: the realisability of cohomology algebras, proposed by N. E. Steenrod in the 1960s, which asks for a characterisation of graded algebras that appear as the cohomology of a certain space; the problem of Moore G-spaces, also proposed by Steenrod, which asks for the characterisation of ZG-modules that appear as the homology of Moore G-spaces; and the problem of realisability of abstract groups proposed by D. Kahn which asks for the characterisation of groups that appear as the group of self-equivalences of simply connected spaces. These three problems have in common the pursuit of spaces that realise an algebraic structure through an homotopy invariant. In this work we focus on Kahn’s problem, which was introduced in the sixties and received quite a lot of attention, even if progress towards a general solution to it was slow at first. Kahn’s problem is the case C = HoTop of the more general group realisability problem, which asks if every group appears as the automorphism group of an object in a given category C. And it is precisely a classical solution to this more general problem in the category of C = Graphs that led to the most important breakthrough so far with relation to Kahn’s problem: in 2014, Costoya-Viruel showed that every finite group is the group of self-homotopy equivalences of a rational space by building a nice functor from a subcategory of Graphs to the category C = CDGAs, algebraic models of rational homotopy types of spaces. Our main goal in this thesis is to expand on that sort of techniques to study further realisability problems. We consider two generalised realisability problems. The first one deals with realising a subgroup of the product of two groups in the context of arrow categories, and the second deals with the realisability of permutation representations. By using ideas of Costoya-Viruel, we first give positive solutions to these problems in the category of Graphs, to then translate them to other frameworks by introducing appropriate functors. We do so in the category of Coalgebras over any field and in the category of CDGAs over any integral domain of characteristic different to 2 and 3. Then, using techniques from Rational Homotopy Theory, we translate the solutions of such problems to C = HoTop. Furthermore, we exploit the interesting properties of our functors to obtain results regarding the isomorphism problem for groups through their faithful representations in both Coalgebras and CDGAs, the realisability of monoids and categories in the context of CDGAs, and the existence of highly connected inflexible and strongly chiral manifolds. Finally, as our solutions to the realisability problem in C = HoTop are obtained using tools of rational homotopy theory, the obtained spaces are not of finite type over Z. With the objective of giving a solution to Kahn’s problem in terms of simpler spaces, we consider the possibility of using the classification of homotopy types of A(n,2)-polyhedra of Whitehead and Baues as our algebraic framework. We are indeed able to use their classification to study self-homotopy equivalences of A(n,2)-polyhedra, but the results we obtain in this regard lead us to think that a positive solution to Kahn’s problem is not possible in this setting.