We prove that every finite group G can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces X. To construct those spaces we introduce a new technique which leads, for example, to the existence of infinitely many inflexible manifolds. Further applications to representation theory will appear in a separate paper.
In this paper we raise the realisability problem in arrow categories. Namely, for a fixed category C and for arbitrary groups H ≤ G 1 × G 2 , is there an object φ : A 1 → A 2 in Arr(C) such that Aut Arr(C) (φ) = H, Aut C (A 1 ) = G 1 and Aut C (A 2 ) = G 2 ? We are interested in solving this problem when C = HoT op * , the homotopy category of simplyconnected pointed topological spaces. To that purpose, we first settle that question in the positive when C = Graphs.Then, we construct an almost fully faithful functor from Graphs to CDGA, the category of commutative differential graded algebras, that provides among other things, a positive answer to our question when C = CDGA and, as long as we work with finite groups, when C = HoT op * . Some results on representability of concrete categories are also obtained.
We prove that the isomorphism type of a large class of groups (containing finite groups, countable Artinian groups and mapping class groups of certain surfaces, among others) is determined by the set of differential graded Q-algebras on which these groups act faithfully.
In this paper we construct an infinite family of homotopically rigid spaces. These examples are then used as building blocks to forge highly connected rational spaces with prescribed finite group of self-homotopy equivalences. They are also exploited to provide highly connected inflexible and strongly chiral manifolds.
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