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Let X be a finite type An2-polyhedron, n ≥ 2. In this paper, we study the quotient group ℰ (X) / ℰ*(X), where ℰ (X) is the group of self-homotopy equivalences of X and ℰ*(X) the subgroup of self-homotopy equivalences inducing the identity on the homology groups of X. We show that not every group can be realised as ℰ (X) or ℰ (X) / ℰ *(X) for X an An2-polyhedron, n ≥ 3, and specific results are obtained for n = 2. [ABSTRACT FROM AUTHOR]

Let the pair ( G , M ) R denote a group G acting on an R -module M , R a unitary ring. We ask for the existence of an R -local space X such that ( E ( X ) , π k ( X ) ) R is equivalent, in a natural way, to ( G , M ) R , for some k ≥ 2 , where E ( X ) denotes the group of homotopy classes of self-homotopy equivalences of X . If such an X exists, we say that X realizes the group action ( G , M ) R . We prove that if G is finite and acts faithfully on a finitely generated Q -module M , there exist infinitely many rational spaces realizing ( G , M ) Q . Our proof relies on providing a positive answer to Kahn's realizability problem for a large class of orthogonal groups that strictly contains finite ones. As a matter of fact, we enlarge the class of groups that is known to be realizable in the classical Kahn's sense. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]