Every finite group is the group of self-homotopy equivalences of an elliptic space

Abstract

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.

Homotopically rigid Sullivan algebras and their applications

“…A decade after, Arkowitz and Lupton came across an example of a minimal Sullivan algebra (equivalently, a rational homotopy type of a space) M with E(M ) trivial, [3,Example 5.1]. This, let us say homotopically rigid algebra, M provides us with motivation and guidance: in our paper [5], we use M as a building block to construct minimal Sullivan algebras, [5,Definition 2.1], with the property that their group of self-homotopy equivalences are isomorphic to an arbitrary finite group G previously fixed, [5,Theorem 1.1]. In other words, every finite group G is realizable through minimal Sullivan algebras that are built upon M ; this is a partial answer to Kahn's realizability problem and we refer to [5] for more details.…”

“…Therefore, inflexible manifolds are extraordinary objects and still not many examples are known. Indeed, all the examples found in literature show low levels of connectivity when observing their minimal Sullivan models [3,1,6,5].…”

“…This, let us say homotopically rigid algebra, M provides us with motivation and guidance: in our paper [5], we use M as a building block to construct minimal Sullivan algebras, [5,Definition 2.1], with the property that their group of self-homotopy equivalences are isomorphic to an arbitrary finite group G previously fixed, [5,Theorem 1.1]. In other words, every finite group G is realizable through minimal Sullivan algebras that are built upon M ; this is a partial answer to Kahn's realizability problem and we refer to [5] for more details. Observe that as an application, for the trivial group G ∼ = {e}, M is then a building block for an infinite family of homotopically rigid minimal Sullivan algebras (equivalently, homotopically rigid rational spaces).…”

Realisability problem in arrow categories

“…Families of elliptic CDGA's. We follow a similar approach as in [4,5] where, to every finite simple graph G, a minimal Sullivan algebra M G is assigned in such a way that the group of self-homotopy equivalences of M G is isomorphic to the automorphism group of G. Our construction in [5, Definition 2.1] was based on a homotopically rigid algebra given in [2], and it was functorial only when restricted to the subcategory of full graph monomorphisms (see [5,Remark 2.8]). However, the morphism ϕ obtained in Theorem 1.2 (see also Corollary 3.7) is not a full monomorphism in general, so our previous construction is useless to answer Question 1.1 in Arr(CDGA).…”

The group of self-homotopy equivalences ofAn2-polyhedra

“…Problems related to E(X) have been extensively studied, deserving a special mention Kahn's realisability problem, which has been placed first to solve in [2] (see also [1,12,13,15]). It asks whether an arbitrary group can be realised as E(X) for some simply connected X, and though the general case remains an open question, it has recently been solved for finite groups, [7]. As a way to approach Kahn's problem, in [10,Problem 19] the question of whether an arbitrary group can appear as the distinguished quotient E(X)/E * (X) is raised.…”