Realization problems for the group of homotopy classes of self-equivalences

Abstract

The group of homotopy classes of homotopy equivalences from a space X to itself, which we write as G(X), has been studies in recent years by Arkowitz and Curjel [1,2], the author [6,7] and several others. It is a natural quotient of the associative H-space made from homotopy-equivalcnces from X to itself under composition, and it has the same position, in the homotopy category, as the group of automorphisms of a group in the category of groups. It also has the practical use of determining how much the choice of k-invariants overdetermines the homotopytype of a space. G(X) is usually a complicated, non-abelian group, but we often have strong information -for example, we know in important cases, that it is finitely-presented (see [7]).Many simple questions are unfortunately not answered. For example: When does G(X) have torsion? When is G(X) finite? One basic problem, due to M.Arkowitz, asked whether there is a reasonable space X with G(X)=Z, the group of integers. The only case where we have the full picture is the fact that G(K(~, n)) = Aut (g).Our goals here are the following:1) We give a characterization of when G(X) is finite, when X is a reasonable space with finite k-invariants. Our results cover H-spaces and generalize results in [1] and [2].2) We construct a space, X, with non-trivial rational homology, for which G(X) is trivial. This shows that there is no reasonable analogue of ~-theory for G(X), even when ~ is a collection of finite groups. It also shows that one cannot expect to deduce the result of 1) above from the corresponding theorem for a product of a finite number of K(GI, 2ni + 1) spaces. We also indicate how one may get a finite complex with the same properties as X.3) We construct a reasonable space X, which has four non-vanishing homotopy groups, for which G(X)= Z.4) We prove two easy propositions which show that the situation in 2) or 3) above cannot happen in the usual stable cases.5) Finally, we indicate various open questions in this area, which should be within reach of present methods.

Realisability problem in arrow categories

“…This question was also addressed by Kahn in the 1960's for C = HoT op * , the homotopy category of simply-connected pointed topological spaces, and it received a remarkable attention. However, the tools for seriously digging into Kahn's question were at the time insufficient and this problem appeared recurrently in lists of open problems and surveys in homotopy theory [1,10,15,16,19]. The impasse ended with a general method by the authors of this paper, [5], that gives a solution to the classical group realisability problem in C = HoT op * for the case of finite groups.…”

The group of self-homotopy equivalences ofAn2-polyhedra

“…Let E(X) denote the group of homotopy classes of self-homotopy equivalences of a space X and E * (X) denote the normal subgroup of self-homotopy equivalences inducing the identity on the homology groups of X. 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].…”

Homotopically rigid Sullivan algebras and their applications

“…An obvious example is X = K(Z/2, n) since E(K(Z/2, n)) ∼ = Aut(Z/2) ∼ = * . The first elaborated example with non trivial rational homology is constructed in [17] by D. Kahn. He expressed in [18] the belief that spaces with trivial group of selfhomotopy equivalences, named homotopically rigid spaces, might play a role in some way of decomposing a space.…”