Isotropy groups of homotopy classes of maps

Abstract

Let aut ⁡ ( X ) \operatorname {aut}(X) be the group of homotopy classes of self-homotopy equivalences of a space X X and let [ f ] ∈ [ X , Y ] [f] \in [X,Y] be a homotopy class of maps from X X to Y Y . The aim of this paper is to prove that under certain nilpotency and finiteness conditions the isotropy group aut ⁡ ( X ) [ f ] \operatorname {aut}{(X)_{[f]}} of [ f ] [f] under the action of aut ⁡ ( X ) \operatorname {aut}(X) on [ X , Y ] [X,Y] is commensurable to an arithmetic group. Therefore aut ⁡ ( X ) [ f ] \operatorname {aut}{(X)_{[f]}} is a finitely presented group by a result of Borel and Harish-Chandra.