Automorphisms of spaces with finite fundamental group

Abstract

Let X X be a finite CW-complex with finite fundamental group. We show that the group aut ( X ) {\text {aut}}(X) of homotopy classes of self-homotopy equivalences of X X is commensurable to an arithmetic group. If in addition X X is an oriented manifold then the subgroup au t t ( X ) {\text {au}}{{\text {t}}_t}(X) of homotopy classes of tangential homotopy equivalences is commensurable to an arithmetic group. Moreover if X X is a smooth manifold of dimension ⩾ 5 \geqslant 5 then the subgroup diff ( X ) {\text {diff}}(X) of aut ( X ) {\text {aut}}(X) the elements of which are represented by diffeomorphisms is also commensurable to an arithmetic group.