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The group of self-equivaQences
(homotopy classes of base point preserving homotopy equivalences) of a certain
class of finite CW-complexes is studied. This cfass includes, in particular, all
closed, connected, n-manifolds M with finite fundamental group such that
πi(M) = 0,1 < i < n. Such complexes are easily seen to be the quotient space of a
fixed point free action of a finite group on a homotopy n-sphere, and include the
Klein-Clifford manifolds.
The main result characterizes this group as a normal subgroup of Aut(π1(X)), for
X in the above mentioned class, consisting of all 𝜃 such that 𝜃 induces either tke
identity map or the inverse map on Hn+1(π1(X);Z) = Zk, k being the order of
π1(X). This leads to a collection of general results on the algebraic structure of the
group of self-equivalences, as well as several explicit calculations, including the
recovery of results due to Olum.
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