Vol. 54, No. 1, 1974

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The group of self-equivalences of certain complexes

David Smallen

Vol. 54 (1974), No. 1, 269–276

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.

Mathematical Subject Classification
Primary: 55D10
Received: 9 March 1973
Revised: 13 July 1973
Published: 1 September 1974
David Smallen