Date: 18 Jan 98 16:54:43 EST
From: Martin.A.Arkowitz@Dartmouth.EDU (Martin A. Arkowitz)

The following may be of interest with regard to the first question of
Chris Phillips.
--Martin

For every odd prime p, Oka [Memoirs Fac. of Sci., Kyushu Univ., 34 (1980),
171-181] has constructed a space X_p whose group of homotopy classes of
homotopy self equivalences is Z_p.  The space X_p is the mapping cone of a map
from a Moore space of type (Z_p, 2p-2) (ie, first homology in dim 2p-2) to one
of type (Z_p, 4p-5).  It then immediately follows that for any non-zero
integral cohomology element x, h*(x) is not equal to -x for every homotopy
equivalence h of X.  This does not answer Phillips's question since X_p has its
first cohomology in dim 2p-1.