From: Greg Kuperberg <greg@matching.math.ucdavis.edu>
Subject: Re: Question 1
Date: Sun, 18 Jan 1998 07:45:23 -0800 (PST)

Chris Phillips, via Claude Schochet, asks:
>         I would like an example of a "nice" (see below) compact Hausdorff
> space X and a torsion class c in H^3 (X; Z) such that there does _not_
> exist any homotopy equivalence h : X ---> X with h^* (c) = -c. "Nice"
> means preferably a compact manifold; next best would be a finite complex.

I think this works:

Let G be the non-abelian group with 21 elements.  G has abelianization
Z/3 with kernel Z/7.  G does not admit an automorphism inverting the Z/3,
because if a is in Z/7, one of the elements in the Z/3 conjugates a to
a^2, while the other conjugates it to a^4.

Let M be a closed 4-manifold with pi_1(M) = G.  Then
H^3(M) = H_1(M) = Z/3, as desired.

I used the same group in:

    http://front.math.ucdavis.edu/math.GT/9712206

This paper is a simplification of examples, first found by my mom,
of locally connected continua X such that for every two points x and
y in X, there is a homoemorphism of X taking x to y, but sometimes no
homeomorphism exchanging x and y.

                                Greg