Is there a *graph manifold* (<https://en.wikipedia.org/wiki/Graph_manifold>) that doesn't admit an orientation reversing involution? If so, what would be a simple example?



A graph manifold without an orientation reversing involution?
$\mathbb{N}il^3$-manifolds and $\widetilde{SL}$-manifolds are orientable
and do not have orientation-reversing self homotopy equivalences.
This is most easily seen algebraically.
The fundamental group $\Gamma$ of the $S^1$-bundle over the torus $T$ with Euler class a generator of $H^2(T;\mathbb{Z})$ is a central extension of $\mathbb{Z}^2$ by $\mathbb{Z}$.
(This is the Heisenberg group of upper triangular matrices in $SL(3,\mathbb{Z})$.) Direct calculation of the automorphisms of $\Gamma$ shows that an automorphism which induces $A\in{GL(2,\mathbb{Z})}$ on the central quotient $\mathbb{Z}^2$ acts by $\det{A}$ on the centre, and hence is orientation-preserving on the extension.


In general, the fundamental groups of such 3-manifolds are virtually
central extensions by $\mathbb{Z}$ of orientable surface groups $S$,
with non-zero extension class in $H^2(S;\mathbb{Z})$.
Moreover, the centre of the group is $\mathbb{Z}$,
and so the extension is preserved under any automorphism of the group.
Considering the effect of an automorphism on the centre and the quotient
shows that the extension class is preserved if and only if the induced
automorphisms of the centre and the central quotient $S$ are both orientation-preserving or both orientation reversing.


An analogous situation holds for $\mathbb{S}^3$-manifolds.
However the argument breaks down for some lens spaces (as observed by Ryan), as these admit self-maps which do not preserve the Seifert fibration.


