If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$
such that the non-wandering set $\Omega(f)$ consists of finitely many
orientable $(\pm)$ attractors derived from expanding maps, then $M$ must be a
rational homology sphere; moreover all those attractors are of topological
dimension $n-2$.

Expanding maps are expanding on (co)homologies.