Only rational homology spheres admit $\mathrm{\Omega }(f)$ to be union of DE attractors

If there exists a diffeomorphism $f$ on a closed, orientable $n$-manifold $M$ such that the non-wandering set $\mathrm{\Omega }(f)$ 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.