  We introduce a generic procedure to reduce a supersymmetric Yang-Mills (SYM)
theory along the Hopf fiber of squashed $S^{2r-1}$ with $U(1)^r$ isometry, down
to the $\mathbb{CP}^{r-1}$ base. This amounts to fixing a Killing vector $v$
generating a $U(1)\subset U(1)^r$ rotation and dimensionally reducing either
along $v$ or along another direction contained in $U(1)^r$. To perform such
reduction we introduce a $\mathbb{Z}_p$ quotient freely acting along one of the
two fibers. For fixed $p$ the resulting manifolds $S^{2r-1}/\mathbb{Z}_p\equiv
L^{2r-1}(p,\pm 1)$ are a higher dimensional generalization of lens spaces. In
the large $p$ limit the fiber shrinks and effectively we find theories living
on the base manifold. Starting from $\mathcal{N}=2$ SYM on $S^3$ and
$\mathcal{N}=1$ SYM on $S^5$ we compute the perturbative partition functions on
$L^{2r-1}(p,\pm 1)$ and, in the large $p$ limit, on $\mathbb{CP}^{r-1}$,
respectively for $r=2$ and $r=3$. We show how the reductions along the two
inequivalent fibers give rise to two distinct theories on the base. Reducing
along $v$ gives an equivariant version of Donaldson-Witten theory while the
other choice leads to a supersymmetric theory closely related to Pestun's
theory on $S^4$. We use our technique to reproduce known results for $r=2$ and
we provide new results for $r=3$. In particular we show how, at large $p$, the
sum over fluxes on $\mathbb{CP}^2$ arises from a sum over flat connections on
$L^{5}(p,\pm 1)$. Finally, for $r=3$, we also comment on the factorization of
perturbative partition functions on non simply connected manifolds.
