@MikeMiller Can you remind me the Morse theory argument for $H_*(\Omega S^n)$? This is what I vaguely remember: I take the energy functional on $\Omega S^n$ which is a Morse functional in an appropriate sense, whose critical points are closed geodesics at a fixed point on $S^n$. These would be a great circle wrapping around $m$ times.
The index is the multiplicity of the conjugate point antipodal to the basepoint, which is $n - 1$ (number of orthogonal directions to the geodesic, in which if I perturb I hit that point again). That gives a total index of $m(n-1)$. This is means $\Omega S^n$ …
Let $S^n/\Gamma_i\,(i=1,2)$ be a $n$-dimensional spherical space form, where $\Gamma_i \subset SO(n+1)$ is a finite subgroup acting freely on $S^n$.
Suppose $S^n/\Gamma_1$ is diffeomorphic to $S^n/\Gamma_2$, can we show they are isometric?
