# The fundamental groups of 3-dimensional spherical space forms

Let $$S^3/\Gamma_i\,(i=1,2)$$ be a $$3$$-dimensional spherical space form, where $$\Gamma_i \subset SO(4)$$ is a finite subgroup acting freely on $$S^3$$. If $$S^3/\Gamma_1$$ is homotopy equivalent to $$S^3/\Gamma_2$$, can we find a $$g \in SO(4)$$ such that $$g\Gamma_1 g^{-1}=\Gamma_2?$$

This is not true in general, in fact it can fail for the lens spaces which are the quotients $$L(p;q) = S^3 / \Gamma(p;q)$$ where $$\Gamma(p;q)$$ is the cyclic subgroup of order $$p$$ in $$SO(4)$$ that is generated by the element described, in complex coordinates, by the formula $$(z_1,z_2) \mapsto (e^{2 \pi i/p} \cdot z_1, e^{2 \pi i q/p} \cdot z_2)$$ The link given explains how to determine when $$L(p;q_1)$$ and $$L(p;q_2)$$ are homotopy equivalent and when they are homeomorphic, and these do not match up. And if they are not homeomorphic then there cannot be any $$g \in SO(4)$$ for which $$g \Gamma(p;q_1) g^{-1} = \Gamma(p;q_2)$$, because the map of $$SO(4)$$ determined by $$g$$ would descend to a homeomorphism between $$L(p;q_1)$$ and $$L(p;q_2)$$.