# Which spherical space forms embed in $S^4$?

Is there any hope of getting a classification of which 3-dimensional spherical space forms are smoothly embeddable in $S^4$? I read that lens spaces cannot embed in $S^4$, but some other spherical space forms can, such as $S^3/Q_8$, where $Q_8 = \{1,-1,i,-i,j,-j,k,-k\}$. I was also wondering if anyone has an idea of whether an infinite number of topologically distinct spherical space forms are embeddable in $S^4$.

• I suspect a complete classification is unknown. Some partial information can be found in mathoverflow.net/questions/104451/… and references therein. – Igor Belegradek Sep 26 '15 at 16:26
• Thank you Igor Belegradek for the reference. There, @Ryan Budney's answer cites Budney and Burton arXiv:0810.2346, which considers the more general question "which 3-manifolds embed smoothly in $S^4$" but confirms that a complete classification is unknown, even for spherical space forms. It seems only $S^3$ and $S^3/Q_8$ are known to embed and lens spaces, $S^3/Q_{40}$ and $S^3/P_{120}$ cannot embed. Is it reasonable to conjecture that only a finite number of spherical space forms are embeddable in $S^4$? If so, are there any general strategies to attack this problem? – Topology Student Sep 26 '15 at 18:10
• See Theorem 2.2 of Crisp and Hillman, which indeed shows that the only spherical manifolds which embed smoothly are $S^3$ and $S^3/Q_8$. The Poincaré dodecahedral space ($S^3/I^\ast$ in their notation) does not embed smoothly since it has non-zero Rochlin invariant (see e.g. Budney's paper). plms.oxfordjournals.org/content/76/3/685 – Ian Agol Sep 26 '15 at 19:47