# Write homogeneous spherical space forms as coset spaces

Hi,

Let $S=G/K$ be a sphere written as coset space. I know there are just few possibilities for $G$, and $K$ due to the classification of compact connected groups that can be transitive on a sphere.

If $F \subset G$ is a finite fixed point free subgroup, can we write $S/F = G / (F \times K)$??

This should be true for real projective spaces, but for example how about lens spaces $S^{2n-1}/\mathbb Z_q$?

thanks

David

• No, generally you can't do this. It's easy enough to work out in the case of lens spaces using linear algebra -- elements of $K$ do not commute with most of the elements of $F$. – Ryan Budney Nov 26 '11 at 21:54
• And in the general case what if $F$ and $K$ commute? For example for finite subgroups of $SU(2)$ – David P Nov 26 '11 at 23:23
• Sorry for adding a comment as answer. Anyway it is enough to have $F$ (finite) normal (hence cenral) in $G$, or just central to well-define a $G$-action on $M/F$, but what about other sufficient cases? For instance, if we see $S^{2n-1} = U(n)/U(n-1)$ and $F$ is the cyclic group acting by multiplying by $e^{2 \pi i/q}$ on each complex coordinates, of course $F$ is central in $G$ and we can say that $S^{2n-1}/F$ is homogeneous under $U(n)$. How about a $SO(2n)$-action? – David P Nov 28 '11 at 21:37

so what are the spherical space forms homogeneous under? Starting from various ways to write the spheres themselves $S^{n-1} = SO(n)/SO(n-1)$, $S^{2n-1} = SU(n)/SU(n-1)$ and $S^{4n-1} = Sp(n)/Sp(n-1)$ ??
Is it enough to have $F$ commute with $K$? (notation as in the first post)