Subject: Re: two postings From: Allen Hatcher Date: Fri, 23 Feb 2007 14:29:55 -0500 Here is a short reply to John Olsen's question. The group SO(3) is the lens space RP3, and SO(3) has a Z_2 + Z_2 subgroup (180 degree rotations about the coordinate axes in R3) which acts freely on SO (3) by left multiplication. Another way to look at this is to take the quaternion group Q_8 acting on S3 and first factor out by its center Z_2. Allen Hatcher > Consider a three dimensional lens space S3/\Z_k with an action of Z_2 > + Z_2. If k is odd this space is a Z_2 cohomology sphere and hence the > action is not free. If k is even is it still true that the action > cannot be free? > > Any references are much appreciated! > > Thank you, > John Olsen > ______________________________ Subject: Re: Question about lens spaces From: Siu Por Lam Date :Sat, 24 Feb 2007 21:38:03 +0000 (GMT) If we think of R4 as quaternions and S3 as the unit quaternions, then S3 is a group. Take Q(8)-the finite quaternion group of order 8, which acts (freely) on S3 using the group structure of S3. Take k=2 and Z/2 to be the subgroup Z={1, -1}. Then S3/Z is RP3 and Q(8)/Z(=the product of two Z/2) is a subgroup of RP3 and hence acts freely on RP3. I hope this helps. Siu Por Lam ___________________________________________________________