Subject: Re: two postings
From: Allen Hatcher <hatcher@math.cornell.edu>
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 <siuporlamlam@yahoo.co.uk>
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

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