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
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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
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