From boud at astro.uni.torun.pl  Wed May  9 09:35:45 2012
From: boud at astro.uni.torun.pl (Boud Roukema)
Date: Wed, 9 May 2012 09:35:45 +0200 (CEST)
Subject: [Cos-top] half-turn space analysis; new spherical 3-manifolds ?
In-Reply-To: <alpine.DEB.1.10.1009300957001.29892@adjani.astro.uni.torun.pl>
References: <alpine.DEB.1.10.1009300957001.29892@adjani.astro.uni.torun.pl>
Message-ID: <alpine.DEB.2.00.1205090922090.368@adjani.astro.uni.torun.pl>

hi cos-top

On Thu, 30 Sep 2010, Boud Roukema wrote:

> (2) http://arxiv.org/abs/1009.5825
> Multipole analysis in cosmic topology
> Authors: Peter Kramer
>
> This seems to claim that the author has found 3 new spherical
> 3-manifolds, "N8, N9, N10". It's not clear to me if he claims that
> they can be given constant curvature, but maybe it's obvious to
> someone who knows the mathematics a bit better.  i had thought that
> the constant curvature spherical 3-manifolds were already completely
> classified.
>
> Are N8, N9, and N10 new constant-curvature spherical 3-manifolds, in
> addition to those in Gausmann et al. 2001 http://arxiv.org/abs/gr-qc/0106033 
> ?

It looks like I forgot to reply on-list. From off-list discussion, it's
clear that these are just specific examples of constant curvature
spherical 3-manifolds. They are claimed to be "new" in the sense of not
having been specifically described in this way before, without claiming
that they are additional to the standard classification.

Aurich, Kramer & Lustig [1] give a direct answer in terms of the
similarly defined N2 and N3:

   N2 is a construction of the lens space L(8,3) - which is globally
   inhomogeneous - by starting at a specifically chosen point, around
   which the Dirichlet/Voronoi domain happens to be ... a cube;

and

   N3 = S^3/D_8^*.

cheers
boud


[1] Aurich, Kramer & Lustig, 2011,  Physica Scripta 84, 055901, arXiv:1107.5214



From sven.lustig at uni-ulm.de  Thu May 10 08:18:03 2012
From: sven.lustig at uni-ulm.de (sven.lustig at uni-ulm.de)
Date: Thu, 10 May 2012 08:18:03 +0200
Subject: [Cos-top] half-turn space analysis; new spherical 3-manifolds	?
In-Reply-To: <alpine.DEB.2.00.1205090922090.368@adjani.astro.uni.torun.pl>
References: <alpine.DEB.1.10.1009300957001.29892@adjani.astro.uni.torun.pl>
	<alpine.DEB.2.00.1205090922090.368@adjani.astro.uni.torun.pl>
Message-ID: <20120510081803.tn17lt740484g4w0@imap.uni-ulm.de>

hi cos-top!

Zitat von Boud Roukema <boud at astro.uni.torun.pl>:

> hi cos-top
>
> On Thu, 30 Sep 2010, Boud Roukema wrote:
>
>> (2) http://arxiv.org/abs/1009.5825
>> Multipole analysis in cosmic topology
>> Authors: Peter Kramer
>>
>> This seems to claim that the author has found 3 new spherical
>> 3-manifolds, "N8, N9, N10". It's not clear to me if he claims that
>> they can be given constant curvature, but maybe it's obvious to
>> someone who knows the mathematics a bit better.  i had thought that
>> the constant curvature spherical 3-manifolds were already completely
>> classified.
>>
>> Are N8, N9, and N10 new constant-curvature spherical 3-manifolds, in
>> addition to those in Gausmann et al. 2001   
>> http://arxiv.org/abs/gr-qc/0106033 ?
>
> It looks like I forgot to reply on-list. From off-list discussion, it's
> clear that these are just specific examples of constant curvature
> spherical 3-manifolds. They are claimed to be "new" in the sense of not
> having been specifically described in this way before, without claiming
> that they are additional to the standard classification.
>
> Aurich, Kramer & Lustig [1] give a direct answer in terms of the
> similarly defined N2 and N3:
>
>   N2 is a construction of the lens space L(8,3) - which is globally
>   inhomogeneous - by starting at a specifically chosen point, around
>   which the Dirichlet/Voronoi domain happens to be ... a cube;
>
> and
>
>   N3 = S^3/D_8^*.
>
> cheers
> boud
>
>
> [1] Aurich, Kramer & Lustig, 2011,  Physica Scripta 84, 055901,
> arXiv:1107.5214
>
> _______________________________________________
> Cos-top mailing list
> Cos-top at cosmo.torun.pl
> http://cosmo.torun.pl/mailman/listinfo/cos-top

In the notation of Peter Kramer N1-N7 are platonic manifolds.
In contrast N8-N11 are orbifolds. These orbifolds are generated from  
platonic manifolds using their discret rotation symmetry.

Best
Sven



From boud at astro.uni.torun.pl  Thu May 10 10:53:31 2012
From: boud at astro.uni.torun.pl (Boud Roukema)
Date: Thu, 10 May 2012 10:53:31 +0200 (CEST)
Subject: [Cos-top] half-turn space analysis; new spherical 3-manifolds ?
In-Reply-To: <20120510081803.tn17lt740484g4w0@imap.uni-ulm.de>
References: <alpine.DEB.1.10.1009300957001.29892@adjani.astro.uni.torun.pl>
	<alpine.DEB.2.00.1205090922090.368@adjani.astro.uni.torun.pl>
	<20120510081803.tn17lt740484g4w0@imap.uni-ulm.de>
Message-ID: <alpine.DEB.2.00.1205101026330.26785@adjani.astro.uni.torun.pl>

hi Sven, cos-top,

On Thu, 10 May 2012, sven.lustig uni-ulm.de wrote:
> Zitat von Boud Roukema <boud astro.uni.torun.pl>:

> In the notation of Peter Kramer N1-N7 are platonic manifolds.
> In contrast N8-N11 are orbifolds. These orbifolds are generated from platonic 
> manifolds using their discret rotation symmetry.

Thanks for the correction - I think you are saying that these are
orbifolds that are not manifolds. Is that right?

>From what I understand (e.g. [1] and discussions with Jeff and Vincent),
   manifold without boundary \Rightarrow orbifold,
but
   orbifold \not\Rightarrow manifold.

So that means that the word descriptions of N8-N10 in
http://arxiv.org/abs/1009.5825 (v1 and published version) are
incorrect in the sense that these are not 3-manifolds, although they
are 3-orbifolds.

Also, do you mean N8-N10? I don't see N11 defined in 1009.5285.

cheers
boud

[1] http://en.wikipedia.org/wiki/Orbifold



From sven.lustig at uni-ulm.de  Fri May 11 07:46:24 2012
From: sven.lustig at uni-ulm.de (sven.lustig at uni-ulm.de)
Date: Fri, 11 May 2012 07:46:24 +0200
Subject: [Cos-top] half-turn space analysis; new spherical 3-manifolds	?
In-Reply-To: <alpine.DEB.2.00.1205101026330.26785@adjani.astro.uni.torun.pl>
References: <alpine.DEB.1.10.1009300957001.29892@adjani.astro.uni.torun.pl>
	<alpine.DEB.2.00.1205090922090.368@adjani.astro.uni.torun.pl>
	<20120510081803.tn17lt740484g4w0@imap.uni-ulm.de>
	<alpine.DEB.2.00.1205101026330.26785@adjani.astro.uni.torun.pl>
Message-ID: <20120511074624.e2u58hyeos48c84k@imap.uni-ulm.de>

hi Boud, cos-top,

yes! N8-N10 that are orbifolds.

In arXiv:1201.1875 N8-N10 are also called orbifolds. In this paper one  
finds the orbifold N11.

Best,
Sven


Zitat von Boud Roukema <boud at astro.uni.torun.pl>:

> hi Sven, cos-top,
>
> On Thu, 10 May 2012, sven.lustig uni-ulm.de wrote:
>> Zitat von Boud Roukema <boud astro.uni.torun.pl>:
>
>> In the notation of Peter Kramer N1-N7 are platonic manifolds.
>> In contrast N8-N11 are orbifolds. These orbifolds are generated   
>> from platonic manifolds using their discret rotation symmetry.
>
> Thanks for the correction - I think you are saying that these are
> orbifolds that are not manifolds. Is that right?
>
> From what I understand (e.g. [1] and discussions with Jeff and Vincent),
>   manifold without boundary \Rightarrow orbifold,
> but
>   orbifold \not\Rightarrow manifold.
>
> So that means that the word descriptions of N8-N10 in
> http://arxiv.org/abs/1009.5825 (v1 and published version) are
> incorrect in the sense that these are not 3-manifolds, although they
> are 3-orbifolds.
>
> Also, do you mean N8-N10? I don't see N11 defined in 1009.5285.
>
> cheers
> boud
>
> [1] http://en.wikipedia.org/wiki/Orbifold
>
> _______________________________________________
> Cos-top mailing list
> Cos-top at cosmo.torun.pl
> http://cosmo.torun.pl/mailman/listinfo/cos-top