A&A 439, 479-485 (2005)

DOI: 10.1051/0004-6361:20042081

## A constraint on any topological lensing hypothesis in the spherical case: it must be a root of the identity

**B. F. Roukema**

Torun Centre for Astronomy, N. Copernicus University, ul. Gagarina 11, 87-100 Torun, Poland

e-mail: boud@astro.uni.torun.pl

(Received 28 September 2004 / Accepted 9 May 2005)

** Abstract **

Three-dimensional catalogues of objects at cosmological
distances can potentially yield candidate
topologically lensed pairs of sets of objects, which would
be a sign of the global topology of the Universe. In the spherical
case (i.e. if curvature is positive), a necessary condition, which
does not exist for either null or negative curvature, can
be used to falsify such hypotheses, *without needing to loop through
a list of individual spherical 3-manifolds*. This condition is that the
isometry between the two sets of objects
must be a root of the identity isometry in the covering space *S*^{3}.
This enables numerical falsification of topological lensing hypotheses
without needing to assume any particular spherical 3-manifold.
By embedding *S*^{3} in euclidean 4-space,
,
this condition can be expressed as the requirement that
*M*^{n} = *I* for an integer *n*, where *M* is the matrix representation of the
hypothesised topological
lensing isometry and *I* is the identity.
Moreover, this test becomes even simpler with the requirement that
the two rotation angles,
, corresponding to the given isometry,
satisfy
.
The calculation of this test involves finding the two eigenplanes of
the matrix *M*.
A GNU General Public Licence numerical package, called eigenplane,
is made available for finding the rotation angles
and eigenplanes of an arbitrary isometry *M* of *S*^{3}.

**Key words:**cosmology: observations

**--**cosmological parameters

**--**cosmic microwave background

**--**quasars: general

**©**

*ESO 2005*