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

Toruń Centre for Astronomy, N. Copernicus University, ul. Gagarina 11, 87-100 Toruń, Poland e-mail: boud@astro.uni.torun.pl

Received:
28
September
2004

Accepted:
9
May
2005

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 .
This enables numerical falsification of topological lensing hypotheses
without needing to assume any particular spherical 3-manifold.
By embedding in euclidean 4-space, ,
this condition can be expressed as the requirement that
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*