  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, 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 R^4, 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 lensing isometry and I is the identity.
Moreover, this test becomes even simpler with the requirement that the two
rotation angles, theta, phi, corresponding to the given isometry, satisfy 2\pi
/ \theta, 2\pi / \phi \in Z. 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 at
http://cosmo.torun.pl/GPLdownload/eigen/ for finding the rotation angles and
eigenplanes of an arbitrary isometry M of S^3.
