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

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, 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.