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

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 . 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 S³.