  We propose an alternative formalism to simulate CMB temperature maps in
$\Lambda$CDM universes with nontrivial spatial topologies. This formalism
avoids the need to explicitly compute the eigenmodes of the Laplacian operator
in the spatial sections. Instead, the covariance matrix of the coefficients of
the spherical harmonic decomposition of the temperature anisotropies is
expressed in terms of the elements of the covering group of the space. We
obtain a decomposition of the correlation matrix that isolates the topological
contribution to the CMB temperature anisotropies out of the simply connected
contribution. A further decomposition of the topological signature of the
correlation matrix for an arbitrary topology allows us to compute it in terms
of correlation matrices corresponding to simpler topologies, for which closed
quadrature formulae might be derived. We also use this decomposition to show
that CMB temperature maps of (not too large) multiply connected universes must
show ``patterns of alignment'', and propose a method to look for these
patterns, thus opening the door to the development of new methods for detecting
the topology of our Universe even when the injectivity radius of space is
slightly larger than the radius of the last scattering surface. We illustrate
all these features with the simplest examples, those of flat homogeneous
manifolds, i.e., tori, with special attention given to the cylinder, i.e.,
$T^1$ topology.
