  This article investigates the signature of the seventeen multi-connected flat
spaces in cosmic microwave background (CMB) maps. For each such space it
recalls a fundamental domain and a set of generating matrices, and then goes on
to find an orthonormal basis for the set of eigenmodes of the Laplace operator
on that space. The basis eigenmodes are expressed as linear combinations of
eigenmodes of the simply connected Euclidean space. A preceding work, which
provides a general method for implementing multi-connected topologies in
standard CMB codes, is then applied to simulate CMB maps and angular power
spectra for each space. Unlike in the 3-torus, the results in most
multi-connected flat spaces depend on the location of the observer. This effect
is discussed in detail. In particular, it is shown that the correlated circles
on a CMB map are generically not back-to-back, so that negative search of
back-to-back circles in the WMAP data does not exclude a vast majority of flat
or nearly flat topologies.

Astrophysics