  We investigate how well the 3D density field of neutral hydrogen in the
Intergalactic Medium (IGM) can be reconstructed using the Lyman-alpha
absorptions observed along lines of sight to quasars separated by arcmin
distances in projection on the sky. We use cosmological hydrodynamical
simulations to compare the topologies of different fields: dark matter, gas and
neutral hydrogen optical depth and to investigate how well the topology of the
IGM can be recovered from the Wiener interpolation method implemented by Pichon
et al. (2001). The global statistical and topological properties of the
recovered field are analyzed quantitatively through the power-spectrum, the
probability distribution function (PDF), the Euler characteristics, its
associated critical point counts and the filling factor of underdense regions.
The local geometrical properties of the field are analysed using the local
skeleton by defining the concept of inter-skeleton distance. At scales larger
than ~1.4 <d_LOS>, where <d_LOS> is the mean separation between lines of sight,
the reconstruction accurately recovers the topological features of the large
scale density distribution of the gas, in particular the filamentary
structures. At scales larger than the intrinsic smoothing length of the
inversion procedure, the power spectrum of the recovered HI density field
matches well that of the original one and the low order moments of the PDF are
well recovered as well as the shape of the Euler characteristic. The integral
errors on the PDF and the critical point counts are indeed small, less than 20%
for <d_LOS>~2.5 arcmin. The small deviations between the reconstruction and the
exact solution mainly reflect departures from the log-normal behaviour that are
ascribed to highly non-linear objects in overdense regions.
