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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 , where 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 ~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., Comment: 20 pages, 18 figures, accepted for publication in MNRAS