We consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in $\mathbb{R}^3$ from its cross-sections with a given set of cutting planes having arbitrary orientations. Under appropriate sampling conditions that are satisfied when the set of cutting planes is dense enough, we prove that the algorithm presented by Liu et al. in [1] preserves the homotopy type of the object. Using the homotopy equivalence, we also show that the reconstructed object is homeomorphic (and isotopic) to the original object. This is the first time that 3D shape reconstruction from cross-sections comes with such theoretical guarantees. [1] L. Liu, C.L. Bajaj, J.O. Deasy, D.A. Low, and T. Ju. Surface reconstruction from non-parallel curve networks. Computer Graphics Forum, 27:155-163, 2008.