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The spherically symmetric dust model of Lemaitre-Tolman can describewormholes, but the causal communication between the two asymptotic regionsthrough the neck is even less than in the vacuum(Schwarzschild-Kruskal-Szekeres) case. We investigate the anisotropicgeneralisation of the wormhole topology in the Szekeres model. The functionE(r, p, q) describes the deviation from spherical symmetry if \partial_r E \neq0, but this requires the mass to be increasing with radius, \partial_r M > 0,i.e. non-zero density. We investigate the geometrical relations between themass dipole and the locii of apparent horizon and of shell-crossings. Wepresent the various conditions that ensure physically reasonablequasi-spherical models, including a regular origin, regular maxima and minimain the spatial sections, and the absence of shell-crossings. We show thatphysically reasonable values of \partial_r E \neq 0 cannot compensate for theeffects of \partial_r M > 0 in any direction, so that communication through theneck is still worse than the vacuum.  We also show that a handle topology cannot be created by identifyinghypersufaces in the two asymptotic regions on either side of a wormhole, unlessa surface layer is allowed at the junction. This impossibility includes theSchwarzschild-Kruskal-Szekeres case.Comment: zip file with LaTeX text + 6 figures (.eps & .ps). 47 pages. Second  replacement corrects some minor errors and typos. (First replacement prints  better on US letter size paper.

You cannot get through Szekeres wormholes: Regularity, topology, and causality in quasispherical Szekeres models