Consider four intersecting open cylinders arranged in the unit cube where the caps of the cylinders are of arbitrariy small radius and 'look' globally as if they coincide with the vertices of the unit cube. Rigorously, the open cylinders are precisely $S^1\times(0,\sqrt{2})$ accumulating to their respective vertices. The matching of vertices to cylinder caps is a bjiection. Glue the caps of arbitrarily small radius together s.t. the top four vertices get glued to the bottom four vertices (they define opposite faces of the cube). Likewise glue the other pairs of faces together, to obtain an immersed surface inside the $3$-torus.
Here is another description (all before the glueing occurs): Partition $\Bbb R^3$ into ten $3$-cells. Delete the one non-compact $3$-cell. As for the other nine compact $3$-cells, delete their 'interior.' Delete the eight $0$-cells.
I think this immersed surface in the $3$-torus is an orbifold or pseudomanifold because it looks like it will degenerate at exactly $1$ point. But I think Dehn surgery can be used to make it a bonafide manifold.
Does Dehn surgery apply here? Can it be used successfully to obtain the surface?
My thinking is to use surgery on this 'bad' point. I've drawn a picture to portray this (forgot to draw that one cylinder on the RHS):
