Consider three 2-torus ($S^1\*S^1$) living in four space. Can I have links of these objects, which is generalization of links of circles in 3D? If so, how can I judge whether three 2-torus are linked or not?



Questions about knot (link) of surface in four dimension
Yes. 


There are a lot of ways to approach this.


Following Roseman, you can take a generic projection of the torus into R^3, with overcrossing and undercrossing data. It will
have double points, triple points and branched points. There is a presentation of
the fundamental group of the complement of the torus using meridians around each facet of the immersed surface
with Wirtinger type relators of from the double curves. If you can show it it isn't
the fundamental group of the complement of the standard torus you are in.


Roseman also developed Reidemeister like moves for these.


Also the number of triple points can be used to detect knottedness.


An alternative approach is due to Carter and Saito, it involves looking at slices of the
knots as a family of R^3's moves through R^4 to give a movie of a link progressing in space with Morse type singularities. There are movie moves similar to the Reidemeister moves. From these you can attempt TQFT type invariants of the knotted surface to detect knottedness.


There is also a theory of braids in 4 dimensions due to Kamada.


However, there are more subtle invariants of knottedness than the fundamental group. You can start looking at characteristic classes of the normal bundle for instance.


