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Step 1: We glue two sets of complement space of $D^2\times T^2$ out of the 4-sphere $S^4$, through their $T^3$ boundary with their three $S^1$ boundaries of $T^3$ cyclic permuted to obtain a new 4-manifold:

$$(S^4 \smallsetminus D^2\times T^2) \cup_{\text{cyclic permutation}} (S^4 \smallsetminus D^2 \times T^2)=(S^2\times S^2),$$

Step 2: Meanwhile we insert two $T^2$ 2-tori along the generator of homology group of the first 4-manifold $(S^4 \smallsetminus D^2\times T^2)$, $$H_2[(S^4 \smallsetminus D^2 \times T^2),\mathbb{Z}]=\mathbb{Z}^2$$ and we insert two additional $T^2$ 2-tori along the two generators of the homology group $H_2[(S^4 \smallsetminus D^2 \times T^2),\mathbb{Z}]=\mathbb{Z}^2$ for the other $(S^4 \smallsetminus D^2\times T^2)$.

Step 3: Now you can imagine that the four 2-tori $T^2$ (inserted in Step 2) sit in $S^2 \times S^2$ (by the gluing procedure in Step 1) are somehow "linked" in the 4-manifold $S^2 \times S^2$.

My question is that how are the four $T^2$ linked (or not) in the glued manifold 4-manifold $S^2 \times S^2$? And are the four $T^2$ linked together?

Or are there any three of $T^2$ linked together? In the sense of the triple linking (defined for example in arXiv:math/0007141)?

Please describe what is the mathematical link of four $T^2$ tori in $S^2 \times S^2$.

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  • $\begingroup$ Define "linked" in your settings. $\endgroup$
    – Alex Degtyarev
    Sep 21, 2016 at 19:44
  • $\begingroup$ This question is exactly about defining the link in the mathematical terms -- such as in terms of the homology groups of sub-manifold -- if I know how to define the "link" rigorously, I would not ask. I give one example, where you can define the triple linking in the cited ref above. $\endgroup$
    – wonderich
    Sep 21, 2016 at 20:26
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    $\begingroup$ Usually "linked" would mean "not in the same isotopy class as the trivial embedding", i.e. you are giving the space of embeddings a base-point. The trivial embedding would be the image of a linear embedding in $\mathbb R^4$, after you embed the $\mathbb R^4$ in your $4$-manifold. $\endgroup$
    – Ryan Budney
    Sep 21, 2016 at 21:17
  • $\begingroup$ In addition to what @RyanBudney said about link vs. unlink, "linked" may refer to a number (or collection thereof) defined via some sort of Alexander like duality. But in a situation as rich as yours, it's your choice what you call the linking number. (Of course, just the homology wouldn't be enough to distinguish links.) $\endgroup$
    – Alex Degtyarev
    Sep 21, 2016 at 21:28
  • $\begingroup$ I can see part of the link for some pairs of $T^2$ 2-tori show the spun Hopf link. It is not arbitrary. I am looking forward to some insightful analysis or Ref exploring the topology of this configuration carefully -- with a great imagination. $\endgroup$
    – wonderich
    Sep 22, 2016 at 2:37

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