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Could someone tell me or give a reference for the minimal triangulation of $S^2\times S^2$ and $S^2\times S^1\times S^1$ ?

Thanks,

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    $\begingroup$ Chapter 4 of arxiv.org/pdf/math/0404465v1.pdf may be of interested to you. I doubt that something is known about minimal triangulations of 4-manifolds, but you might ask the author of that preprint. $\endgroup$
    – ThiKu
    Commented Mar 7, 2016 at 10:40
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    $\begingroup$ I believe the minimal triangulation of $S^2 \times S^2$ has 6 $4$-dimensional simplices. It depends on what your "rules" for triangulations are. I'm using unordered delta complexes. Do you want a simplicial triangulation? You'll need many more simplices in that case. $\endgroup$
    – Ryan Budney
    Commented Apr 20, 2016 at 6:24
  • $\begingroup$ This other paper of Lutz is more relevant to your question: arxiv.org/pdf/math/0506372.pdf. In particular, Theorem 40 says that the vertex-minimal triangulations of $S^2\times S^2$ have 11 vertices. It is not unique but Lutz conjectures that the $f$-vector $(11, 55 , 150, 170, 68)$ is component-wise minimal. (I am assuming you want a simplicial triangulation, as opposed to Ryan Budney's answer), $\endgroup$
    – Francisco Santos
    Commented May 7, 2017 at 9:02

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