The join of two abstract simplicial complexes $K$ and $L$, denoted $K\star L$ is defined as a simplicial complex on the base set $V(K)\dot{\cup} V(L)$ whose simplices are disjoint union of simplices of $K$ and of $L$ (here $V(K)$ denotes the base set of $K$, which we take to be finite.)

I was wondering if there exist generalization of the join in the following sense:

Suppose $X_1\star X_2 \star \ldots \star X_n $ is the join of finitely many simplicial complexes. If we take one point from each $X_i$ in the join, we obtain a simplex spanned by the points.

Is there an analogous $n$-ary operation $\mathcal{J}(X_1,\ldots , X_n)$ that gives a simplicial complex on $V(X_1)\dot{\cup}V(X_2)\dot{\cup} \ldots \dot{\cup} V(X_n)$, but for which selecting one point from each $X_i$ doesn't give a simplex? (e.g it could give a sphere)

Would such an object make sense? (e.g. be well-defined, not hopelessly horrible topologically)

I would also be glad to hear about similar generalizations in other contexts (as would be, for instance, generalizations to $r$-th secant varieties of affine varieties, provided the generalizations exist).

If such a generalization exists, what are typical situations where it is considered?

(I have been sloppy in several respects and I apologize for that. I hope the geometric idea of the generalization is more or less clear though.)

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    $\begingroup$ It would be helpful if you could explain why do you need such a weirdo. $\endgroup$ – Anton Petrunin May 7 '13 at 14:04
  • $\begingroup$ I expected to give myself an answer to that question after getting some replies. $\endgroup$ – Camilo Sarmiento May 7 '13 at 15:40

Well, the deleted join has been studied. Somewhat informally, that's where you take the join of a simplicial complex $\Delta$ with itself, then delete the faces $\sigma_1 \cup \sigma_2$ in $\Delta * \Delta$ such that $\sigma_1 \cap \sigma_2 \neq \emptyset$ in $\Delta$. The deleted join is nice from the Borsuk-Ulam point of view, because it admits a free $\mathbb{Z}_2$ action by exchanging the two copies of $\Delta$.

A related construction is the Bier sphere, where you replace the second copy of $\Delta$ with the combinatorial Alexander dual.

Matousek discusses both deleted joins and Bier spheres in his book Using the Borsuk-Ulam Theorem.

  • $\begingroup$ That's a good example, but in the deleted join, a pair of points from each copy may or may not span a simplex, depending on whether the points are "different" or not. What I am looking for is a case where the thing spanned by taking one point from each copy is the same (regardless of the identity of the points), and is not a ball. In terms of the r-th secant variety, the "generalization" I'm looking for is when the r copies of the variety are not embedded in a skew way. Do you know if such a "weirdo" shows up anywhere? $\endgroup$ – Camilo Sarmiento May 7 '13 at 15:38
  • $\begingroup$ If I understand what you're looking for correctly, it suffices to delete the faces consisting of one point from every $X_i$. Since one can specify a simplicial complex either by the maximal faces or else by the minimal non-faces, this complex is well-defined. (You're just adding some new minimal non-faces to the join.) The induced subcomplex on such a vertex subset (consisting of one point from each complex) will be a simplex boundary, hence a sphere. I haven't seen these studied anywhere, but that doesn't mean that they haven't been. $\endgroup$ – Russ Woodroofe May 7 '13 at 20:18

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