"Almost embedding" the complete 2-dimensional complex $\mathcal K_7^2$ into $\Bbb R^4$

Question

Let $\mathcal K_7^2$ be the complete 2-dimensional simplicial complex on seven vertices, i.e. it has all $7\choose 2$ edges and all $7\choose 3$ 2-simplices (and no higher-dimensional simplices).

I once read that $\mathcal K_7^2$ can be "almost embedded" into $\Bbb R^4$ (say PL) in the sense that the mapping is injective except for a single intersection between two 2-simplices with disjoint boundary. I can't find a source for this anymore and can't think of an argument or construction.

So my question is: is this correct, and what would be a source or quick argument for this claim?

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I know that $\mathcal K_7^2$ with a single 2-simplex removed can be embedded into $\Bbb R^4$ (and there is a quick argument for it). So if it helps, we can assume this.

Answer 1

I was made aware of an elegant solution.

I first need to recall why $\mathcal K_7^2-\Delta$ (that is, $\mathcal K_7^2$ with a single 2-simplex removed) can be embedded into $\Bbb R^4$. This construction was brought to my attention by Tâm Nguyên-Phan:

Note that the complete graph $K_3\simeq \Bbb S^1$, and the complete complex $\mathcal K_4^2\simeq\Bbb S^2$ are topologically spheres. Their join $K_3\star \mathcal K_4^2\simeq \Bbb S^1\star \Bbb S^2\simeq \Bbb S^4$ is a 4-sphere. Note further that $\mathcal K_7^2-\Delta\subset K_3\star \mathcal K_4^2$ is a proper subcomplex, where the missing 2-simplex $\Delta$ comes from fact that there is no 2-simplex filling $K_3$. This provides an embedding of $\mathcal K_7^2-\Delta$ into $\Bbb S^4$. By removing a point from $\Bbb S^4$ that is not in $\mathcal K_7^2-\Delta$ we obtain an embedding into $\Bbb R^4$.

The next part is due to Agelos Georgakopoulos:

In the above construction, choose a 2-simplex $\sigma\subset\mathcal K_4^2$ and an interior point $x\in\operatorname{int}(\sigma)$. Let $C_x$ be the cone over $K_3$ with apex at $x$ and note that $C_x\subset K_3\star \mathcal K_4^2$. Moreover, $C_x$ is topologically a disk and intersects $\mathcal K_7^2-\Delta\subset K_3\star \mathcal K_4^2$ only in the single point $x$. This gives the desired embedding of $\mathcal K_7^2$ into $\Bbb S^4$ with a single self-intersection between the 2-simplices $C_x$ and $\sigma$.