Consider the product $\Delta\times\Delta^{\vee}\subset(\mathbb{R}^{d})^{*}\times\mathbb{R}^{d}$. The complex $\Sigma$ — our prospective base space — will be a subdivision of

(1)

$$|\Sigma|=\{(m,n)\in\Delta\times\Delta^{\vee}\ :\ \langle m,n\rangle=1\}$$

which is the union of all sets of the form $F\times F^{\vee}$, where $F$ runs over all proper faces of $\Delta$, and $F^{\vee}$ denotes the face (of $\Delta^{\vee}$) dual to $F$. The triangulations $S$ and $T$ induce a subdivision of $\Delta\times\Delta^{\vee}$ into products of simplices, which restricts to a subdivision of $|\Sigma|$.

The vertices of $\Sigma$ correspond to pairs $(\widehat{\sigma},\widehat{\tau})$ of (barycenters of) simplices $\sigma\in S$ and $\tau\in T$ with $\langle\sigma,\tau\rangle=1$. Under this correspondence, the cells of $\Sigma$ correspond to pairs of chains in the face posets of $S$ and $T$.

We postpone the proof, and proceed with definitions.

For $d=2$, $D$ is empty. For $d=3$, $D$ will be a finite set of $\leq 24$ points (with equality if $S$ and $T$ use all lattice points). For $d=4$, $D$ will be the first subdivision of a trivalent graph. For all $d$, the complement $\Sigma\backslash D$ is homotopy equivalent to a bipartite graph.

First, we introduce some notation that will be used later on. There are two piecewise linear projections

$$p_{1}\colon\Sigma\rightarrow\operatorname{bsd}(S)\ \text{ and }\ p_{2}\colon\Sigma\rightarrow\operatorname{bsd}(T).$$

For a vertex $v\in\operatorname{vert}(S)$ or $w\in\operatorname{vert}(T)$, define $U_{v}$ respectively $V_{w}$ to be the preimages

$$U_{v}=p_{1}^{-1}(\operatorname{star}_{\operatorname{bsd}(S)}(v))\ \text{ and }\ V_{w}=p_{2}^{-1}(\operatorname{star}_{\operatorname{bsd}(T)}(w))$$

of open stars in the barycentric subdivisions. Thus, $U_{v}$ is an open regular neighborhood of the contractible set $v\times(\operatorname{carrier}_{\Delta}v)^{\vee}$ in $\Sigma$, and $V_{w}$ is an open regular neighborhood of $(\operatorname{carrier}_{\Delta^{\vee}}w)^{\vee}\times w$. We will abbreviate the collections by $\mathcal{U}=(U_{v})_{v\in\operatorname{vert}(S)}$ and $\mathcal{V}=(V_{w})_{w\in\operatorname{vert}(T)}$. For future reference, define $\overline{\mathcal{U}}=(\overline{U}_{v})_{v\in\operatorname{vert}(S)}$ and $\overline{\mathcal{V}}=(\overline{V}_{w})_{w\in\operatorname{vert}(T)}$, as well as $\partial\mathcal{U}=\bigcup\partial U_{v}$ and $\partial\mathcal{V}=\bigcup\partial V_{w}$. Observe that with these definitions $\bigcup\mathcal{U}\cup\bigcup\mathcal{V}=\Sigma\backslash D$, and $\partial\mathcal{U}\cap\partial\mathcal{V}=D$.