1.2. Maximal polyhedral complexes and their decomposition into primitive pieces [04R5]

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1.2. Maximal polyhedral complexes and their decomposition into primitive pieces

Definition 4.

We call $\Pi$ a dual $\Delta$ -complex if it corresponds to the convex polyhedron $\Delta\subset\mathbb{R}^{n+1}$ by Proposition 1.4. We call $\Pi$ a maximal polyhedral complex if the elements of the corresponding subdivision from Corollary 1.5 are simplices of volume $\frac{1}{(n+1)!}$ (a so-called unimodular lattice triangulation).

Proposition 1.7.

The minimal positive volume of a lattice polyhedron in $\mathbb{R}^{n+1}$ is $\frac{1}{(n+1)!}$ . Any lattice polyhedron of volume $\frac{1}{(n+1)!}$ can be identified with the standard simplex $\Delta_{1}$ (see (1)) by an element of $ASL_{n+1}(\mathbb{Z})$ .

Here $ASL_{n+1}(\mathbb{Z})$ stands for the group of affine-linear transformations of $\mathbb{R}^{n+1}$ whose rotation part belongs to $SL_{n+1}(\mathbb{Z})$ .

Proof.

We may assume that our lattice polyhedron is a simplex, since otherwise we can triangulate it to smaller polyhedra. Fix one of its vertex and consider the $(n+1)$ integer vectors connecting it to other vertices. The volume of the simplex is equal to the determinant of the sublattice generated by these vectors divided by $(n+1)!$ . ∎

Example 3.

Clearly, a dual $\Delta_{1}$ -complex (see (1)) is necessarily maximal. The complexes from Figure 1 are maximal dual $\Delta$ -complexes for the polyhedra $\Delta$ pictured on Figure 3.

Proposition 1.8.

Any dual $\Delta_{1}$ -complex is the result of a translation of $\Sigma_{n}$ in $\mathbb{R}^{n+1}$ .

Proof.

Such a complex $\Pi$ is determined by a function $v:\Delta_{1}\cap\mathbb{Z}^{n+1}\to\mathbb{R}$ , i.e. by $n+2$ numbers $a_{1},\dots,a_{n+1},b\in\mathbb{R}$ . Recall (see Example 2) that $\Pi$ is the corner locus $L_{v}(x_{1},\dots,x_{n+1})=\max\{x_{j}-a_{j},-b\}$ . If $a_{j}=b=0$ for all $j$ than $\Pi=\Sigma_{1}$ . Adding the same real number to all numbers does not change $\Pi$ . Changing $a_{j}$ by $t$ results in a translation by $t$ in the direction of $x_{j}$ . ∎

Remark 1.9.

Not for every $\Delta$ there exists maximal dual $\Delta$ -complex. E.g. a lattice simplex in $\mathbb{R}^{3}$ , whose vertices are $(1,0,0)$ , $(0,1,0)$ , $(1,1,0)$ and $(0,0,n)$ , cannot be further subdivided. On the other hand, a maximal dual $\Delta$ -complex is, of course, not unique.

Proposition 1.10.

If $\Pi$ is a maximal dual $\Delta$ -complex then $\Pi$ is homotopy equivalent to the bouquet of $\#(\operatorname{Int}\Delta\cap\mathbb{Z}^{n+1})$ copies of $S^{n}$ .

Proof.

Because of its maximality, the polyhedron $\Pi$ is dual to a unimodular triangulation of $\Pi$ . Such a triangulation cannot be further subdivided and therefore its vertices are all the lattice points of $\Delta$ . Therefore, $\Pi$ is homotopy equivalent to $\operatorname{Int}\Delta\smallsetminus\mathbb{Z}^{n+1}$ . ∎

Here is a way to canonically cut a maximal complex $\Pi\subset\mathbb{R}^{n+1}$ into standard-looking subsets $U_{j}$ . We define the cutting locus $\Xi$ as the following simplicial complex that is partially dual to $\Pi$ . The vertices of $\Xi$ are the baricenters of all bounded $k$ -cells, $k>0$ from $\Pi$ . The simplices of $\Xi$ have the baricenters of positive-dimensional cells $F_{k}\subset\Pi$ in the embedded towers $F_{1}\subset\dots\subset F_{l}$ as its vertices. Note that $\Xi\subset\Pi$ is a finite simplicial $(n-1)$ -complex.

Definition 5.

The connected components of $\Pi\smallsetminus\Xi$ are called the primitive pieces of $\Pi$ . We denote them with $U_{j}$ . These open sets are parametrized by the vertices of $\Pi$ or, equivalently, by the $(n+1)$ -simplices of the triangulation of $\Delta$ .

Proposition 1.11.

For each $U_{j}$ there exists $M_{j}\in ASL_{n+1}(\mathbb{Z})$ such that $M_{j}(U_{j})\subset\Sigma_{n}$ is an open set in the primitive complex $\Sigma_{n}$ from Example 1.

Proof.

This proposition also follows from the duality with a unimodular triangulation $\mathcal{D}$ of $\Delta$ . Let $U_{j}$ be a primitive piece. It corresponds to a simplex of volume $\frac{1}{(n+1)!}$ in $\mathcal{D}$ . There is an element of $SL_{n+1}(\mathbb{Z})$ which takes this simplex to the standard simplex $\Delta_{1}^{n+1}$ (see (1)). Then the image of $U_{j}$ by the adjoint to the inverse of this element is contained in a dual $\Delta_{1}$ -complex. Such a complex is the result of a translation of $\Sigma_{n}$ by Proposition 1.8. ∎

Recall that a polyhedral complex $\Pi$ is called generic at a point $x\in\Pi$ from an open $k$ -cell if there exists a neighborhood isomorphic to $\mathbb{R}^{k}\times\Sigma_{n-k}$ .

Thus, Proposition 1.10 implies that a maximal dual $\Delta$ -complex is a generic polyhedron. In topology such polyhedra often appear as the so-called special spines of smooth manifolds. In the next section we see that $\Pi$ can be compactified so as to become a spine of the polyhedron $\Delta$ after puncturing it in the interior lattice points.