Abstract isometric polyhedral gluings and the chain metric

Shape and face posets. Let $(P,\le )$ be a poset (Partial order and partially ordered set) with least element $\mathrm{\varnothing }$ such that:

(a) for every $p\in P$ the principal down-set $P_{\le p}:=\{r\in P:r\le p\}$ is finite and, when $p\mathrm{\ne }\mathrm{\varnothing }$, is isomorphic as a poset to the face poset of a nonempty compact convex polyhedral cell (Finite convex cell complex and linear subdivision, Face poset and order complex);

(b) every two elements $p,q\in P$ have a greatest lower bound $p\wedge q$ in $P$, that is, an element of $P$ that is a lower bound of both and is larger than every such lower bound.

The elements of $P$ are called faces, $p\le q$ is read "$p$ is a face of $q$", and $\mathrm{\varnothing }$ is the empty face.

Isometric polyhedral gluing. An isometric polyhedral gluing of shape $P$ consists of the following data.

(i) Cells and face isometries. For every $p\in P\setminus \{\mathrm{\varnothing }\}$ a nonempty compact convex polyhedral cell $C_p$ in a finite-dimensional Euclidean affine space, and for every pair $p\le q$ of nonempty faces a nonempty face $F_{p,q}$ of $C_q$ together with an affine isometry $h_{p,q}$ from the affine span of $C_p$ onto the affine span of $F_{p,q}$ (Isometry, isometric embedding, and the subspace metric on a subset) which carries $C_p$ onto $F_{p,q}$, subject to:

• $h_{p,p}={\mathrm{id}}_{C_p}$ for every $p$;
• the cocycle condition $h_{q,r}\circ h_{p,q}=h_{p,r}$ whenever $p\le q\le r$;
• for every $q$ the assignment $p\mapsto F_{p,q}=h_{p,q}(C_p)$ is an isomorphism of posets from $P_{\le q}\setminus \{\mathrm{\varnothing }\}$ onto the set of nonempty faces of $C_q$, ordered by inclusion.

(ii) Quotient and intersection condition. Let $X$ be the quotient of the disjoint union ${⨆}_{p\in P\setminus \{\mathrm{\varnothing }\}}{C}_p$ by the equivalence relation generated by $x\sim h_{p,q}(x)$ for $x\in C_p$ and $p\le q$, and let ${\iota }_p\text{\hspace{0.17em}⁣}:C_p\to X$ denote the quotient map. The intersection condition is:

• each ${\iota }_p$ is injective; and
• for all nonempty faces $p,q$ one has ${\iota }_p(C_p)\cap {\iota }_q(C_q)={\iota }_{p\wedge q}(C_{p\wedge q})$, where the right-hand side denotes the empty subset of $X$ when $p\wedge q=\mathrm{\varnothing }$.

(iii) Weak topology. A subset $U\subseteq X$ is declared open if and only if $U\cap {\iota }_p(C_p)$ is relatively open in ${\iota }_p(C_p)$ for every nonempty face $p$ (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

Standing hypotheses. (H1) Connectedness: $X\mathrm{\ne }\mathrm{\varnothing }$ and $X$ is connected. (H2) Local finiteness: every point of $X$ lies in ${\iota }_p(C_p)$ for only finitely many $p$. (H3) Finite shapes: the cells $C_p$ fall into only finitely many isometry classes. By (H1) at least one cell exists, so by (H3) the maximum $D:=\max \{\dim C_p:p\in P\setminus \{\mathrm{\varnothing }\}\}$ of the dimensions of the cells is a well-defined natural number.

Chains and the chain metric candidate. Write $d_p$ for the Euclidean metric on the affine span of $C_p$. Let $x,y\in X$. A chain from $x$ to $y$ is a finite sequence $x=x_0,x_1,\dots ,x_m=y$ of points of $X$ such that for each $i\in \{1,\dots ,m\}$ some cell contains both $x_{i-1}$ and $x_i$; for $m=0$ the chain is the one-term sequence $x$. Its length is $\mathrm{\ell }(x_0,\dots ,x_m):={\sum }_{i=1}^m{d}_{p_i}(x_{i-1},x_i),$ where for each $i$ the face $p_i$ is any nonempty face with $x_{i-1},x_i\in {\iota }_{p_i}(C_{p_i})$; this number does not depend on these choices. The chain metric candidate of the gluing is $d(x,y):=\inf \{\mathrm{\ell }(x_0,\dots ,x_m):x_0,\dots ,x_m\text{ is a chain from }x\text{ to }y\}\mathrm{.}$ Because $X$ is connected, every two points of $X$ are joined by a chain (this and the independence of $\mathrm{\ell }$ from the chosen faces are proved in The chain metric is a metric, its topology is the weak topology, and the space is proper and complete ↗), so the infimum is taken over a nonempty set of real numbers, and $0\le d(x,y)<\mathrm{\infty }$. This finiteness follows from (H1) and the gluing data; it is not an additional hypothesis.

What is asserted, and what is not. For all $x,y,z\in X$ one has $d(x,x)=0$ (the one-term chain), $d(x,y)=d(y,x)$ (reverse a chain) and $d(x,z)\le d(x,y)+d(y,z)$ (concatenate chains), and these three facts need no hypothesis beyond the definitions. No other metric axiom, no agreement of $d$ with the weak topology, and no completeness, properness or geodesic property is asserted here: those are the conclusions of the theorems of this page, under (H1)–(H3).

Caveat on cell metrics. If $x,y$ lie in a common cell then the one-step chain gives $d(x,y)\le d_p(x,y)$, but equality can fail, because a chain may leave the cell and return with smaller total length (Bridson–Haefliger I.7.6). Nothing above asserts that the chain metric restricts to the Euclidean metric of a cell, and no cell is assumed to be geodesic for $d$.