Namespaces
Variants
Actions

Difference between revisions of "Polyhedron, abstract"

From Encyclopedia of Mathematics
Jump to: navigation, search
m
(TeX)
Line 1: Line 1:
 
{{DEF}}
 
{{DEF}}
{{TEX|want}}
+
{{TEX|done}}
  
The union of a locally finite family of convex polytopes in a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736701.png" />. By a convex polytope one understands the intersection of a finite number of closed half-spaces if this intersection is bounded. Local finiteness of the family means that each point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736702.png" /> has a neighbourhood that intersects only with a finite number of the polytopes. A compact polyhedron is the union of a finite number of convex polytopes. The dimension of a polyhedron is the maximum dimension of the constituent polytopes. Any open subset of an (abstract) polyhedron, in particular any open subset of a Euclidean space, is a polyhedron. Other polyhedra are: the [[Cone|cone]] and the [[Suspension|suspension]] over a compact polyhedron. Simple examples (a cone over an open interval) show that the join of a compact and a non-compact polyhedron need be not a polyhedron. The name subpolyhedron of a polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736703.png" /> is given to any polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736704.png" /> lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736705.png" />. Sometimes one restricts the consideration to closed subpolyhedra. Each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736706.png" /> in a polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736707.png" /> has in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736708.png" /> a neighbourhood that is a cone in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p0736709.png" /> with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367010.png" /> and with a compact base. This property is characteristic: Any subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367011.png" /> each point of which has a conical neighbourhood with a compact base is a polyhedron.
+
The union of a locally finite family of convex polytopes in a certain $\mathbb{R}^n$. By a convex polytope one understands the intersection of a finite number of closed half-spaces if this intersection is bounded. Local finiteness of the family means that each point in $\R^n$ has a neighbourhood that intersects only with a finite number of the polytopes. A compact polyhedron is the union of a finite number of convex polytopes. The dimension of a polyhedron is the maximum dimension of the constituent polytopes. Any open subset of an (abstract) polyhedron, in particular any open subset of a Euclidean space, is a polyhedron. Other polyhedra are: the [[Cone|cone]] and the [[Suspension|suspension]] over a compact polyhedron. Simple examples (a cone over an open interval) show that the join of a compact and a non-compact polyhedron need be not a polyhedron. The name subpolyhedron of a polyhedron $Q$ is given to any polyhedron $P$ lying in $Q$. Sometimes one restricts the consideration to closed subpolyhedra. Each point $a$ in a polyhedron $P\in\R^n$ has in $P$ a neighbourhood that is a cone in $\R^n$ with vertex $a$ and with a compact base. This property is characteristic: Any subset in $\R^n$ each point of which has a conical neighbourhood with a compact base is a polyhedron.
  
Any compact polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367012.png" /> can be split up into a finite number of closed simplices in such a way that any two simplices either do not intersect or else intersect in a common face. In the case of a non-compact polyhedron it is required that the family of simplices should be locally finite. This decomposition is called a rectilinear triangulation of the polyhedron. Any two triangulations of a given polyhedron have a common subdivision. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367013.png" /> is a closed subpolyhedron of a polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367014.png" />, then any triangulation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367016.png" /> can be extended to a certain triangulation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367017.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367018.png" />. In that case it is said that the resulting pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367019.png" /> of geometrical simplicial complexes triangulates the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367020.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367021.png" /> of a polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367022.png" /> into a polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367023.png" /> is called a piecewise-linear mapping (or a pl-mapping) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367024.png" /> is simplicial with respect to certain triangulations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367026.png" /> (cf. [[Simplicial mapping|Simplicial mapping]]). An equivalent definition is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367027.png" /> is piecewise linear if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367028.png" /> is locally conical, i.e. if each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367029.png" /> has a conical neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367030.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367031.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367034.png" />. For a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367035.png" /> to be piecewise linear it is necessary and sufficient that its graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367036.png" /> is a polyhedron.
+
Any compact polyhedron $P$ can be split up into a finite number of closed simplices in such a way that any two simplices either do not intersect or else intersect in a common face. In the case of a non-compact polyhedron it is required that the family of simplices should be locally finite. This decomposition is called a rectilinear triangulation of the polyhedron. Any two triangulations of a given polyhedron have a common subdivision. If $P$ is a closed subpolyhedron of a polyhedron $Q$, then any triangulation $K$ of $P$ can be extended to a certain triangulation $L$ of $Q$. In that case it is said that the resulting pair $(L,K)$ of geometrical simplicial complexes triangulates the pair $(Q,P)$. A mapping $f$ of a polyhedron $P\subset\R^n$ into a polyhedron $Q\subset\R^n$ is called a piecewise-linear mapping (or a pl-mapping) if $f$ is simplicial with respect to certain triangulations of $P$ and $Q$ (cf. [[Simplicial mapping|Simplicial mapping]]). An equivalent definition is that $f$ is piecewise linear if $f$ is locally conical, i.e. if each point $a\in P$ has a conical neighbourhood $N=a^*L$ such that $f(\lambda a+\mu x)=\lambda f(a)+\mu f(x)$ for any $x\in L$ and $\lambda,\mu\geq0$, $\lambda+\mu=1$. For a mapping $f$ to be piecewise linear it is necessary and sufficient that its graph $\Gamma_f\subset\R^n\times\R^n$ is a polyhedron.
  
A superposition of piecewise-linear mappings is piecewise linear. The inverse mapping of an invertible piecewise-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367037.png" /> is piecewise linear. In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367038.png" /> is called a pl-homeomorphism.
+
A superposition of piecewise-linear mappings is piecewise linear. The inverse mapping of an invertible piecewise-linear mapping $f$ is piecewise linear. In that case $f$ is called a pl-homeomorphism.
  
The category whose objects are polyhedra (and polyhedral pairs) and whose morphisms are pl-mappings is denoted by PL or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367039.png" /> (see also [[Piecewise-linear topology|Piecewise-linear topology]]). The category PL is one of the basic objects and tools of research in topology. The role of the category PL is particularly great in [[Algebraic topology|algebraic topology]] and in the [[Topology of manifolds|topology of manifolds]], because the class of polyhedra is fairly wide.
+
The category whose objects are polyhedra (and polyhedral pairs) and whose morphisms are pl-mappings is denoted by PL or by $\mathcal{P}$ (see also [[Piecewise-linear topology|Piecewise-linear topology]]). The category PL is one of the basic objects and tools of research in topology. The role of the category PL is particularly great in [[Algebraic topology|algebraic topology]] and in the [[Topology of manifolds|topology of manifolds]], because the class of polyhedra is fairly wide.
  
For example, each [[Differentiable manifold|differentiable manifold]] can be represented in a natural way as a polyhedron. Each [[Continuous mapping|continuous mapping]] of one polyhedron into another can be approximated arbitrary closely by a pl-mapping. Therefore the category PL is a good approximation to the category of all topological spaces and continuous mappings. On the other hand, the triangulation of a polyhedron enables one to use methods from combinatorial topology. Many algebraic invariants (for example, the [[Homology group|homology group]] or [[Cohomology ring|cohomology ring]]) can be constructed and effectively calculated by decomposition into simplices. The question whether all homeomorphic polyhedra are pl-homeomorphic is called the Hauptvermutung and the answer is negative: For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367040.png" /> there exist homeomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367041.png" />-dimensional polyhedra that are not pl-homeomorphic [[#References|[3]]]. There also exist different pl-structures on certain closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367042.png" />-manifolds. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367043.png" />, homeomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367044.png" />-dimensional polyhedra are pl-homeomorphic. A polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367045.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367046.png" />-dimensional pl-manifold if each point in it has a neighbourhood that is pl-homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367047.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367048.png" />. Any rectilinear triangulation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367049.png" /> of a pl-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367050.png" /> is combinatorial. This means that the star at each of its vertices is combinatorially equivalent to a simplex. The Hauptvermutung for polyhedra that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367051.png" />-dimensional topological manifolds naturally splits up into two hypotheses: the hypothesis that any triangulation of such a polyhedron is combinatorial and the Hauptvermutung for pl-manifolds. One of the major achievements in modern topology is that a negative answer has been obtained to both hypotheses for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367052.png" /> [[#References|[4]]], [[#References|[5]]]. The two hypotheses are true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367053.png" />.
+
For example, each [[Differentiable manifold|differentiable manifold]] can be represented in a natural way as a polyhedron. Each [[Continuous mapping|continuous mapping]] of one polyhedron into another can be approximated arbitrary closely by a pl-mapping. Therefore the category PL is a good approximation to the category of all topological spaces and continuous mappings. On the other hand, the triangulation of a polyhedron enables one to use methods from combinatorial topology. Many algebraic invariants (for example, the [[Homology group|homology group]] or [[Cohomology ring|cohomology ring]]) can be constructed and effectively calculated by decomposition into simplices. The question whether all homeomorphic polyhedra are pl-homeomorphic is called the Hauptvermutung and the answer is negative: For $n\geq5$ there exist homeomorphic $n$-dimensional polyhedra that are not pl-homeomorphic [[#References|[3]]]. There also exist different pl-structures on certain closed $4$-manifolds. For $n\leq 3$, homeomorphic $n$-dimensional polyhedra are pl-homeomorphic. A polyhedron $M$ is called an $n$-dimensional pl-manifold if each point in it has a neighbourhood that is pl-homeomorphic to $\R^n$. or $\R^n_+$ Any rectilinear triangulation $T$ of a pl-manifold $M$ is combinatorial. This means that the star at each of its vertices is combinatorially equivalent to a simplex. The Hauptvermutung for polyhedra that are $n$-dimensional topological manifolds naturally splits up into two hypotheses: the hypothesis that any triangulation of such a polyhedron is combinatorial and the Hauptvermutung for pl-manifolds. One of the major achievements in modern topology is that a negative answer has been obtained to both hypotheses for $n\geq 5$ [[#References|[4]]], [[#References|[5]]]. The two hypotheses are true for $n\leq 3$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367054.png" /> be a compact subpolyhedron of a polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367055.png" /> and let the pair of geometrical simplicial complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367056.png" /> triangulate the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367057.png" /> in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367058.png" /> is a complete subcomplex. This means that each simplex of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367059.png" /> with vertices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367060.png" /> also lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367061.png" />; this can always be achieved by passing to a derived [[Subdivision|subdivision]]. The polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367062.png" /> consisting of all closed simplices of a derived subdivision <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367063.png" /> having vertices in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367064.png" /> is called a regular neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367065.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367066.png" />, and the same applies to its image under any pl-homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367067.png" /> into itself that leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367068.png" /> invariant. For any two regular neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367070.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367071.png" /> there exists a pl-isotopy that leaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367072.png" /> invariant, namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367073.png" />, which deforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367074.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367075.png" />, i.e. is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367076.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367077.png" />. One says that the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367078.png" /> is obtained by an elementary polyhedral collapse of a polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367079.png" /> if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367080.png" /> the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367081.png" /> is pl-homeomorphic to the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367082.png" />. The polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367083.png" /> polyhedrally collapses to its subpolyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367084.png" /> (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367085.png" />) if one can pass from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367086.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367087.png" /> by a finite sequence of elementary polyhedral collapses. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367088.png" />, then in a certain triangulation of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367089.png" /> the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367090.png" /> can be obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367091.png" /> by a sequence of elementary combinatorial collapses each of which consists in deleting a principal simplex along with its free face. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367092.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367093.png" />-dimensional pl-manifold, then any regular neighbourhood of a compact polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367094.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367095.png" />-dimensional pl-manifold and collapses polyhedrally to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367096.png" />. This property is characteristic: If the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367097.png" />-dimensional pl-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367098.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p07367099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670100.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670101.png" /> is a regular neighbourhood in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670102.png" />. Any regular neighbourhood of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670103.png" /> of a compact pl-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670104.png" /> is pl-homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670105.png" />.
+
Let $P$ be a compact subpolyhedron of a polyhedron $Q$ and let the pair of geometrical simplicial complexes $(L,K)$ triangulate the pair $(Q,P)$ in such a way that $K$ is a complete subcomplex. This means that each simplex of $L$ with vertices in $K$ also lies in $K$; this can always be achieved by passing to a derived [[Subdivision|subdivision]]. The polyhedron $N$ consisting of all closed simplices of a derived subdivision $L'$ having vertices in $K$ is called a regular neighbourhood of $P$ in $Q$, and the same applies to its image under any pl-homeomorphism of $Q$ into itself that leaves $P$ invariant. For any two regular neighbourhoods $N_1$ and $N_2$ of $P$ there exists a pl-isotopy that leaves $P$ invariant, namely $h_t:N_1\times I\to Q$, which deforms $N_1$ to $N_2$, i.e. is such that $h_0(N_1)=N_1$ and $h_1(N_1)=N_2$. One says that the polyhedron $P$ is obtained by an elementary polyhedral collapse of a polyhedron $P_1\supset P$ if for some $n\geq0$ the pair $\left(\overline{P_1\backslash P},\overline{P_1\backslash P}\cap P\right)$ is pl-homeomorphic to the pair $(I^n\times I,I^n\times\{0\})$. The polyhedron $P_1$ polyhedrally collapses to its subpolyhedron $P$ (denoted by $(P_1\downarrow P)$) if one can pass from $P_1$ to $P$ by a finite sequence of elementary polyhedral collapses. If $P_1\downarrow P$, then in a certain triangulation of the pair $(P_1,P)$ the polyhedron $P$ can be obtained from $P_1$ by a sequence of elementary combinatorial collapses each of which consists in deleting a principal simplex along with its free face. If $Q$ is an $n$-dimensional pl-manifold, then any regular neighbourhood of a compact polyhedron $P\subset Q$ is an $n$-dimensional pl-manifold and collapses polyhedrally to $P$. This property is characteristic: If the $n$-dimensional pl-manifold $N\subset Q$ is such that $P\subset\mathm{Int}N$ and $N\downarrow P$, then $N$ is a regular neighbourhood in $P$. Any regular neighbourhood of the boundary $\partial M$ of a compact pl-m
 +
anifold $M$ is pl-homeomorphic to $\partial M\times I$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670107.png" /> be closed subpolyhedra of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670108.png" />-dimensional pl-manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670111.png" />. It is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670112.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670113.png" /> are in general position if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670114.png" />. Any closed subpolyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670115.png" /> may be moved into general position by an arbitrarily small isotopy (cf. [[Isotopy (in topology)|Isotopy (in topology)]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670116.png" />. This means that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670117.png" /> there exists an (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670118.png" />-pl)-isotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670119.png" /> such that the polyhedra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670121.png" /> are in general position. Sometimes one includes conditions of transversality type in the definition of general position. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670122.png" />, one can ensure that for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670123.png" /> and a certain neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670124.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670125.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670126.png" />, the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670127.png" /> will be pl-homeomorphic to the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670128.png" />.
+
Let $P$ and $Q$ be closed subpolyhedra of an $n$-dimensional pl-manifold $M$, $\dim P=p$, $\dim Q=q$. It is said that $P$ and $Q$ are in general position if $\dim(P\cap Q)\leq p+q-n$. Any closed subpolyhedra $P,Q\subset\mathrm{Int}M$ may be moved into general position by an arbitrarily small isotopy (cf. [[Isotopy (in topology)|Isotopy (in topology)]]) in $M$. This means that for any $\epsilon>0$ there exists an ($\epsilon$-pl)-isotopy $h_t:M\to M$ such that the polyhedra $P$ and $Q_1=h_1(Q)$ are in general position. Sometimes one includes conditions of transversality type in the definition of general position. For example, if $p+q=n$, one can ensure that for each point $a\in P\cap Q_1$ and a certain neighbourhood $U$ of the point $a$ in $M$, the triple $(U,U\cap P,U\cap Q_1)$ will be pl-homeomorphic to the triple $(\R^p\times \R^q,\R^p\times\{0\},\{0\}\times\R^q)$.
  
A curved or topological polyhedron is a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670129.png" /> equipped with a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670130.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670131.png" /> is a polyhedron. The images of the simplices in some triangulation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670132.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670133.png" /> form a curvilinear triangulation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670134.png" />. It is also said that the homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670135.png" /> defines a pl-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670136.png" />. Two pl-structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670138.png" />, coincide if the homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670139.png" /> is piecewise linear, and they are isotopic if the homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670140.png" /> is isotopic to a piecewise-linear one, while they are equivalent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670141.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670142.png" /> are pl-homeomorphic. For any differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670143.png" /> there exists a pl-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670144.png" /> compatible with the differentiable structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670145.png" />. This means that for each closed simplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670146.png" /> of some triangulation of the polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670147.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670148.png" /> is differentiable and does not have singular points. Any two such pl-structures in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670149.png" /> are isotopic. All the concepts defined for a polyhedron (triangulation, subpolyhedron, regular neighbourhood, and general position) can be transferred by means of the homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670150.png" /> to the curvilinear polyhedron <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670151.png" />.
+
A curved or topological polyhedron is a topological space $X$ equipped with a homeomorphism $f:P\to X$, where $P$ is a polyhedron. The images of the simplices in some triangulation $T$ of $P$ form a curvilinear triangulation of $X$. It is also said that the homeomorphism $f$ defines a pl-structure on $X$. Two pl-structures $f_i:P_i\to X$, $i=1,2$, coincide if the homeomorphism $f_2^{-1}f_1$ is piecewise linear, and they are isotopic if the homeomorphism $f_2^{-1}f_1$ is isotopic to a piecewise-linear one, while they are equivalent if $P_1$ and $P_2$ are pl-homeomorphic. For any differentiable manifold $M$ there exists a pl-structure $f:P\to M$ compatible with the differentiable structure on $M$. This means that for each closed simplex $\sigma$ of some triangulation of the polyhedron $P$ the mapping $f|_{\sigma}:\sigma\to M$ is differentiable and does not have singular points. Any two such pl-structures in $M$ are isotopic. All the concepts defined for a polyhedron (triangulation, subpolyhedron, regular neighbourhood, and general position) can be transferred by means of the homeomorphism $f:P\to X$ to the curvilinear polyhedron $X$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Combinatorial topology" , Graylock , Rochester  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.P. Rourke,  B.J. Sanderson,  "Introduction to piecewise-linear topology" , Springer  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Milnor,  "Two complexes which are homeomorphic but combinatorially distinct"  ''Ann. of Math.'' , '''74''' :  3  (1961)  pp. 575–590</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.C. Kirby,  L.C. Siebenmann,  "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press  (1977)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.D. Edwards,  "The double suspension of a certain homology 3-sphere is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670152.png" />"  ''Notices Amer. Math. Soc.'' , '''22''' :  2  (1975)  pp. A-334</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,  "Combinatorial topology" , Graylock , Rochester  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.P. Rourke,  B.J. Sanderson,  "Introduction to piecewise-linear topology" , Springer  (1972)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Milnor,  "Two complexes which are homeomorphic but combinatorially distinct"  ''Ann. of Math.'' , '''74''' :  3  (1961)  pp. 575–590</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.C. Kirby,  L.C. Siebenmann,  "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press  (1977)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  R.D. Edwards,  "The double suspension of a certain homology 3-sphere is $S^5$"  ''Notices Amer. Math. Soc.'' , '''22''' :  2  (1975)  pp. A-334</TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
Recent developments include: imbedding of topological manifolds as polyhedra with convex (or non-convex) faces in a Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670153.png" />, in particular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670154.png" /> (e.g., polyhedral realizations of regular mappings (i.e. analogues of the regular polyhedra)); polyhedra of given genus with minimal number of vertices or edges or faces; colouring problems; and polyhedral realizations of famous configurations in geometry or topology.
+
Recent developments include: imbedding of topological manifolds as polyhedra with convex (or non-convex) faces in a Euclidean $E^n$, in particular in $E^3$ (e.g., polyhedral realizations of regular mappings (i.e. analogues of the regular polyhedra)); polyhedra of given genus with minimal number of vertices or edges or faces; colouring problems; and polyhedral realizations of famous configurations in geometry or topology.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U. Brehm,  W. Kühnel,  "A polyhedral model for Cartan's hypersurfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670155.png" />"  ''Mathematika'' , '''33'''  (1986)  pp. 55–61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Grünbaum,  "Regular polyhedra - old and new"  ''Aequat. Math.'' , '''16'''  (1977)  pp. 1–20</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. McMullen,  Ch. Schulz,  J.M. Wills,  "Polyhedral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670156.png" />-manifolds in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670157.png" /> with unusually large genus"  ''Israel J. of Math.'' , '''46'''  (1983)  pp. 127–144</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Schulte,  J.M. Wills,  "A polyhedral realization of Felix Klein's map <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073670/p073670158.png" /> on a Riemann surface of genus 3"  ''J. London Math. Soc.'' , '''32'''  (1985)  pp. 539–547</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  U. Brehm,  "Maximally symmetric polyhedral realizations of Dyck's regular map"  ''Mathematika'' , '''34'''  (1987)  pp. 229–236</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.R. Munkres,  "Elementary differential topology" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L.C. Glaser,  "Geometrical combinatorial topology" , '''1–2''' , v. Nostrand  (1970)</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  U. Brehm,  W. Kühnel,  "A polyhedral model for Cartan's hypersurfaces in $S^4$"  ''Mathematika'' , '''33'''  (1986)  pp. 55–61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Grünbaum,  "Regular polyhedra - old and new"  ''Aequat. Math.'' , '''16'''  (1977)  pp. 1–20</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. McMullen,  Ch. Schulz,  J.M. Wills,  "Polyhedral 2-manifolds in $E^3$ with unusually large genus"  ''Israel J. of Math.'' , '''46'''  (1983)  pp. 127–144</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Schulte,  J.M. Wills,  "A polyhedral realization of Felix Klein's map$\{3,7\}_8$ on a Riemann surface of genus 3"  ''J. London Math. Soc.'' , '''32'''  (1985)  pp. 539–547</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  U. Brehm,  "Maximally symmetric polyhedral realizations of Dyck's regular map"  ''Mathematika'' , '''34'''  (1987)  pp. 229–236</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.R. Munkres,  "Elementary differential topology" , Princeton Univ. Press  (1963)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L.C. Glaser,  "Geometrical combinatorial topology" , '''1–2''' , v. Nostrand  (1970)</TD></TR></table>

Revision as of 14:04, 25 April 2012


This page is deficient and requires revision. Please see Talk:Polyhedron, abstract for further comments.

The union of a locally finite family of convex polytopes in a certain $\mathbb{R}^n$. By a convex polytope one understands the intersection of a finite number of closed half-spaces if this intersection is bounded. Local finiteness of the family means that each point in $\R^n$ has a neighbourhood that intersects only with a finite number of the polytopes. A compact polyhedron is the union of a finite number of convex polytopes. The dimension of a polyhedron is the maximum dimension of the constituent polytopes. Any open subset of an (abstract) polyhedron, in particular any open subset of a Euclidean space, is a polyhedron. Other polyhedra are: the cone and the suspension over a compact polyhedron. Simple examples (a cone over an open interval) show that the join of a compact and a non-compact polyhedron need be not a polyhedron. The name subpolyhedron of a polyhedron $Q$ is given to any polyhedron $P$ lying in $Q$. Sometimes one restricts the consideration to closed subpolyhedra. Each point $a$ in a polyhedron $P\in\R^n$ has in $P$ a neighbourhood that is a cone in $\R^n$ with vertex $a$ and with a compact base. This property is characteristic: Any subset in $\R^n$ each point of which has a conical neighbourhood with a compact base is a polyhedron.

Any compact polyhedron $P$ can be split up into a finite number of closed simplices in such a way that any two simplices either do not intersect or else intersect in a common face. In the case of a non-compact polyhedron it is required that the family of simplices should be locally finite. This decomposition is called a rectilinear triangulation of the polyhedron. Any two triangulations of a given polyhedron have a common subdivision. If $P$ is a closed subpolyhedron of a polyhedron $Q$, then any triangulation $K$ of $P$ can be extended to a certain triangulation $L$ of $Q$. In that case it is said that the resulting pair $(L,K)$ of geometrical simplicial complexes triangulates the pair $(Q,P)$. A mapping $f$ of a polyhedron $P\subset\R^n$ into a polyhedron $Q\subset\R^n$ is called a piecewise-linear mapping (or a pl-mapping) if $f$ is simplicial with respect to certain triangulations of $P$ and $Q$ (cf. Simplicial mapping). An equivalent definition is that $f$ is piecewise linear if $f$ is locally conical, i.e. if each point $a\in P$ has a conical neighbourhood $N=a^*L$ such that $f(\lambda a+\mu x)=\lambda f(a)+\mu f(x)$ for any $x\in L$ and $\lambda,\mu\geq0$, $\lambda+\mu=1$. For a mapping $f$ to be piecewise linear it is necessary and sufficient that its graph $\Gamma_f\subset\R^n\times\R^n$ is a polyhedron.

A superposition of piecewise-linear mappings is piecewise linear. The inverse mapping of an invertible piecewise-linear mapping $f$ is piecewise linear. In that case $f$ is called a pl-homeomorphism.

The category whose objects are polyhedra (and polyhedral pairs) and whose morphisms are pl-mappings is denoted by PL or by $\mathcal{P}$ (see also Piecewise-linear topology). The category PL is one of the basic objects and tools of research in topology. The role of the category PL is particularly great in algebraic topology and in the topology of manifolds, because the class of polyhedra is fairly wide.

For example, each differentiable manifold can be represented in a natural way as a polyhedron. Each continuous mapping of one polyhedron into another can be approximated arbitrary closely by a pl-mapping. Therefore the category PL is a good approximation to the category of all topological spaces and continuous mappings. On the other hand, the triangulation of a polyhedron enables one to use methods from combinatorial topology. Many algebraic invariants (for example, the homology group or cohomology ring) can be constructed and effectively calculated by decomposition into simplices. The question whether all homeomorphic polyhedra are pl-homeomorphic is called the Hauptvermutung and the answer is negative: For $n\geq5$ there exist homeomorphic $n$-dimensional polyhedra that are not pl-homeomorphic [3]. There also exist different pl-structures on certain closed $4$-manifolds. For $n\leq 3$, homeomorphic $n$-dimensional polyhedra are pl-homeomorphic. A polyhedron $M$ is called an $n$-dimensional pl-manifold if each point in it has a neighbourhood that is pl-homeomorphic to $\R^n$. or $\R^n_+$ Any rectilinear triangulation $T$ of a pl-manifold $M$ is combinatorial. This means that the star at each of its vertices is combinatorially equivalent to a simplex. The Hauptvermutung for polyhedra that are $n$-dimensional topological manifolds naturally splits up into two hypotheses: the hypothesis that any triangulation of such a polyhedron is combinatorial and the Hauptvermutung for pl-manifolds. One of the major achievements in modern topology is that a negative answer has been obtained to both hypotheses for $n\geq 5$ [4], [5]. The two hypotheses are true for $n\leq 3$.

Let $P$ be a compact subpolyhedron of a polyhedron $Q$ and let the pair of geometrical simplicial complexes $(L,K)$ triangulate the pair $(Q,P)$ in such a way that $K$ is a complete subcomplex. This means that each simplex of $L$ with vertices in $K$ also lies in $K$; this can always be achieved by passing to a derived subdivision. The polyhedron $N$ consisting of all closed simplices of a derived subdivision $L'$ having vertices in $K$ is called a regular neighbourhood of $P$ in $Q$, and the same applies to its image under any pl-homeomorphism of $Q$ into itself that leaves $P$ invariant. For any two regular neighbourhoods $N_1$ and $N_2$ of $P$ there exists a pl-isotopy that leaves $P$ invariant, namely $h_t:N_1\times I\to Q$, which deforms $N_1$ to $N_2$, i.e. is such that $h_0(N_1)=N_1$ and $h_1(N_1)=N_2$. One says that the polyhedron $P$ is obtained by an elementary polyhedral collapse of a polyhedron $P_1\supset P$ if for some $n\geq0$ the pair $\left(\overline{P_1\backslash P},\overline{P_1\backslash P}\cap P\right)$ is pl-homeomorphic to the pair $(I^n\times I,I^n\times\{0\})$. The polyhedron $P_1$ polyhedrally collapses to its subpolyhedron $P$ (denoted by $(P_1\downarrow P)$) if one can pass from $P_1$ to $P$ by a finite sequence of elementary polyhedral collapses. If $P_1\downarrow P$, then in a certain triangulation of the pair $(P_1,P)$ the polyhedron $P$ can be obtained from $P_1$ by a sequence of elementary combinatorial collapses each of which consists in deleting a principal simplex along with its free face. If $Q$ is an $n$-dimensional pl-manifold, then any regular neighbourhood of a compact polyhedron $P\subset Q$ is an $n$-dimensional pl-manifold and collapses polyhedrally to $P$. This property is characteristic: If the $n$-dimensional pl-manifold $N\subset Q$ is such that $P\subset\mathm{Int}N$ and $N\downarrow P$, then $N$ is a regular neighbourhood in $P$. Any regular neighbourhood of the boundary $\partial M$ of a compact pl-m anifold $M$ is pl-homeomorphic to $\partial M\times I$.

Let $P$ and $Q$ be closed subpolyhedra of an $n$-dimensional pl-manifold $M$, $\dim P=p$, $\dim Q=q$. It is said that $P$ and $Q$ are in general position if $\dim(P\cap Q)\leq p+q-n$. Any closed subpolyhedra $P,Q\subset\mathrm{Int}M$ may be moved into general position by an arbitrarily small isotopy (cf. Isotopy (in topology)) in $M$. This means that for any $\epsilon>0$ there exists an ($\epsilon$-pl)-isotopy $h_t:M\to M$ such that the polyhedra $P$ and $Q_1=h_1(Q)$ are in general position. Sometimes one includes conditions of transversality type in the definition of general position. For example, if $p+q=n$, one can ensure that for each point $a\in P\cap Q_1$ and a certain neighbourhood $U$ of the point $a$ in $M$, the triple $(U,U\cap P,U\cap Q_1)$ will be pl-homeomorphic to the triple $(\R^p\times \R^q,\R^p\times\{0\},\{0\}\times\R^q)$.

A curved or topological polyhedron is a topological space $X$ equipped with a homeomorphism $f:P\to X$, where $P$ is a polyhedron. The images of the simplices in some triangulation $T$ of $P$ form a curvilinear triangulation of $X$. It is also said that the homeomorphism $f$ defines a pl-structure on $X$. Two pl-structures $f_i:P_i\to X$, $i=1,2$, coincide if the homeomorphism $f_2^{-1}f_1$ is piecewise linear, and they are isotopic if the homeomorphism $f_2^{-1}f_1$ is isotopic to a piecewise-linear one, while they are equivalent if $P_1$ and $P_2$ are pl-homeomorphic. For any differentiable manifold $M$ there exists a pl-structure $f:P\to M$ compatible with the differentiable structure on $M$. This means that for each closed simplex $\sigma$ of some triangulation of the polyhedron $P$ the mapping $f|_{\sigma}:\sigma\to M$ is differentiable and does not have singular points. Any two such pl-structures in $M$ are isotopic. All the concepts defined for a polyhedron (triangulation, subpolyhedron, regular neighbourhood, and general position) can be transferred by means of the homeomorphism $f:P\to X$ to the curvilinear polyhedron $X$.

References

[1] P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian)
[2] C.P. Rourke, B.J. Sanderson, "Introduction to piecewise-linear topology" , Springer (1972)
[3] J. Milnor, "Two complexes which are homeomorphic but combinatorially distinct" Ann. of Math. , 74 : 3 (1961) pp. 575–590
[4] R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977)
[5] R.D. Edwards, "The double suspension of a certain homology 3-sphere is $S^5$" Notices Amer. Math. Soc. , 22 : 2 (1975) pp. A-334


Comments

Recent developments include: imbedding of topological manifolds as polyhedra with convex (or non-convex) faces in a Euclidean $E^n$, in particular in $E^3$ (e.g., polyhedral realizations of regular mappings (i.e. analogues of the regular polyhedra)); polyhedra of given genus with minimal number of vertices or edges or faces; colouring problems; and polyhedral realizations of famous configurations in geometry or topology.

References

[a1] U. Brehm, W. Kühnel, "A polyhedral model for Cartan's hypersurfaces in $S^4$" Mathematika , 33 (1986) pp. 55–61
[a2] B. Grünbaum, "Regular polyhedra - old and new" Aequat. Math. , 16 (1977) pp. 1–20
[a3] P. McMullen, Ch. Schulz, J.M. Wills, "Polyhedral 2-manifolds in $E^3$ with unusually large genus" Israel J. of Math. , 46 (1983) pp. 127–144
[a4] E. Schulte, J.M. Wills, "A polyhedral realization of Felix Klein's map$\{3,7\}_8$ on a Riemann surface of genus 3" J. London Math. Soc. , 32 (1985) pp. 539–547
[a5] U. Brehm, "Maximally symmetric polyhedral realizations of Dyck's regular map" Mathematika , 34 (1987) pp. 229–236
[a6] J.R. Munkres, "Elementary differential topology" , Princeton Univ. Press (1963)
[a7] L.C. Glaser, "Geometrical combinatorial topology" , 1–2 , v. Nostrand (1970)
How to Cite This Entry:
Polyhedron, abstract. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Polyhedron,_abstract&oldid=25374
This article was adapted from an original article by S.V. Matveev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article