Polyhedron, abstract

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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 $P\subset\mathrm{Int}N$ 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)