A polyhedron (beware remark 1) is a topological space made up of very simple bits ‘glued’ together. The ‘bits’ are simplices of different dimensions. An abstract simplicial complex is a neat combinatorial way of giving the corresponding ‘gluing’ instructions, a bit like to plan of a construction kit!
A simplicial complex (sometimes called an abstract simplicial complex) consists of
such that the simplices satisfy the following conditions:
if is a simplex and , , then is also a simplex;
every singleton , , is a simplex.
We say is a face of . If has elements it is said to be a -simplex. The set of -simplices of is denoted by . The dimension of is the largest such that is non-empty.
The word ‘polyhedron’ is used here as it is often used by algebraic topologists, as a space described by a simplicial complex. Elsewhere in mathematics, it might mean a finite union of finite intersections of sets in Euclidean space defined by linear inequalities, usually assumed compact, and often with other assumptions as well (e.g., connected or convex). The various usages have a long history, as recounted in detail in Lakatos’s Proofs and Refutations.
Any (undirected simple) graph gives a simplicial complex. The usual definition of graph is that it is an ordered pair where is a set of vertices, and a set of (unordered) pairs of vertices. This is the simplest form of graph; it is undirected, edges do not have a ‘start’ and ‘finish’, (or ‘head’ and ‘tail’) and ‘simple’, in as much as there can be at most one edge between a given pair of vertices. (The case of a ‘multigraph’ where there can be multiple edges between vertices, and perhaps loops at a vertex, does not correspond to a simplicial complex, but does give a simplicial set.) A graph is a 1-dimensional simplicial complex.
Given any two sets and , and a relation , there are two simplicial complexes that encode information on the relation. These are generalisations of the nerve and the Vietoris complex. They are studied in detail in Dowker's theorem.
Buildings: An important class of simplicial complexes is provided by the notion of building, due to Jacques Tits.
Simplicial complexes are, in some sense, special cases of simplicial sets, but only ‘in some sense’.
To get from a simplicial complex to a fairly small simplicial set, you pick a total order on the set of vertices. Without an order on the vertices, you cannot speak of the face of a simplex, which is an essential feature of a simplicial set! The degeneracies are obtained by repeating an element when listing the vertices of a simplex. If , with then, for instance, .
If you do not want to pick an order then you can still form a simplicial set where to each -simplex of the original simplicial complex will correspond to simplices of that associated simplicial set. The result is more unwieldy, but can be useful under some circumstances as it defines a functor from the category of simplicial complexes to that of simplicial sets. This is very important when discussion group actions on simplicial complexes and how this transfers to the associated simplicial set.
Simplicial sets are essentially (that is, up to equivalence) presheaves on the simplex category of finite nonempty totally ordered sets, whereas simplicial complexes may be regarded as concrete presheaves on the category of finite nonempty sets and functions between them. This works as follows: given a simplicial complex, , define a presheaf whose values are sets of functions :
This defines an evident functor
that is full and faithful. The essential image is the subcategory of concrete presheaves, where a presheaf is concrete if the canonical map
is an injection. The point is that a morphism of concrete presheaves is uniquely determined from the function between their underlying sets (i.e., the underlying-set functor on concrete presheaves is faithful, so that a concrete presheaf is a set equipped with extra structure – that’s what makes it “concrete”).
(Equivalently but somewhat more elaborately, the category of concrete presheaves is the same as the full subcategory of concrete sheaves on with respect to the trivial topology, where the only covering sieve is the maximal sieve.)
which has a right adjoint. (See local topos.) It follows from this characterization that the category of simplicial complexes is a quasitopos, and in particular is locally cartesian closed. The category of simplicial sets on the other hand is a topos.
An abstract simplicial complex is a combinatorial gadget that models certain aspects of a spatial configuration. Sometimes it is useful, perhaps even necessary, to produce a topological space from that data in a simplicial complex.
To each simplicial complex , one can associate a topological space called the polyhedron of often also called the geometric realisation of and denoted . (This is essentially a special case of the geometric realisation of a simplicial sets.)
This can be constructed by taking a copy of a standard topological -simplex for each -simplex of and then ‘gluing’ them together according to the face relations encoded in .
We therefore first need the definition of a standard -simplex
The standard (topological) -simplex is (usually) taken to be the convex hull of the basis vectors in .
The geometric realisation of a simplicial complex, is then constructed by taking, for each abstract -simplex, , a copy, of such a standard topological -simplex, and then ‘gluing’ faces together, so whenever is a face of we identify with the corresponding face of . This space is usually denoted .
As a set, is constructed as follows:
is the set of all functions from to the closed interval such that
is a simplex of ;
There are two commonly used topologies on the set . The first is the metric topology: we put a metric on by
, when endowed with the metric space topology, will be denoted . Notice that when is finite, this gives as a subspace of the metric space (which is usually of much higher dimension than might seem geometrically significant in a given context).
The second topology is the coherent topology: each geometric simplex consists of all supported in , and is given the subspace topology inherited as a subset of ; then the coherent topology on is the largest topology for which all inclusions are continuous. This topological space is normally denoted just , reflecting the fact that the coherent topology is regarded as the default topology to put on the set .
Note that if is an inclusion of simplices in , then there is an induced subspace inclusion . The space may then be characterized as the colimit in of the diagram consisting of geometric simplices and inclusions between them, so that a function is continuous if and only if its restriction to each simplex is continuous. In particular, the identity function is continuous, so that the coherent topology contains the metric topology (and is often strictly larger).
If a topological space can be described up to homeomorphism as the geometric realization of a simplicial complex, we say it is triangulable, and a triangulation of a space is a simplicial complex together with a homeomorphism . (This is discussed in a bit more detail in the entry on classical triangulation.
There is another stronger notion of triangulation used by geometric topologists: a piecewise-linear (PL) structure on a topological manifold is given by a PL atlas, where the transition functions are piecewise-linear homeomorphisms. (A homeomorphism is piecewise linear if its graph is the intersection of with a semilinear set , meaning that is given by a finite Boolean combination of solution sets of linear inequalities.)
All smooth manifolds are triangulable and, in fact, admit PL structures.
All topological manifolds in dimensions 2 and 3 admit PL structures, and are in fact smoothable (admit a smooth manifold structure).
The manifold? does not admit a triangulation, much less a PL structure.
In dimensions , the -fold suspension of the Poincaré sphere is homeomorphic to the -sphere, hence is triangulable, but it does not admit a PL structure.
The following statement may seem obvious, but it requires careful proof:
As an important step:
The basic technique is to use subdivision.
A standard textbook reference is
An exposition is in