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classical triangulation

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Classical Triangulations

Aim

The aim of this entry is to describe some of the classical versions of important concepts which are needed elsewhere. This may serve as an entry point for someone versed in a more classical version of algebraic topology, or being adept at the nPOV, and its ramifications, needs to bridge the gap to more classical ideas to understand some more classically written source. (It may also be useful if some classically written source is not at hand when you need it!) The exposition will be fairly ‘classical’ with asides to explain the significance for later developments and for connections to the nPOV.

Triangulations à l’ancienne

Let XX be a polyhedron (in the sense of polyhedral space), i.e. a space homeomorphic to the geometric realisation of a simplicial complex.

Definition

A classical triangulation of XX is a pair (K,f)(K,f), where KK is a simplicial complex and f:|K|Xf : |K|\to X is a homeomorphism.

(In this context we will often drop the term ‘classical’ referring to ‘triangulation’ if there is little risk of confusion.)

Classical subdivision

The older form of subdivision involved the geometric realisation in the following way:

Definition

If KK is a simplicial complex, a (classical) subdivision of KK is a simplicial complex, K K^\prime, such that

a) the vertices of K K^\prime are (identified with) points of |K||K|;

b) if s s^\prime is a simplex of K K^\prime there is a simplex, ss of KK such that s |s|s^\prime \subset |s|; and

c) the mapping from |K ||K^\prime| to |K||K|, that extends the mapping of vertices of K K^\prime to the corresponding points of |K||K|, is a homeomorphism.

The interpretation, in simplicial complex, of the points of |K||K| as convex combinations of the vertices, allows an interpretation to be ascribed to K K^\prime. The general question of the meaning of ‘refinements’ that will be examined later may need a deeper examination of this subdivision process as it is a simple case of such a refinement.

Properties of Classical Subdivisions

  • Any subdivision of a subdivision of KK is a subdivision of KK.

  • If KK' and KK'' are subdivisions of KK then there is a subdivision KK''' of KK that is a subdivision of both KK' and KK''.

These statements thus assert that the subdivisions of a simplicial complex KK form a directed set with respect to the partial ordering defined by the relation of subdivision. (We will return to this later in this entry.)

Given any simplex, sKs\in K (or put more pedantically sS(K)s \in S(K)), and using the Canonical Construction of |K||K|, we can assign an open simplex, s\langle s \rangle, to ss. We first recall, from simplicial complex, that |K||K| is constructed as follows:

|K||K| is the set of all functions from V(K)V(K) to the closed interval [0,1][0,1] such that

  • if α|K|\alpha \in |K|, the set
{vV(K)α(v)0}\{v \in V(K) \mid \alpha(v) \neq 0\}

is a simplex of KK;

  • for each vV(K)v\in V(K),
    αV(K)α(v)=1.\sum_{\alpha \in V(K)} \alpha (v) = 1.

(We then give this set a topology (see simplicial complex).)

Definition

For sKs\in K, the open simplex, s|K|\langle s \rangle\subset |K| is defined by

s={α|K|α(v)0vs}.\langle s \rangle = \{ \alpha \in |K| \mid \alpha(v) \neq 0 \Leftrightarrow v\in s\}.

Beware although a closed simplex will be a closed subset of |K||K|, an open simplex need not be open in |K||K|. However every s\langle s \rangle is and open set of |s||s|. (see Spanier, p. 112, for a discussion.)

Barycentric subdivision (classical geometric forms)

The barycentric subdivision is one of the best known and most useful natural subdivisions available in general. (Other are also used, for instance, the middle edge or ordinal subdivision.) The barycentric subdivision has the good property that it exists without recourse to the realisation process, although usually introduced via that process. It is in that form that it is discussed in subdivision. Here we give the ‘classical’ form and go from that towards the other functorial form.

Definition

If σ={v 0,,v q}K q\sigma = \{ v_0, \ldots, v_q\} \in K_q, the set of qq-simplices of a simplicial complex, KK, then its barycentre, b(σ)b(\sigma), is the point

b(σ)= 0iq1q+1v i|K|.b(\sigma) = \sum_{0\leq i \leq q}\frac{1}{q + 1} v_i \in |K|.

The barycentric subdivision, sdKsd K, of KK is the simplicial complex whose vertices are the barycentres of the simplices of KK and whose simplices are finite non-empty collections of barycentres of simplices, which are totally ordered by the face relation of KK, i.e., by inclusion when considered as subsets of V(K)V(K).

As b(σ)b(\sigma) is completely determined by σ\sigma, this can be rephrased as:

Definition: (Alternative form)

The barycentric subdivision, sdKsd K, of KK is the simplicial complex specified by

  • the vertices of sdKsd K are the simplices of KK;
  • the subset {σ 0,σ q}\{\sigma_0, \ldots \sigma_q\} is a simplex of sdKsd K if there is a total order on its elements (which will be assumed to be the given order, as written) such that
σ 0σ q.\sigma_0\subset \ldots \subset\sigma_q.

Triangulations and Covers

We now need a bit more terminology:

Definition

Given any vertex vv of KK, its star is defined by

st(v)={α|K|α(v)0}.st( v) = \{ \alpha \in |K| \mid \alpha(v)\neq 0\}.

The set, st(v)st(v), is open in |K||K|. We have

st(v)={svisavertexofs},st(v) = \bigcup \{\langle s \rangle\mid v is a vertex of s\},

the union of the interiors of those simplices that have ss as a vertex. These vertex stars give an open cover, 𝒰\mathcal{U}, of |K||K| and the following classical result tells us that the nerve N(𝒰)N(\mathcal{U}) of this covering is KK itself (up to isomorphism):

Proposition (cf. Spanier, p. 114)

Let 𝒰={st(v)vV(K)}\mathcal{U} = \{st(v)\mid v \in V(K)\}. The vertex map ϕ\phi from KK to N(𝒰)N(\mathcal{U}) defined by

ϕ(v)=st(v)\phi (v) = \langle st(v)\rangle

is a simplicial isomorphism

ϕ:KN(𝒰).\phi : K \cong N(\mathcal{U}).

category: motivation

Revised on December 5, 2010 18:33:34 by Tim Porter (95.147.236.206)