Let E be a linear space. The convex hull C =⟨A⟩of a non-empty finite set A ⊂E is called a (convex)linear cell (or just acell). An n-dimensional cell is called ann-cell. The following set is called theradial interiorofC:
rintC={
x∈C∀y∈C,∃δ >0 such that (1 +δ)x−δy∈C} . The set∂C =C\rintC is called theradial boundaryofC. ThefaceofC atx∈C is defined as the following set:
Cx={
y∈C∃δ >0 such that (1 +δ)x−δy∈C} .
Then, rintC={x∈C|Cx=C}. According to Proposition 1.4.1, every finite-dimensional flat has the unique topology and is homeomorphic to Euclidean space. Then, the radial interior rintC and the radial boundary∂C are equal to the topological interior intC and the topological boundary bdC of C in the flat hull ofC with respect to this topology.
Using affine functionals, we characterize cells as follows:
Proposition 1.5.1 A non-degenerate C ⊂E is a cell if and only if flC is [4.1.8]
finite-dimensional, x+R+(y−x)̸⊂C for each distinct points x, y∈C, and there are finitely many non-constant affine functionals f1, . . . , fk : flC →R such thatC= flC∩∩k
i=1fi−1(R+), whererintC= flC∩∩k
i=1fi−1((0,∞)).18 Every cellC has the smallest finite subset C(0) whose convex hull ⟨C(0)⟩ is equal to C. Then, C(0) ⊂ ∂C. Each point of C(0) is called a vertex of C. When C(0) ={v1, . . . , vn} is affinely independent,C is called a simplex and dimC = n−1 (n = cardC(0)). An n-dimensional simplex is called an n-simplex. A cellD is said to be a face of C if D =Cx for some x ∈C, and denoted by D 6C or C >D. A faceD 6C is called a proper face of C if D ̸= C, denoted by D < C or C > D. Then, ∂C = ∪
D<CD. An n-dimensional face is called ann-face. For simplexesσ and τ,τ 6σ if and only ifτ(0) ⊂σ(0). When a cellCis representedC= flC∩∩k
i=1fi−1(R+) for some non-constant affine functionalsf1, . . . , fk: flC→R, iffi−10 (0)∩C̸=∅, thenD=C∩fi−10 (0) is a face ofC, where
fi−10 (0)∩C=∅ ⇒ C= flC∩ ∩
i̸=i0
fi−1(R+).
IfC̸= flC∩∩
i̸=i0fi−1(R+), then D=C∩fi−10 (0) is a proper face ofC and flD= flC∩fi−10 (0).
18Proposition 4.1.8 of [GAGT] does not mention about rintC, but this last equality is easily verified.
1.5 Cell Complexes and Simplicial Complexes 21 Indeed, takex1 ∈rintC and x2 ∈flC∩∩
i̸=i0fi−1((0,∞))\rintC. Then, we havex0∈ ⟨x1, x2⟩ ∩fi−10 (0). Observe
x∈flC∩fi−10 (0)∩ ∩
i̸=i0
fi−1((0,∞)) = rintD
⊂D= flC∩fi−10 (0)∩ ∩
i̸=i0
fi−1(R+)⊂C.
Hence, it follows that D ⊂ Cx ̸= C. For each y ∈ C \D and t > 0, sincefi0(x0) = 0 and fi0(y) >0, it follows that fi0((1 +t)x0−ty) <0, hence (1 +t)x0−ty̸∈C, that meansy̸∈Cx0. Therefore,Cx0 ⊂D. Thus, D=Cx0 is a proper face ofC. Moreover, flD⊂flC∩fi−10 (0) because flD is the smallest flat containingD. Conversely, for eachz∈flC∩fi−10 (0), we can choose so smallδ >0 that (1−δ)x0+δz∈fi−1((0,∞)) for everyi̸=i0
becausex0∈∩
i̸=i0fi−1((0,∞)). Hence, (1−δ)x0+δz∈D, which implies
z∈flD by [GAGT, Proposition 3.2.1]. Thus, flD= flC∩fi−10 (0). ⊓⊔ [4.1.8]
Two simplex σ and τ are joinable(or σ is joinable to τ) if σ∩τ = ∅ andσ(0)∪τ(0) is affinely independent. In this case,⟨σ(0)∪τ(0)⟩is a simplex of dimension dimσ+ dimτ+ 1. This simplex is called thejoinofσandτand denoted by στ. For a proper faceτ < σ, the simplexτ′ =⟨σ(0)\τ(0)⟩is also a face of σandσis the join ofτ andτ′, i.e.,σ=τ τ′. This face is called the opposite faceofσto a faceτ < σ.
A collectionK of (convex linear) cells inE is called a (convex linear) cell complexifK satisfies the following two conditions:
(C1) IfC∈K andD6C thenD∈K;
(C2) For eachC, D∈K withC∩D̸=∅,C∩D6C (andC∩D6D).
Under condition (C1), condition (C2) is equivalent to each of the following conditions:
(C2′) For eachC, D∈K,C∩rintD̸=∅ impliesD6C.
(C2′′) For eachC̸=D∈K, rintC∩rintD=∅.
A subcollection Lof a cell complex K is called asubcomplex ifLis a cell complex. For a cellC, the following is cell complexes:
F(C) ={
DD6C}
and F(∂C) ={
DD < C} ,
where F(∂C) is a subcomplex of F(C). A cell complex consisting of sim-plexes is called asimplicial complex. For a simplexσ,F(σ) andF(∂σ) are simplicial complexes.
For a cell (simplicial) complexK, the set |K|=∪ K=∪
C∈KC is called thepolyhedron19ofK. The topology for|K|is defined as follows:
U ⊂ |K|is open in|K| ⇔ ∀C∈K, U∩Cis open inC.
19The plural ispolyhedra
22 1 Preliminaries and Background Results
This topology is called theWhitehead(orweak)topology. Apolyhedron (resp. atopological polyhedron) is defined as a spaceP such thatP=|K|
(resp.P ≈ |K|) for some cell complexK. A simplicial complexK is called a triangulationof a polyhedron (resp. a topological polyhedron)P ifP=|K|
(resp. P ≈ |K|), where it is also said that P is triangulated by K or K triangulatesP. A subspaceQof a polyhedron (or a topological polyhedron) P is called a subpolyhedron20 of P if there exists a pair (K, L) of a cell complex and its subcomplex such that P = |K| and Q = |L| (or (P, Q) ≈ (|K|,|L|)).
[4.2.2]
[4.7.12]
Proposition 1.5.2 Every (topological) polyhedron is perfectly normal and paracompact.
[4.2.1] Proposition 1.5.3 Every subpolyhedron of a polyhedron P is a closed sub-space of P.
For a cell (simplicial) complexK, K(0) =∪
C∈KC(0) is called theset of vertices (or the 0-skeleton) of K. It is said that K isfinite,infinite, or countable depending on cardK (equivalently cardK(0)). For each n ∈ N, the subcomplexK(n)={C∈K|dimC6n}is calledn-skeletonofK. The dimension ofK is defined by dimK = supC∈KdimC and K is said to be n-dimensionalif dimK=n. Then, dimK6nif and only if K=K(n). It is said thatKisfinite-dimensional(denoted by dimK <∞) if dimK6n for some n∈N. Otherwise,K is said to beinfinite-dimensional (denoted by dimK=∞).
The subcomplex St(C, K) =∪
C6D∈KF(D) ofKis calledstaratC∈K, where |St(C, K)|= st(rintC, K) for eachC ∈K. In particular, for a vertex v∈K(0), St(v, K) =∪
C∈K[v]F(C) and|St(v, K)|= st(v, K). For each point x∈ |K|, letcK(x)∈K be the smallest cell containingx(sox∈rintcK(x)), which is called the carrier of x in K. Then, |St(cK(x), K)| = st(x, K) =
∪K[x] = ∪
C∈K[x]C is a (closed) neighborhood of x in |K|. On the other hand,OK(x) =∪
C∈K[x]rintC is an open neighborhood ofxin|K|, which is called theopen staratx(with respect toK). We have the following special covers of|K|:
SK ={
|St(v, K)|v∈K(0)}
and OK ={
OK(v)v∈K(0)} . In the case K is a simplicial complex, the link at σ∈ K can be defined as follows:
Lk(σ, K) ={
τ∈St(σ, K)τ∩σ=∅} . Then, everyτ∈Lk(σ, K) is joinable toσ,|St(σ, K)|=∪
τ∈Kστ, and St(σ, K) =F(σ)∪{
σ′τ σ′6σ, τ ∈Lk(σ, K)} .
20The plural issubpolyhedra
1.5 Cell Complexes and Simplicial Complexes 23 A cell complexK is said to belocally finite or locally finite-dimen-sional depending on whether the star St(v, K) at every vertex v ∈ K(0) is finite or finite-dimensional. Concerning the topology of the polyhedron |K|, the following is elementary:
Proposition 1.5.4 Let K be a cell complex.
[4.2.6]
[4.2.16(2)]
[4.2.16(1)]
(1) |K| is compact metrizable if and only if K is finite;
(2) |K| is (locally compact) metrizable if and only ifK is locally finite;
(3) |K| is separable if and only ifK is countable.
A full (simplicial) complex is a simplicial complex such that K(0) is affinely independent and ⟨v1, . . . , vn⟩ ∈ K for any finite distinct vertices v1, . . . , vn ∈ K(0). For any simplex σ, F(σ) is a full complex. A subcom-plexLof a simplicial complexKsaid to befull inKor afull subcomplex of K provided that σ ∈ L for any σ ∈ K with σ(0) ⊂ L(0). Let K be an n-dimensional simplicial complex and σ ∈ K an n-simplex. Then, F(∂σ) is not a full complex nor a full subcomplex ofK, but it is a full subcomplex of the (n−1)-skeletonK(n−1).
A cell complex K′ is called a subdivision of K (or it is said that K′ subdividesK) if the following conditions are satisfied:
(S1) Each cell ofKis covered by finitely many cells ofK′;
(S2) Each cell ofK′ is contained in some cell ofK(i.e., K′ refinesK).
Then, |K′| =|K| as spaces. Evidently, if K′ is a subdivision of K and K′′
is a subdivision of K′, K′′ is a subdivision of K. A simplicial complex K′ subdividingKis called asimplicial subdivision.
Theorem 1.5.5 Let K be a cell complex with an order on K(0) such that C(0) has the maximum vC for each cell C ∈ K (e.g., a total order on K(0) satisfies this condition). Then, the following is a simplicial subdivision with K′(0)=K(0):
K′ ={
⟨vC1, . . . , vCn⟩C1>· · ·> Cn∈K} .
Proof. First of all, it should be remark that the representation of simplexes of K′ in the above is not unique. For i < j, it is possible that vCi =vCj. In this case,vCi=vCfor any faceC < Ci withCj6C. It should also remarked thatvCi, . . . , vCn∈Ci(0) for eachi= 1, . . . , n. Then, in the above definition of K′, we can tack on the following additional conditions toC1>· · ·> Cn∈K:
• vC1 >· · · > vCn and vCi ̸∈C(0) for Ci > C > Ci+1, i = 1, . . . , n, where the casei=nmeansCn ={vCn}.
The above second condition is equivalent to the following:
• EachCi is the smallest cell ofK containingvCi, . . . , vCn.
24 1 Preliminaries and Background Results
Indeed, Cn ={vCn} is the smallest cell of K containingvCn If Ci+1 is the smallest cell ofK containingvCi+1, . . . , vCn, thenCi is the smallest cell ofK containing vCi, . . . , vCn because vCi ̸∈ C(0) for Ci > C > Ci+1. By induc-tion, each Ci is the smallest cell ofK containingvC1, . . . , vCn. The converse implication is trivial.
To show thatK′ is a simplicial subdivision ofK, we have the conditions (C1), (C2′′), (S1), and (S2), where (C1) and (S2) are obvious. For (S1), it is enough to show that |K| is covered by K′ because each cell of K has only finite vertices. Thus, to see (C2′′) and (S1), it suffices to show the following:
(*) For each pointx∈ |K|, there exists a unique sequenceC1>· · ·> Cn∈K such that vC1 > · · · > vCn, each Ci is the smallest cell of K containing vC1, . . . , vCn, andx∈rint⟨vC1, . . . , vCn⟩.
(Uniqueness) Let C1 > · · · > Cn ∈ K be the sequence in (*). Then,
⟨vC1, . . . , vCn⟩ ⊂ (C1)x by the definition of (C1)x. From the minimality of C1, it follows thatC1 = (C1)x, sox∈rintC1 [GAGT, Proposition 3.2.5(8)].
[3.2.5(8)]
Therefore, C1 = cK(x). When n > 1, x2 ∈ ∂C1 is uniquely determined so that x ∈ rint⟨vC1, x2⟩. We also have x′2 ∈ rint⟨vC2, . . . , vCn⟩ such that x ∈ rint⟨vC1, x′2⟩. Since ⟨vC2, . . . , vCn⟩ ⊂ C2 ⊂ ∂C1, it follows that x2 = x′2 ∈ rint⟨vC2, . . . , vCn⟩. Then, by the same argument asx, we have C2 =cK(x2).
By iterating this operation, we havexi∈∂Ci−1such thatxi−1∈rint⟨vCi, xi⟩, which implies that xi ∈ rint⟨vCi, . . . , vCn⟩ and Ci = cK(xi). Thus, a finite sequenceC1>· · ·> Cn ∈K in (*) is uniquely determined.
(Existence) For each x ∈ |K|, write x = x1 and let C1 = cK(x1) ∈ K (the carrier of x1). When dimC1 = 0, C1 = {x1} = {vC1} and x = x1 ∈ rint⟨vC1⟩. Otherwise, we have x2 ∈ ∂C1 such that x1 ∈ rint⟨vC1, x2⟩. Let C2 =cK(x2)< C1. Then, vC2 < vC1 because vC1 ̸∈C2(0). If vC1 ∈ C(0) for some C1> C > C2, thenx1∈ ⟨vC1, x2⟩ ⊂C, which contradicts the fact that C < C1=cK(x1). Therefore,vC1 ̸∈C(0)forC1> C > C2. When dimC2= 0, C2={x2}={vC2}andx∈rint⟨vC1, vC2⟩. Otherwise, we havex3∈∂C2 such that x2∈rint⟨vC2, x3⟩, which implies
x∈rint⟨vC1, x2⟩ ⊂rint⟨vC1, vC2, x3⟩.
Let C3 =cK(x3)< C2. Then, vC2 > vC3 because vC2 ̸∈C3(0). IfvC2 ∈C(0) for some C2 > C > C3, then x2 ∈ ⟨vC2, x3⟩ ⊂ C, which contradicts the fact that C < C2 = cK(x2). Therefore, vC2 ̸∈ C(0) for C2 > C > C3 By iterating this operation until dimCn = 0, we can obtain a finite sequence of cellsC1> C2>· · ·> Cn={vCn}withxi∈Ci(x1=x,xn=vCn) such that vC1 >· · ·> vCn, vCi ̸∈C(0) forCi > C > Ci+1, andx∈rint⟨vC1, . . . , vCn⟩.
As saw in the above, eachCiis the smallest cell ofK containingvC1, . . . , vCn
if and only ifvCi ̸∈C(0) forCi> C > Ci+1,i= 1, . . . , n. ⊓⊔
Note The above subdivisionK′is the same as the one obtained in [GAGT, Theorem 4.2.10]. Indeed, in the proof of [GAGT, Theorem 4.2.10], the sim-[4.2.10]
1.5 Cell Complexes and Simplicial Complexes 25 plicial subdivision ofKis defined as∪
n∈NLn, whereL1 =K(1) andLn is a simplicial subdivision ofK(n)inductively defined as follows:
Ln=Ln−1∪{
vCσC∈K(n)\K(n−1), σ∈Ln−1, withσ⊂∂CandvC̸∈cK(ˆσ)(0)}
. Then, Ln satisfies the conditions that L(0)n = K(0), Ln−1 ⊂ Ln, and vcK(ˆσ) ∈ σ(0) for each σ ∈ Ln, where ˆσ is the barycenter of σ. To see that this subdivision is equal to the subdivisionK′ of Theorem 1.5.5, sup-poseLn−1 ⊂ K′. Let C ∈ K(n)\K(n−1) and σ ∈ Ln−1 with σ ⊂ ∂C andvC ̸∈cK(ˆσ)(0). By the assumption,σis written asσ=⟨vC1, . . . , vCn⟩, where C1 >· · · > Cn ∈ K, vC1 > · · · > vCn, and each Ci is the small-est cell of K containing vC1, . . . , vCn. In the above proof (Uniqueness), let x = ˆσ. Then, C1 = cK(ˆσ) and rintσ ⊂ rintC1, hence C > C1 and vC> vC1. Thus,vCσ=⟨vC, vC1, . . . , vCn⟩ ∈K′. Therefore,Ln⊂K′. Since {rintσ|σ∈∪
n∈NLn}=|K|, it follows that∪
n∈NLn=K′.
A cell complex K with such an oder as in Theorem 1.5.5 is called an ordered cell complex. In the case K is simplicial complex, such an order gives a total order on the vertices of each simplex in K. Then, an ordered simplicial complex is a simplicial complex K with an order on K(0) such that σ(0) is totally ordered for eachσ∈K.
The following lemma is useful to construction simplicial subdivision of cell complexes:
[4.6.1]
Lemma 1.5.6 Let C be an n-cell and v0 ∈ rintC. Given a triangulationL of ∂C (that is a subdivision ofF(∂C)), define
K=L∪ {v0} ∪{
v0σσ∈L} .
Then, K is a triangulation ofC (that is a subdivision ofF(C)) such that L is a full subcomplex of K.
Using Lemma 1.5.6 in the skeletonwise construction, we can obtain the following:
[4.6.2]
Proposition 1.5.7 LetKbe a cell complex andL′be a simplicial subdivision of a subcomplex L ⊂ K. Given vC ∈ rintC for each C ∈ K\L, there is a simplicial subdivisionK′ ofK such that L′ is a full subcomplex ofK′ and
K′(0) =L′(0)∪K(0)∪{ vC
C∈K\L} .
In the caseL=∅ in Proposition 1.5.7 above, the obtained subdivision is written as follows:
K′ ={
⟨vC1, . . . , vCn⟩C1<· · ·< Cn∈K} ,
which is called a derived subdivision of K. For a simplicial complex K, as vσ ∈ rintσ, σ∈K, we can take thebarycenter σˆ =∑n
i=1n−1vi, where
26 1 Preliminaries and Background Results
σ=⟨v1, . . . , vn⟩(n= dimσ−1). Then, the obtained derived subdivisionK′ ofK is called thebarycentric subdivisionofK and denoted by SdK.
For a simplicial complexKandx∈ |K| \K(0), we can define the following simplicial subdivision ofK
Kx= (K\K[x])∪{
xσσ∈St(cK(x))\K[x]} ,
wherexσis the join ofxandσ, i.e.,xσ=⟨{x}∪σ(0)⟩. The operationK→Kx
(orKxitself) is called astarringofK atx. A subdivision obtained by finite starrings is known as a stellar subdivision. In general, (Kx)y ̸= (Ky)x
for distinct two points x, y ∈ |K| \K(0). When K is finite, every derived subdivision ofK is a stellar subdivision.
The following theorem is well known and very important (e.g., the proof of the paracompactness of polyhedra is depend on this theorem). Almost all textbooks mentioning or citing the result do not treat the proof. For the proof, we were used to refer the original article by J.H.C. Whitehead. But now, the complete proof is found in [GAGT, Section 4.7].
[4.7.11] Theorem 1.5.8 (J.H.C. Whitehead) LetKbe an arbitrary simplicial com-plex. For any open coverU of|K|,K has a simplicial subdivisionK′ such that SK′ ≺ U.
The following theorem is widely used and proved in many textbook:
[4.3.3] Theorem 1.5.9 (Homotopy Extension Theorem) Let L be a subcom-plex of a cell comsubcom-plexK andh:|L| ×I→X be a homotopy into an arbitrary spaceX. If h0 extends to a map f :|K| →X, thenhextends to a homotopy
¯h: |K| ×I→X with ¯h0 =f. Moreover, if h is a U-homotopy for an open coverU of X, then¯hcan be taken as aU-homotopy.
Each cell complexKitself is a cover of the polyhedron|K|. It is said that two map f, g:X → |K| are contiguousiff and g are K-close, that is, for eachx∈X, there is a cellC∈K such that f(x), g(x)∈C. Then, every two contiguous maps f, g : X → |K| are K-homotopic, i.e., f ≃K g because a K-homotopyh:X×I→ |K|can be defined as follows:
h(x, t) = (1−t)f(x) +tg(x) for each x∈X andt∈I.
Such a homotopy is called the straight line homotopy.
Letf :C→Dbe a map from a cellC to another cellD. It is said thatf isaffineif it satisfies the following
f((1−t)x+ty) = (1−t)f(x) +tf(y) for each x, y∈C andt∈I.
For affine maps in more general settings, refer to [GAGT, Sections 3.1 and 3.2].
1.5 Cell Complexes and Simplicial Complexes 27 Let K and L be cell complexes. A map f : |K| → |L| is said to be piecewise linear (PL) if there is a subdivision K′ of K such that f|C is affine for eachC∈K′. In this definition, we can takeK′ so thatf(K′)≺L, that is, for eachC∈K′,f(C) is contained in some cell inL[GAGT, Lemma
4.4.3]. From this fact, it easily follows that the composition of PL maps is also [4.4.3]
PL [GAGT, Proposition 4.4.4]. There is the following characterization of PL [4.4.4]
maps:
[4.4.2]
Theorem 1.5.10 LetKandLbe cell complexes. A mapf :|K| → |L|is PL if and only if the graphG(f) ={(x, f(x))|x∈ |K|}is a polyhedron. ⊓⊔
It should be remarked that the image of a PL map is, in general, not a polyhedron [GAGT, Remark 4 on p.158]. For a homeomorphismf :|K| → |L|, if f is PL, then the inverse f−1 is also PL by Theorem 1.5.10 above. A homeomorphism f :|K| → |L|being PL is called a piecewise linear (PL) homeomorphism. Note that the composition of PL homeomorphisms is also PL. It is said that the polyhedra|K|and|L|arePL homeomorphicor|K|is PL homeomorphicto|L|if there exists a PL homeomorphismf :|K| → |L|.
Now, letKandLbe simplicial complexes. A mapf :|K| → |L|is called a simplicialmap fromK toL(or with respect toKandL) iff|σis affine and f(σ)∈Lfor eachσ∈K, where dimf(σ)6dimσ. Evidently,f(K(0))⊂L(0) andf′(K) ={f(σ)|σ∈K} is a subcomplex ofL. Whenσ=⟨v1, . . . , vn⟩ ∈ K, we havef(σ) =⟨f(v1), . . . , f(vn)⟩ ∈Land
f( ∑n i=1tvi
)=
∑n i=1
tif(vi) fort1, . . . , tn∈Iwith
∑n i=1
ti= 1,
where it is possible that f(vi) = f(vj) fori ̸= j. A simplicial map from K to Lis also written as f :K →L. Obviously, every simplicial map is PL. It might be expected that every PL mapf :|K| → |L|is simplicial with respect to some simplicial subdivisions ofK andLbut this is not true. In fact, there exists a PL map f : |K| → |L| that is not simplicial with respect to any simplicial subdivisions ofK andL[GAGT, Remark 6 on p.162]. However, we have the following:
[4.6.5]
Theorem 1.5.11 LetKandLbe cell complexes. A proper mapf :|K| → |L|
is PL if and only iff is simplicial with respect to some simplicial subdivisions of K andL. ⊓⊔
For a bijectionf : |K| → |L|, if f is simplicial with respect to K and L, then the inverse f−1 is a simplicial map fromL to K. A simplicial bijection f :K→Lis called asimplicial isomorphism. It is said thatK andLare simplicially isomorphic or K issimplicially isomorphic to L (denoted by K ≡ L) if there is a simplicial isomorphism f : K → L. Obviously, ≡ is an equivalence relation among simplicial complexes. A weaker equivalence
28 1 Preliminaries and Background Results
relation can be defined among them. It is said thatK andL are combina-torially equivalentorKiscombinatorially equivalenttoL(denoted by K∼=L) if they have simplicial subdivisions which are simplicially isomorphic.
From the following theorem, it follows that∼= is an equivalence relation.
[4.4.8] Theorem 1.5.12 Two simplicial complexes K and L are combinatorially equivalent if and only if the polyhedra|K| and|L| are PL homeomorphic.
LetK be a simplicial complex. There exist mapsβvK :|K| →I,v∈K(0), such that everyβvK is affine on each simplex ofK,
x= ∑
v∈K(0)
βvK(x)v, ∑
v∈K(0)
βvK(x) = 1, and cK(x)(0)={
v∈K(0)βvK(x)>0} ,
whereβvK(x) is called thebarycentric coordinateofxatvwith respect to K. Note that (βvK)−1((0,1]) =OK(v) for everyv∈K(0). The injectionβK :
|K| → ℓ1(K(0)) defined by βK(x) = (βKv (x))v∈K(0) is called the canonical representationofK. Observe thatβK(v) =evis the unit vector ofℓ1(K(0)) for eachv∈K(0). We define the metric ρK on|K|as follows:
ρK(x, y) =∥βK(x)−βK(y)∥= ∑
v∈K(0)
βvK(x)−βvK(y).
The topology on|K|induced by the metricρKis called themetric topology.
The space |K| with this topology is denoted by |K|m. Then, id|K| : |K| →
|K|m is continuous but not a homeomorphism in general.
[4.5.6] Proposition 1.5.13 For a simplicial complex K, the metric topology on|K|
coincides with the Whitehead topology (i.e.,|K|m=|K|as spaces) if and only if K is locally finite.
[4.5.9] Theorem 1.5.14 For a simplicial complexK, the following are equivalent:
(a) |K|m is completely metrizable;
(b) K contains no infinite full complexes as subcomplexes;
(c) ρK is complete.
[4.9.6] Theorem 1.5.15 For every simplicial complex K, the identity φ = id :
|K| → |K|m is a homotopy equivalence with a homotopy inverseψ:|K|m→
|K| such that ψφ≃K id andφψ≃K id,21 where ψφ≃OK id andφψ≃OK id are also valid. These homotopies are realized by the straight-line homotopy.
21Both mapsψφandφψare contiguous to id.