For simplicial complexesK andL, the following implications are trivial:
K≡L⇒K∼=L⇒ |K| ≈ |L|.
Although it goes without saying that the converse of the first implication does not hold, the converse of the second does not either. It should be noted that
|K|=|L| impliesK∼=Lby Theorems 3.2.11 and 3.2.10. The converse of the second implication is calledHauptvermutung(the fundamental conjecture).
It took a long time to find finite simplicial complexesKandLsuch thatK∼=L but |K| ≈ |L|. It is known that this conjecture does not hold even if |K| and|L|aren-manifolds (i.e., there exists an n-manifold that has topological triangulationsK∼=L).9
Remark 6.By Theorem 3.4.8, it might be expected that every PL map f :
|K| → |L|is simplicial with respect to some simplicial subdivisions ofKandL.
However, this is not the case. For example, letKbe the natural triangulation of R+, that is, K =ω∪ {[n−1, n]| n ∈N}, and let L=I ={0,1,I}. We define a PL map f:|K| → |L| as follows:
f(2n) = 2−n−1 and f(2n+ 1) = 1−2−n−1 for eachn∈ω,
andf is affine on each [n, n+ 1]. Since every subdivision ofK containsω as vertices but every subdivision ofLhas only finitely many vertices, then f is not simplicial with respect to any simplicial subdivisions ofKandL.
In §3.6, it will be proved that every proper PL map f : |K| → |L| is simplicial with respect to some simplicial subdivisions ofKandL. According to Proposition 3.2.6, a (PL) map f : |K| → |L| is proper if and only if, for eachσ∈L, there is a finite subcomplex Kσ⊂K such thatf−1(σ)⊂ |Kσ|.
3.5 The Metric Topology of Polyhedra 165
βvK(x) =
z(i) if v=vi, i= 1, . . . , n+ 1, 0 otherwise.
Thus, we have mapsβvK :|K| →I,v∈K(0), which are affine on each simplex ofK. It follows from the definition that
v∈K(0)
βvK(x) = 1 for eachx∈ |K|,
where {v ∈K(0) | βvK(x)= 0} =cK(x)(0), i.e., βvK(x)>0 ⇔v ∈cK(x)(0). Namely, (βKv )v∈K(0) is a partition of unity on|K|with suppβvK =|St(v, K)|. In fact, we have
(βKv )−1((0,1]) =OK(v) for everyv∈K(0). Then, eachx∈ |K|is uniquely represented in the form
x=
v∈K(0)
βvK(x)v,
whereβvK(x) is called thebarycentric coordinateofxatvwith respect to K. The injectionβK :|K| →ℓ1(K(0)) defined byβK(x)(v) =βKv (x) is called the canonical representationof K. Observe that βK(v) =ev is the unit vector ofℓ1(K(0)). For eachσ∈K, the restrictionβK|σis affine. It should be noted thatβK(K) ={βK(σ)|σ∈K}is a simplicial complex in the Banach spaceℓ1(K(0)) andβK:K→βK(K) is a simplicial isomorphism.
Now, we define the metricρK on|K|as follows:
ρK(x, y) =βK(x)−βK(y)1=
v∈K(0)
|βvK(x)−βvK(y)|,
where · 1 is the norm for ℓ1(K(0)). Note that ρK(v, v′) = 2 for each pair of distinct vertices v, v′ ∈ K(0). The topology on |K| induced by this metric ρK is called the metric topology. The space |K| with this topology is denoted by |K|m. This space is homeomorphic to the subspace βK(|K|) =|βK(K)|of the Banach spaceℓ1(K(0)) becauseβK is an isometry.
The space |K|m (or the metric space (|K|, ρK)) is called a metric polyhe-dron. Note thatℓ1(K(0))⊂RK(0), and that the topology ofβK(|K|) inherited from ℓ1(K(0)) coincides with the one inherited from the product spaceRK(0) because βK(|K|) is contained in the unit sphere ofℓ1(K(0)) (cf. Proposition 0.2.4). Hence, the metric topology on |K| is the coarsest topology such that allβvK :|K| →I(v∈K(0)) are continuous. Then, we have the following:
Fact For an arbitrary spaceX, eachf :X→ |K|m is continuous if and only ifβvKf is continuous for everyv∈K(0).
Since every βvK : |K| → I is continuous (with respect to the Whitehead topology), the identity id|K|:|K| → |K|mis continuous, hence the Whitehead topology and the metric topology are identical on each simplex ofK.
The open starOK(v) at eachv∈K(0)is open in|K|mbecause it is simply (βvK)−1((0,1]) (=|K|\(βvK)−1(0)). Hence,OK ∈cov(|K|m). For eachx∈ |K|, we have
OK(x) =
σ∈K[x]
rintσ=
v∈cK(x)(0)
OK(v)⊂ |St(cK(x), K)|.
Hence, the open starOK(x) is an open neighborhood ofxin |K|m. For each 0< r1, we have the following open neighborhood ofxin |K|m:
OK(x, r) = (1−r)x+rOK(x) =
(1−r)x+ryy ∈OK(x) . Then, OK(x, r) ⊂ BρK(x,2r). Indeed, for each y ∈ OK(x), since cK(x) cK(y) andβvK is affine oncK(y), it follows that
ρK((1−r)x+ry, x) =
v∈K(0)
βvK((1−r)x+ry)−βvK(x)
=
v∈K(0)
rβvK(y)−βKv (x)=rρK(x, y)<2r.
As a consequence, we have the following:
3.5.1 Proposition Let K be a simplicial complex and x ∈ |K|. Then, {OK(x, r)|0< r1}is an open neighborhood basis ofxin |K|m. ⊓⊔
The following proposition can be easily proved:
3.5.2 Proposition For every subcomplexL of a simplicial complex K, the metricρL is the restriction of ρK and |L|m is a closed subspace of |K|m. ⊓⊔
Moreover, we have:
3.5.3 Proposition For a simplicial complexKand eachx∈ |K|, the closure of OK(x) in |K|m coincides with |St(cK(x), K)|. In particular, for a vertex v∈K(0),|St(v, K)|is the closure ofOK(v)in|K|m.
Proof. According to Proposition 3.5.2,|St(cK(x), K)|is closed in|K|m. Then, it suffices to show that (1−t)y+tx∈OK(x) for eachy∈ |St(cK(x), K)|and 0< t1. For eachv∈cK(x)(0), sinceβvK(x)>0, it follows that
βvK((1−t)y+tx) = (1−t)βvK(y) +tβvK(x)>0, i.e., (1−t)y+tx∈(βvK)−1((0,1]) =OK(v).
Hence, (1−t)y+tx∈
v∈cK(x)(0)OK(v) =OK(x). ⊓⊔
3.5 The Metric Topology of Polyhedra 167 Thus, with respect to the metric topology as well as the Whitehead topology, SK = OclK and (βKv )v∈K(0) is a partition of unity on |K|m with suppβvK=|St(v, K)|.
Using the metric topology instead of the Whitehead (weak) topology, Proposition 3.3.4 can be generalized as follows:
3.5.4 Proposition LetKbe a simplicial complex andXan arbitrary space.
If two mapsf, g:X → |K|mare contiguous (with respect toK) thenf ≃K g by the straight-line homotopyh:X×I→ |K|m, that is,
h(x, t) = (1−t)f(x) +tg(x) for each(x, t)∈X×I.
Proof. It suffices to verify the continuity ofh. This follows from the continuity ofβKv h:X×I→I,v∈K(0), where
βvKh(x, t) = (1−t)βvKf(x) +tβvKg(x).
LetKandLbe simplicial complexes. Each simplicial mapf :K→Lcan be represented as follows:
f(x) =f v∈K(0)βvK(x)v
=
v∈K(0)
βKv (x)f(v) =
u∈L(0)v∈f−1(u)
βKv (x)u, that is,βuL(f(x)) =
v∈f−1(u)βvK(x) for each x∈ |K|andu∈L(0). Then, it is easy to show the following:
3.5.5 Proposition Let f : K → L be a simplicial map between simplicial complexes. Then, ρL(f(x), f(y)) ρK(x, y) for each x, y ∈ |K|, hence f :
|K|m→ |L|m is continuous. Whenf is injective,f : (|K|, ρK)→(|L|, ρL)is a closed isometry, so it is a closed embedding. Particularly, iff is bijective (i.e., f is a simplicial homeomorphism), it is a homeomorphism. As a consequence,
K≡L⇒ |K|m≈ |L|m.
For a finite simplicial complexK, since|K|is compact (Proposition 3.2.6), it follows that id|K| : |K| → |K|m is a homeomorphism, that is, the metric topology of |K| coincides with the Whitehead topology. More generally, we can prove the following theorem (cf. 3.2.16(2)):
3.5.6 Theorem For a simplicial complexK, the metric topology of |K| co-incides with the Whitehead topology (i.e.,|K|m=|K|as spaces) if and only ifK is locally finite.
Proof. If|K|m=|K|as spaces then|K|is metrizable, soKis locally finite by 3.2.16(2). To show the converse, letϕ= id :|K| → |K|m. Assume thatKis lo-cally finite, that is, St(v, K) is finite for eachv∈K(0). Then, eachϕ||St(v, K)| is a homeomorphism, soϕ|OK(v) is also a homeomorphism. SinceOK is an open cover of both|K|m and|K|, it follows thatϕis a homeomorphism. ⊓⊔
Concerning subdivisions, we have the following result:
3.5.7 Proposition LetK ⊲ K′ be simplicial complexes. Then,ρK(x, y) ρK′(x, y)for eachx, y∈ |K|=|K′|, henceid :|K′|m→ |K|mis continuous.
Proof. For eachx∈ |K|=|K′| andv∈K(0), we have βvK(x) =βvK w∈K′(0)βKw′(x)w
=
w∈K′(0)
βKw′(x)βKv (w).
Then, for eachx, y∈ |K|=|K′|, ρK(x, y) =
v∈K(0)
βvK(x)−βvK(y)
v∈K(0)w∈K′(0)
βwK′(x)−βwK′(y)βvK(w)
=
w∈K′(0)
βKw′(x)−βwK′(y)=ρK′(x, y).
In contrast to the Whitehead topology,|K′|m=|K|mfor some subdivision K′⊳K. Such an example can be defined inℓ1 as follows:
K=
0, ei, 0,eii∈N and K′=
0, 2−iei, ei, 0,2−iei, 2−iei,eii∈N
⊳K,
where eachei∈ℓ1is defined byei(i) = 1 andei(j) = 0 forj =i(Figure 3.4).
Then,|K|m is simply the hedgehogJ(N). The set{2−iei|i∈N} is closed in
|K′|m but limi→∞2−iei= 0 in|K|m.
e1
e2
ei
2−iei
2−1e1
0 2−2e2
Fig. 3.4.|K′|m=|K|m
For each simplicial map f : K → L, both maps f : |K| → |L| and f : |K|m → |L|m are continuous (Proposition 3.5.5). Moreover, recall that every PL map f : |K| → |L| is continuous. However, a PL mapf : |K|m→
3.5 The Metric Topology of Polyhedra 169
|L|m is not continuous even if f is bijective. In fact, consider K′ ⊳ K in the above example and let L = K. We define a PL map f : |K| → |L| by f(0) = 0, f(ei) = ei and f(2−iei) = 12ei, where 12ei is the barycenter of 0,ei. Then,f :|K|m→ |L|mis not continuous, because 2−iei→0 in |K|m butf(2−iei) =12ei→f(0) = 0 in|L|m.
It is inconvenient that the metric topology is changed by subdivisions and that PL maps are not continuous with respect to the metric topology.
However, concerning product simplicial complexes, the metric topology has the advantage of the Whitehead topology.
3.5.8 Theorem For each pair of ordered simplicial complexesKand L,
|K×sL|m=|K|m× |L|m as spaces.
Proof. The projections pr1:|K×sL|m→ |K|mand pr2:|K×sL|m→ |L|m are simplicial, so they are continuous. Therefore, id :|K×sL|m→ |K|m×|L|m is continuous.
We will prove the continuity of id : |K|m× |L|m → |K×sL|m at each (x, y)∈ |K|m× |L|m. To this end, we need the data ofβ(u,v)K×sL(x, y). Note that the carriercK×sL(x, y) is contained in the cell cK(x)×cL(y). Let
cK(x) =u1, . . . , un, u1<· · ·< un and cL(y) =v1, . . . , vm, v1<· · ·< vm,
and define 0 =a0< a1<· · ·< an= 1 and 0 =b0< b1<· · ·< bm= 1 as ak =
k
i=1
βuKi(x) and bk =
k
i=1
βvLi(y).
In this case, we can write
{a0, . . . , an, b0, . . . , bm}={c0, . . . , cℓ}
such that 0 =c0 < c1 <· · · < cℓ = 1, where max{n, m} ℓ < m+n and ℓ
k=1(ck−ck−1) =cℓ−c0= 1. For eachk= 1, . . . , ℓ, let ai(k)−1< ck ai(k) and bj(k)−1< ckbj(k), and define (¯uk,¯vk) = (ui(k), vj(k)). Then, we have
(¯u1,v¯1), . . . ,(¯uℓ,v¯ℓ) ∈K×sL,
which is the carrier of (x, y), andβ(¯Ku×k,¯svLk)(x, y) =ck−ck−1 because
ℓ
k=1
(ck−ck−1)(¯uk,¯vk)
= n
i=1i(k)=i
(ck−ck−1)ui(k),
m
j=1j(k)=j
(ck−ck−1)vj(k)
= n
i=1
(ai−ai−1)ui,
m
j=1
(bj−bj−1)vj
= n
i=1
βuKi(x)ui,
m
j=1
βvLj(y)vj
= (x, y).
For eachε >0, chooseδ >0 so that 2δ < ck−ck−1 for everyk= 1, . . . , ℓ and 2(ℓ+ 1)δ < ε. Then,βKui(x)> 2δ for i = 1, . . . , n and βvLj(y) > 2δ for j= 1, . . . , m. Now, let (x′, y′)∈ |K|×|L|withρK(x, x′)< δandρL(y, y′)< δ.
To show that ρK×sL((x, y),(x′, y′))< ε, let
cK(x′) =u′1,· · · , u′n′, u′1<· · ·< u′n′ and cL(y′) =v′1,· · ·, vm′ ′, v1′ <· · ·< v′m′. In the same way asxandy, definea′k=k
i=1βuK′
i(x′),b′k=k i=1βLv′
i(y′), and write
{a′0, . . . , a′n′, b′0, . . . , b′m′}={c′0, . . . , c′ℓ′},
where 0 =c′0< c′1<· · ·< c′ℓ′ = 1. For eachk= 1, . . . , ℓ′, let a′i′(k)−1< c′k a′i′(k)andb′j′(k)−1< c′k b′j′(k), and define (¯u′k,v¯k′) = (u′i′(k), v′j′(k)). Then,
(¯u′1,¯v′1), . . . ,(¯u′ℓ′,v¯′ℓ′)=cK×sL(x′, y′) andβK(¯u×′ sL
k,¯vk′)(x′, y′) =c′k−c′k−1. For eachi= 1, . . . , n,
βKui(x′)βuKi(x)− |βuKi(x)−βuKi(x′)|>2δ−ρK(x, x′)> δ >0, which implies that u1<· · ·< un is a subsequence ofu′1<· · ·< u′n′, that is, we can take 1p(1)<· · ·< p(n)n′ to writeui=u′p(i). Observe that
|a′p(k)−ak|
k
i=1
|βuKi(x)−βuKi(x′)|+
u∈{u1,...,un}
βKu(x′) ρK(x, x′)< δ and
|a′p(k)−1−ak−1|
k−1
i=1
|βuKi(x)−βuKi(x′)|+
u∈{u1,...,un}
βuK(x′) ρK(x, x′)< δ.
Similarly, we can take 1q(1)<· · ·< q(m)m′ to writevj =v′q(j). Then,
3.5 The Metric Topology of Polyhedra 171
|b′q(j)−bj|< δ and |b′q(j)−1−bj−1|< δ.
On the other hand, for eachk= 1, . . . , ℓ, becauseai(k)−1< ckai(k)and bj(k)−1< ckbj(k), we have
ck = min{ai(k), bj(k)} and ck−1= max{ai(k)−1, bj(k)−1}. Then, it follows that
b′q(j(k))−1< bj(k)−1+δck−1+δ < ck−δai(k)−δ < a′p(i(k)). Similarly,a′p(i(k))−1< b′q(j(k)). Hence, there is somer(k) = 1, . . . , ℓ′ such that
c′r(k)−1= max{a′p(i(k))−1, b′q(j(k))−1}
< c′r(k)= min{a′p(i(k)), b′q(j(k))}.
This means thatp(i(k)) =i′(r(k)) andq(j(k)) =j′(r(k)), which implies that (¯u′r(k),v¯′r(k)) = (u′i′(r(k)), vj′′(r(k))) = (u′p(i(k)), vq(j(k))′ )
= (ui(k), vj(k)) = (¯uk,¯vk) andβK(¯u×sL
k,¯vk)(x′, y′) =c′r(k)−c′r(k)−1. Observe that ck−δ= min
ai(k)−δ, bj(k)−δ
< c′r(k)= min
a′p(i(k)), b′q(j(k))
< ck+δ= min
ai(k)+δ, bj(k)+δ and ck−1−δ= max
ai(k)−1−δ, bj(k)−1−δ
< c′r(k)−1= max
a′p(i(k))−1, b′q(j(k))−1
< ck−1+δ= max
ai(k)−1+δ, bj(k)−1+δ . Moreover, it should be noted that
i′(r)=i
(c′r−c′r−1) =a′i−a′i−1=βuK′ i(x′)
j′(r)=j
(c′r−c′r−1) =b′j−b′j−1=βvL′
j(y′) and r∈ {r(1), . . . , r(ℓ)} ⇔
i′(r)∈ {p(1), . . . , p(n)}, j′(r)∈ {q(1), . . . , q(m)}. Then, it follows that
r∈{r(1),...,r(ℓ)}
(c′r−c′r−1)
i∈{p(1),...,p(n)}i′(r)=i
(c′r−c′r−1) +
j∈{q(1),...,q(m)}}j′(r)=j
(c′r−c′r−1)
=
i∈{p(1),...,p(n)}
βKu′
i(x′) +
j∈{q(1),...,q(m)}
βvL′ j(y′)
=
u∈{u1,...,un}
βuK(x′) +
v∈{v1,...,vm}
βvL(y′)
< ρK(x, x′) +ρL(y, y′)<2δ.
Consequently, we have the following estimation:
ρK×sL((x, y),(x′, y′)) =
ℓ
k=1
(ck−ck−1)−(c′r(k)−c′r(k)−1)
+
r∈{r(1),...,r(ℓ)}
(c′r−c′r−1)
<
ℓ
k=1
|ck−c′r(k)|+
ℓ
k=1
|ck−1−c′r(k)−1|+ 2δ
<2(ℓ+ 1)δ < ε.
This completes the proof. ⊓⊔
For a simplicial complexK, we can characterize the complete metrizability of|K|m as follows:
3.5.9 Theorem For a simplicial complexK, the following are equivalent:
(a) |K|mis completely metrizable;
(b) Kcontains no infinite full complexes as subcomplexes;
(c) ρK is complete.
Proof. Since (c)⇒(a) is obvious, we show (a) ⇒(b)⇒(c).
(a)⇒(b): Assume thatK contains an infinite full complex as a subcom-plex. Then, we have a countably infinite full complexL⊂K. Because|K|mis completely metrizable, its closed subspace|L|mis also completely metrizable.
However, |L| is the union of countably many simplexes that have no inte-rior points. This contradicts the Baire Category Theorem 1.5.1. Therefore,K contains no infinite full complexes.
(b) ⇒ (c): Let (xi)i∈N be a ρK-Cauchy sequence in |K|m. Since βK : (|K|, ρK)→ℓ1(K(0)) is an isometry, (βK(xi))i∈Nis Cauchy inℓ1(K(0)), hence we haveλ= limi→∞βK(xi)∈ℓ1(K(0)). Observe
3.5 The Metric Topology of Polyhedra 173
v∈K(0)
λ(v) =λ1= lim
i→∞βK(xi)1= 1.
Let A = {v ∈ K(0) | λ(v) > 0}. Each finite subset F ⊂ A is contained in cK(xi)(0) for sufficiently large i ∈N, henceF ∈ K. Thus,K contains the full complex∆(A) as a subcomplex. It follows from (b) thatAis finite. Using the above argument, we haveA ∈K, which means
x=
v∈A
λ(v)v∈ A ⊂ |K|.
Therefore,ρK is complete. ⊓⊔
There is another admissible metric on|K|mthat is widely adopted because it allows for easy estimates of distances.
3.5.10 Another Metric on a Polyhedron
(1) For a simplicial complexK, the following metric is admissible for|K|m: ρ∞K(x, y) =βK(x)−βK(y)∞
= sup
v∈K(0)|βvK(x)−βvK(y)|
ρK(x, y) ,
where · ∞is the norm forℓ∞(K(0)).
Sketch of Proof.For the continuity of id : (|K|, ρ∞K)→ (|K|, ρK), see Proposition 0.2.4.
(2) Proposition 3.5.7 is not valid for the metric ρ∞K, that is, the inequality ρ∞K(x, y)ρ∞K′(x, y) does not hold for someK′⊳K.
Example.Consider the following simplicial complexes inÊ: K={0,±1,0,±1}, K′={0,±12,±1,0,±12,±12,±1}.
Then,K′Kbutρ∞K(−43,34) =34 > ρ∞K′(−34,34) = 12. (3) For a simplicial complexK, the following are equivalent:
(a) K is finite-dimensional;
(b) ρK is uniformly equivalent to ρ∞K; (c) ρ∞K is complete.
Sketch of Proof.(a)⇒(b) and (c): For eachx, y∈ |K|, ρ∞K(x, y) ρK(x, y) 2(dimK+ 1)ρ∞K(x, y).
Then, applying Theorem 3.5.9, we have (c).
(b) or (c) ⇒ (a): If dimK = ∞, then we can inductively choose n-simplexesσn∈K,n∈Æ, so thatσn∩σm=∅ifn=m. Then, for any n < m,ρK(ˆσn,σˆm) = 2 and ρ∞K(ˆσn,ˆσm) = 1/(n+ 1). This is contrary to both (b) and (c).