3
Simplicial Complexes and Polyhedra
In this chapter, we introduce and demonstrate the basic concepts and prop-erties of simplicial complexes. The importance and usefulness of simplicial complexes lies in the fact that they can be used to approximate and explore (topological) spaces. A polyhedron is the underlying space of a simplicial complex, which has two typical topologies, the so-called weak (Whitehead) topology and the metric topology. The paracompactness of the weak topol-ogy will be shown. We show that every completely metrizable space can be represented as the inverse limit of locally finite-dimensional polyhedra with the metric topology. In addition, we give a proof of the Whitehead–Milnor Theorem on the homotopy type of simplicial complexes. We also prove that a map between polyhedra is a homotopy equivalence if it induces isomorphisms between their homotopy groups.
* * *
This chapter is based on Chapters 1 and 2. In particular, we employ the theory of convex sets and the related concepts discussed in Chapter 2.
of the flat hull flC, i.e., dimC = dim flA < ∞(cf. §2.2). An n-dimensional cell is called ann-cell(or alinearn-cell). Obviously, every simplex is a cell.
The affine image of a cell is always a cell and every cell is the affine image of some simplex. A 0-cell and a cell are the same as a 0-simplex and a 1-simplex, respectively. If cardA =n and dimA=n−1, thenA is affinely independent, hence Ais a simplex. Whenv1, . . . , vn+1is ann-simplex, it follows that v1, . . . , vn+1 are affinely independent, hence they are vertices of the simplex.
The radial interior and the radial boundary of a cellC are simply called theinteriorand the boundaryof C. Recall that they are defined without topology, that is,
rintC=
x∈C∀y ∈C, ∃δ >0 such that (1 +δ)x−δy∈C and∂C=C\rintC(cf. §2.2).2 Forx=y∈E,
∂x, y={x, y}, rintx, y=x, y \ {x, y}. Then, we can also write as follows:
rintC=
x∈C∀y∈C, ∃z∈C such that x∈rinty, z .
According to Proposition 2.5.1, the flat hull flChas the unique topology such that the following operation is continuous:
flC×flC×R∋(x, y, t)→(1−t)x+ty∈flC,
and flC is affinely homeomorphic toRn, where n= dimC= dim flC. With respect to this topology, as is shown in Proposition 2.5.5,
rintC= intflCC and ∂C= bdflCC.
Moreover, (C, ∂C) ≈ (Bn,Sn−1) (Corollary 2.5.6). For every cell (or every simplex)C, we always consider the topology to be inherited from this unique topology of flC.3
For the standardn-simplex ∆n ⊂Rn+1,
2According to Proposition 2.5.8,C itself is equal to the radial closure rclC={x∈E| ∃y∈C such that ∀t∈I, (1 +t)x−ty∈C}.
3In fact, as seen in Proposition 2.5.8, C itself has the unique topology such that the following operation is continuous:
C×C×I∋(x, y, t)→(1−t)x+ty∈C.
3.1 Simplexes and Cells 137 rint∆n =
z∈∆n z(i)>0 for every i= 1, . . . , n+ 1 and
∂∆n =
z∈∆n z(i) = 0 for some i= 1, . . . , n+ 1 .
For an n-simplex σ = v1, . . . , vn+1 ⊂ E, there exists the natural affine homeomorphismδσ: fl∆n→flσdefined by
δσ(z) =
n+1
i=1
z(i)vi for eachz∈fl∆n.
Then,σ=δσ(∆n), rintσ=δσ(rint∆n), and∂σ=δσ(∂∆n). Thebarycenter ˆ
σofσis defined as follows:
ˆ σ=δσ
1
n+ 1, . . . , 1 n+ 1
=
n+1
i=1
1 n+ 1·vi.
Whenv1, . . . , vn+1 are not affinely independent, the map defined asδσ is not a homeomorphism, but we do have the following result:
3.1.1 Proposition For every finite subsetA={v1, . . . , vn} ⊂E, rintA= ni=1z(i)vi
z∈rint∆n−1 . Proof. Take z0 ∈ rint∆n−1 and let x0 =n
i=1z0(i)vi ∈ A. For each x∈ rintA, we have x1 ∈ A and 0 < δ < 1 such that x = (1−δ)x0+δx1. Writex1=n
i=1z1(i)vi, z1∈∆n−1. Then,z = (1−δ)z0+δz1∈rint∆n−1 (Proposition 2.2.3) and
x= (1−δ)
n
i=1
z0(i)vi+δ
n
i=1
z1(i)vi =
n
i=1
z(i)vi.
Conversely, for eachz ∈rint∆n−1, we show n
i=1z(i)vi ∈rintA. Each y∈ Acan be written asy=n
i=1z0(i)vifor somez0∈∆n−1. On the other hand, we have z1∈ ∆n−1 and 0< δ < 1 such thatz= (1−δ)z0+δz1. Let y1=n
i=1z1(i)vi∈ A. Then, it follows that
n
i=1
z(i)vi = (1−δ)
n
i=1
z0(i)vi+δ
n
i=1
z1(i)vi= (1−δ)y+δy1. This means thatn
i=1z(i)vi∈rintA. ⊓⊔
3.1.2 Proposition Each cell C ⊂ E has the smallest finite set C(0) such that C(0)=C (i.e., A=C impliesC(0) ⊂A). In addition,C(0) ⊂∂C if dimC >0.
Proof. By the definition of a cell, we can easily find a minimal finite setC(0) such that C(0) = C, i.e., B = C if B C(0). We have to show that A=C implies C(0) ⊂A. Assume that C(0) ⊂A. Let C(0) ={v1, . . . , vn}, wherev1∈Aandvi=vj ifi=j. We can write
v1=
m
i=1
z(i)xi, x1, . . . , xm∈A, z∈rint∆m−1, wherexi=xj ifi=j. Since v1∈A, it follows thatm2. Define
v′ = (z(1) +ε)x1+ (z(2)−ε)x2+
m
i=3
z(i)xi,
v′′= (z(1)−ε)x1+ (z(2) +ε)x2+
m
i=3
z(i)xi,
whereε >0 is chosen so thatz(1)±ε, z(2)±ε∈(0,1). Then,v′ =v′′because v′−v′′= 2ε(x1−x2)= 0. Sincev′, v′′∈C, we can write
v′ =
n
i=1
z′(i)vi, v′′=
n
i=1
z′′(i)vi, z′, z′′∈∆n−1, and therefore
v1=12v′+12v′′=
n
i=1
1
2z′(i) +12z′′(i) vi.
Recall thatv1∈ v2, . . . , vn. Then, it follows that 12z′(1) +12z′′(1) = 1. Since z′(1), z′′(1)∈I, we havez′(1) =z′′(1) = 1. Hence,v′ =v′′=v1, which is a contradiction. Thus,C(0) is the smallest finite set such thatC(0)=C.
The additional assertion easily follows from Proposition 3.1.1 and the min-imality ofC(0). ⊓⊔
In Proposition 3.1.2, each point ofC(0) is called avertex of C; namely, C(0)is the set of vertices ofC. Note that ifσ=v1, . . . , vn+1is ann-simplex thenσ(0)={v1, . . . , vn+1}. Thus, we have the following:
3.1.3 Corollary A cell C ⊂ E is a simplex if and only if C(0) is affinely independent. ⊓⊔
It is said that two simplexesσandτ arejoinable(orσisjoinabletoτ) ifσ∩τ =∅ andσ(0)∪τ(0) is affinely independent. In this case,σ(0)∪τ(0) is a simplex of (dimσ+ dimτ+ 1)-dimension, which is denoted by στ and called thejoinofσandτ. Whenσ={v},{v}τ is simply denoted byvτ.
The face of a cellCat x∈C is defined as in§2.2, that is, Cx=
y∈C∃δ >0 such that (1 +δ)x−δy∈C .
3.1 Simplexes and Cells 139 Recall rintC={x∈C|Cx=C}, hence∂C={x∈C|Cx=C}. Moreover, x∈rintCx (2.2.5(8)) and Cx =Cy for every y ∈rintCx (2.2.5(10)). Recall that an extreme point ofC is a pointx∈C such thatCx={x}.
3.1.4 Proposition For each cellC⊂E and x∈C, the following hold:
(1) Cx is a cell withCx(0) =C(0)∩Cx;
(2) xis a vertex ofCif and only if it is an extreme point ofC, i.e.,x∈C(0)⇔ Cx={x}.
Proof. (1): To see thatCxis a cell, it suffices to show thatCx=C(0)∩Cx. Since Cx is convex (Proposition 2.2.5(7)), we have C(0)∩Cx ⊂ Cx. Each y∈Cxcan be written as
y=
n
i=1
z(i)vi, v1, . . . , vn ∈C(0), z∈rint∆n−1. Chooseδ∈(0,1) so that (1 +δ)x−δy∈C. For eachi= 1, . . . , n, let
xi= (1−δ+δz(i))x+
j=i
δz(j)vj ∈C.
Then, it follows that 1 +12δz(i)
x−12δz(i)vi =12
(1 +δ)x−δy
+12xi ∈C, which means thatvi∈Cx. Hence,y∈ C(0)∩Cx.
SinceCx=C(0)∩Cx, it follows thatCx(0)⊂C(0)∩Cx and Cx(0)∪(C(0)\Cx)=Cx∪(C(0)\Cx)
="
(C(0)∩Cx)∪(C(0)\Cx)#
=C(0)=C.
The latter impliesC(0)∩Cx⊂Cx(0). Hence,C(0)∩Cx=Cx(0).
(2): IfCx={x}thenx∈Cx(0) ⊂C(0) by (1). Conversely, ifx∈C(0) then x∈C(0)∩Cx=Cx(0)by (1). Sincex∈rintCx(Proposition 2.2.5(8)), we have dimCx= 0 by Proposition 3.1.2, which impliesCx={x}. ⊓⊔
A cellDis call afaceof a cellC(denoted byDCorCD) ifD=Cx
for somex∈C. IfDCandD=C,Dis called aproper faceofC(denoted by D < C orC > D). An n-dimensional face is called ann-face. A face of a simplex σis also a simplex (cf. Proposition 3.1.4(1)) andv1, . . . , vkσ forv1, . . . , vk ∈σ(0). Note thatσis the joinσ0σ1 of any two disjoint facesσ0
andσ1 withσ(0) =σ0(0)∪σ1(0), whereσi is called theopposite faceof σto σ1−i (i = 0,1). Moreover, it follows that τ0τ1 σ =σ0σ1 for each τi σi
(i= 0,1).
3.1.5 Proposition For simplexesσandτ, τ σ⇔τ(0)⊂σ(0).
Proof. The implication ⇒is a direct consequence of Proposition 3.1.4(1). If τ(0) ⊂σ(0) thenτ ⊂σ. Take anyx∈rintτ. Then, τ ⊂σx by the definition ofσx. For eachy∈σ\τ, we have distinctv1, . . . , vn∈σ(0) andz∈rint∆n−1 such that y = n
i=1z(i)vi and v1 ∈ τ(0). Then, τ ∩rinty, y′ = ∅ for any y′∈σ, hencey∈σx. Thus, we haveτ=σx. ⊓⊔
3.1.6 Proposition For each cellC⊂E, the following hold:
(1) ∅ =A⊂C(0)⇒ A(0)=A;
(2) IfA⊂C(0) is not a singleton then C(0)∩rintA=∅; (3) DC⇒D=C∩flD andD(0)=C(0)∩D=C(0)∩flD;
(4) DC⇒Dx=Cxfor eachx∈D, henceD′C for eachD′D;
(5) DC⇒D=Cxfor eachx∈rintD;
(6) D, D′C, D∩rintD′=∅ ⇒D′ D;
(7) D < C⇒dimD <dimC;
(8) ∂C=
{D|D < C}=
{D|D < C, dimD= dimC−1}.
Proof. By virtue of Propositions 2.2.5(7) and 3.1.4(1), we have (3). For (4) and (5), we refer to Propositions 2.2.5(9) and 2.2.5(10), respectively. It is easy to obtain (6) from (4) and (5). For (7), it follows from (3) thatD < Cimplies flD flC, so dimD < dimC. Statements (1), (2), and (8) remain to be proved.
(1): First, note thatA(0) ⊂A⊂C(0). LetB= (C(0)\A)∪A(0)⊂C(0). Since A⊂ A ⊂ B, we have C(0) ⊂ B ⊂C, hence B=C. Therefore, B=C(0) by Proposition 3.1.2. This meansA(0) =A.
(2): Assume that A ⊂ C(0) contains at least two vertices and rintA contains somev ∈C(0). SinceA\ {v} =∅, it follows from Proposition 3.1.1 that v∈ A\ {v}. This impliesC(0)=C(0)\ {v}, which contradicts the definition ofC(0). Thus, C(0)∩rintA=∅.
(8): Since∂C={x∈C|Cx=C}, we have the first equality. To prove the second equality, it suffices to show that eachx∈∂Cis contained in an (n− 1)-face ofC, wheren= dimC. LetDbe a maximal proper face ofC containing x. Then, dimDn−1 by (7). Assume dimD < n−1. Since rintDmisses any other proper face ofC by (6), we have∂C\rintD=
{D′ |D=D′ < C}, which is a compact set in the flat flC given the unique topology (Proposition 2.5.1). Take x0 ∈ rintD. Since flC is affinely homeomorphic to Rn, x0 has a convex neighborhood V in flC such that V ∩(∂C \rintD) = ∅. Since x0∈rintD⊂∂C, we can findx1∈C\Dsuch that (1 +t)x0−tx1∈C∪flD for everyt >0. Choosings, t >0 sufficiently small, we have
y= (1−s)x0+sx1∈V ∩rintC, z= (1 +t)x0−tx1∈V \(C∪flD).
3.1 Simplexes and Cells 141 Consequently,x0∈rinty, z. Note that
dim fl(D∪ {y})n−1< n= dim flC.
Hence, there exists a v ∈ (V ∩C)\fl(D∪ {y}). Since z ∈ fl(D∪ {y}) and v ∈ fl(D∪ {y}), it follows thatv, z ∩flD =∅, so v, z ∩D = ∅. On the other hand, v, z ∩∂C =∅ because v ∈C andz ∈C. Sincev, z ⊂V and V ∩∂C⊂rintD, we havev, z ∩D=∅, which is a contradiction. Therefore, dimD=n−1. ⊓⊔
3.1.7 Proposition For eachn-cellC ⊂E and eachk-faceD < C (k < n), there exist facesD =Dk < Dk+1<· · ·< Dn =C such thatdimDi =ifor kin.
Proof. The case n−k = 1 is obvious. When n−k >1, let x∈ rintD. By Proposition 3.1.6(8), we have (n−1)-faceC′ < C such that x∈ C′. Then, Cx′ =Cx =D by Propositions 3.1.6(4) and (5). Hence, D < C′. The result can be obtained by induction. ⊓⊔
Using affine functionals, we characterize cells as follows:
3.1.8 Proposition Let∅ =C ⊂E be non-degenerate. In order forC to be a cell, it is necessary and sufficient thatdim flC <∞,x+R+(y−x)⊂Cfor each pair of distinct pointsx, y∈C, and there are finitely many non-constant affine functionalsf1, . . . , fk: flC→Rsuch thatC=k
i=1fi−1(R+).
Proof. (Necessity) Let n= dimC. By virtue of Proposition 3.1.6(8), we can write ∂C = k
i=1Di, where each Di is an (n−1)-face of C. Because flDi
is a hyperplane in flC, there is an affine functional fi : flC →R such that flDi=fi−1(0) (Proposition 2.1.3(1)). Then,C⊂fi−1(R+) orC⊂fi−1(−R+).
Replacing fi with −fi if C ⊂fi−1(−R+), we may assume that C ⊂ fi(R+) for every i= 1, . . . , k. Thus, we haveC⊂k
i=1fi−1(R+). Suppose that there is az∈k
i=1fi−1(R+)\C. By taking y∈rintC, we havex∈rinty, z ∩∂C.
Then, x is contained in some Di. Since fi(x) = 0 and fi(y) >0, it follows that fi(z)<0, which is a contradiction. Therefore,C=k
i=1fi−1(R+).
(Sufficiency) First, note that C =k
i=1fi−1(R+) is convex and rintC = coreflCC =k
i=1fi−1((0,∞)), hence ∂C = C\rintC =k
i=1(C∩fi−1(0)).
Moreover, C∩fi−10 (0) = ∅ implies C =
i=i0fi−1(R+), that is, fi0(x) 0 for everyx ∈
i=i0fi−1(R+). Indeed, assume that fi0(x) <0 for some x∈
i=i0fi−1(R+). Take any pointy∈C. Becausefi0(y)0, we havez∈ x, y such that fi0(z) = 0. Then, z ∈ k
i=1fi−1(R+) = C, so C∩fi−10 (0) = ∅, which is a contradiction. Thus, we may assume thatC∩fi−1(0)=∅for every i= 1, . . . , k.
Now, by induction onn= dim flC, we shall show thatCis a cell. For each i= 1, . . . , k, letDi =C∩fi−1(0)=∅. Then, as observed above,∂C=k
i=1Di.
Since flDi⊂fi−1(0) and dimfi−1(0) =n−1 (Proposition 2.1.3(2)), eachDi=
j=i(fj|flDi)−1(R+) is a cell by the inductive assumption. Thus, we have a finite set A = k
i=1D(0)i ⊂ ∂C. Consequently, ∂C = k
i=1Di ⊂ A ⊂ C.
Take any pointv ∈A⊂∂C. For each x∈rintC, v+R+(x−v)⊂C, hence there is a y ∈ ∂C such that x ∈ v, y. Then,x ∈ A because v, y ∈ A. Therefore,C=Ais a cell. ⊓⊔
Later, we will use the following results, which are easily proved.
3.1.9 Additional Results for Cells
(1) For each cellC⊂Eand each flatF⊂EwithC∩F =∅, the intersection C∩F is also a cell.
(2) For every two cellsC, D⊂EwithC∩D=∅, the intersectionC∩Dis also a cell with (C∩D)x=Cx∩Dxfor eachx∈C∩D. If rintC∩rintD=∅, then rint(C∩D) = rintC∩rintD.
(3) Let f :C → E′ be an affine map from a cell C ⊂E into another linear spaceE′. Then,f−1(D) is a cell for every cellD⊂E′ withD∩f(C)=∅, where f−1(D)x = Cx ∩f−1(Df(x)) for each x ∈ f−1(D). When f is injective,f(Cx) =f(C)f(x) for eachx∈C.
Sketch of Proof.For the above three items, apply the characterization 3.1.8 (cf. Proposition 2.2.2 for (3)). The statements about faces in (2) and (3) are the same as 2.2.7(1) and (4), respectively. The statement about the radial interior in (2) is 2.2.7(2).
(4) For every two cells C, D ⊂ E, C×D is also a cell with rintC×D = rintC×rintD and (C×D)(x,y)=Cx×Dy for each (x, y)∈C×D.
Sketch of Proof.Note thatC×D=C(0)×D(0)and see 2.2.7(3).