Elementary Geometric Topology
5.2 Simplicial complexes and combinatorial manifolds
8. Show thatTg is a 2 to 1 covering space ofUg+1.
9. By considering the universal cover of K(2) (see Example 5.1.8) show that the universal cover ofT2 is homeomorphic toR2. Deduce that the universal covers of all the path connected closed surfaces exceptS2 andRP2 are homeomorphic toR2. (In the terminology of Ch. 7, all such surfaces are aspherical.)
10. LetGact freely and cocompactly on a path connected orientable open surfaceS.
Prove thatH1(S;Z) is finitely generated as aZG-module ifGis finitely presented, and it is not finitely generated as aZG-module ifGis finitely generated but does not have typeF P2 overZ.
130 5 Elementary Geometric Topology
inherited fromW, is a CW complex. The details are an exercise; see also [51, pp. 171–172]. This|K|is thegeometric realization ofK.
Ifφ:K→Lis a simplicial map, there is an associated map|φ|:|K| → |L| which maps the vertex of|K| corresponding to v ∈ VK to the vertex of|L| corresponding toφ(v)∈VL, and is affine on each|σ|. Clearly|φ|is continuous, and ifφis a simplicial isomorphism,|φ|is a homeomorphism.
The notations|σ| and|K| are sometimes used in a slightly different way.
Assume (i)VK is a subset ofRN such that the vertices of each simplex ofK are affinely independent. Withσas above, define|σ|to be the (closed) convex hull ofσinRN. Write|σ◦|for
)n i=0
tivi|0< ti<1 and n i=0
ti= 1
*
, theopen convex hull ofσ. Assume (ii) that wheneverσ=τ∈SK,|σ◦| ∩ |τ◦|=∅. Define
|K|=
{|σ| |σ∈SK}and|K|n=
{|σ| |σis ak-simplex ofKandk≤n}. Then |K|(with topology inherited from RN) and{|σ◦| | σ∈ SK} satisfy all but one of the requirements in Proposition 1.2.14 for (|K|,{|K|n}) to be a CW complex: the sole problem is that the topology which|K| inherits from RN might not agree with the weak topology with respect to {|σ| |σ ∈SK}. Assume (iii) that{|σ| |σ∈SK}is a locally finite family of subsets of the space
|K|(where |K|has the inherited topology). If K satisfies these assumptions (i), (ii) and (iii), we callK a simplicial complex in RN, and we call |K| its underlying polyhedron. The cell |σ| of |K| is often called a simplex of |K|; context prevents this double use of the word from causing problems. Note that|K| might not be closed inRN; it is closed iff the set of simplexes|σ| is a locally finite family of subsets ofRN (rather than of|K|).
A spaceZ istriangulableif there is an abstract simplicial complexKsuch thatZ is homeomorphic to|K|, andK is called8a triangulation ofZ.
Example 5.2.1.The half-open interval (0,1] in R is triangulable: take VK = )1
n|n∈N
*
and SK to be the set of pairs )1
n, 1 n+ 1
*
together with VK. But the subspaceVK∪ {0}ofRis not triangulable.
Proposition 5.2.2.IfK is a simplicial complex inRN, the weak topology on
|K|with respect to{|σ| |σ∈SK}agrees with the topology inherited fromRN. Moreover,(|K|,{|K|n})is a countable locally finite CW complex.
Proof. |K|weak is a CW complex. Suppose it is not locally finite. Then for some simplex σ of K, there is an infinite collection {τα} of simplexes of K with|σ|∩|τα| =∅for allα. Since|σ|is compact, (iii), above, implies that there are open sets (in RN) U1,· · ·, Uk whose union contains|σ| and meets only finitely many of{|τα|}. This is a contradiction. So|K|weakis locally compact and Hausdorff. The “identity” map|K|weak→ |K|inheritedis continuous, and
8 The word “triangulation” is also used for a homeomorphism h : |K| → Z, or sometimes just for the CW complex|K|.
|K|inherited is clearly locally compact and Hausdorff. By 1.2.11, |K|weak is Hausdorff. By 10.1.8, below,|K|weakis locally compact, and by 10.1.6, below, this implies that the “identity” is a homeomorphism. Thus the two topologies
agree.
Remark 5.2.3.Conveniently, |σ◦| as defined above coincides withe◦ as defined in Sect. 1.2 whene=|σ|is a cell of the CW complex|K|.
LetK and Lbe abstract simplicial complexes: Lis asubcomplex of K if VL ⊂VK andSL ⊂SK. Then|L|is a subcomplex of|K|in the sense of Sect.
1.2. We say thatLis afull subcomplex ofKif, in addition, any set of vertices ofLwhich is a simplex ofK is a simplex ofL. Then|L|is a full subcomplex of|K|in the sense of Sect. 1.5.
If L is a subcomplex ofK, the simplicial neighborhood of L in K is the subcomplex N(L), orNK(L), generated by all simplexes of K which have a vertex inL. Then|N(L)|=N(|L|) in the sense of Sect. 11.4.
If K and L are abstract simplicial complexes, their join, K∗L, is the abstract simplicial complex with9 VK∗L =VKVL andSK∗L =SKSL {στ | σ∈ SK,τ ∈ SL}. If K [resp. L] = ∅, it is implied thatK∗L =L [resp.K].
IfX andY are non-empty spaces, their topological join is the adjunction spaceX∗Y = (XY)∪f(X×I×Y) wheref :X×{0,1}×Y →XY is defined by f(x,0, y) = x and f(x,1, y) = y. To see what this means geometrically, consider an equivalent definition:X∗Y is the quotient space (X×I×Y)/∼ where the equivalence relation∼is generated by: (x,0, y1)∼(x,0, y2) for all y1, y2∈Y; and (x1,1, y)∼(x2,1, y) for allx1, x2∈X. ThusX∗Y contains a “line segment” joining eachx∈X to eachy∈Y; two such “line segments”
meet, if at all, at one end point; these “line segments” vary continuously inx and iny.
Example 5.2.4.Let X = {((1−a), a,0,0) ∈ R4 | 0 ≤ a ≤ 1} and let Y = {(0,0,(1−b), b)∈R4|0≤b≤1}. ThenX∗Y can be identified with {((1−t)(1−a),(1−t)a, t(1−b), tb)∈ R4 |0 ≤a, b, t≤1}. Fixing a andb we get a line segment joining ((1−a), a,0,0)∈X to (0,0,(1−b), b)∈Y, and the various line segments meet as described above. In this example,X∗Y is the standard 3-simplex inR4.
IfA⊂X andB ⊂Y are non-empty closed sets, we considerA∗B to be a subspace ofX∗Y, namely the image ofA×I×B inX∗Y. The quotient and inherited topologies agree for reasons explained in Sect. 1.1.
IfX is the empty space,X∗Y is defined to beY. Similarly, ifY is empty, X ∗Y is X. If, in the last paragraph, A [resp. B] is empty, this suggests identifying the resultingA∗Bwith the obvious copy ofB[resp.A] inX∗Y, and this will always be understood.
9 IfVK∩VL=∅thesymbol could be replaced by∪.
132 5 Elementary Geometric Topology
We seek a relationship between|K| ∗ |L|and|K∗L|. For this, we consider
|K| ⊂W1 and|L| ⊂W2, as above, where W1 and W2 are real vector spaces each having the finite topology. Let W = W1 ×W2, again with the finite topology. Then |K∗L| ⊂ W. Identifying W1 with W1× {0} and W2 with {0} ×W2, we have|K| ⊂ |K∗L| ⊃ |L|. Define a functionφ:|K| ×I× |L| →
|K∗L| by the formulaφ(x, t, y) = (1−t)x+ty where addition and scalar multiplication take place inW.
Proposition 5.2.5.If K or L is locally finite then φ is continuous, and φ induces a homeomorphism Φ:|K| ∗ |L| → |K∗L|.
Proof. RegardIas a CW complex in the usual way. By 1.2.19,|K|×I×|L|is a CW complex, so in order to show thatφis continuous we need only show that for anyσ∈SK andτ∈SL,φ|:|σ| ×I× |τ| →W is continuous. The domain and the image of this restriction lie in finite-dimensional subspaces ofW, so φ|is certainly continuous. One easily checks that φ respects the equivalence relation∼(see the alternative definition ofX∗Y, above) and that the induced map Φ : |K| ∗ |L| → |K∗L| is injective. Moreover, φ(|σ| ×I× |τ|) clearly equals |στ| so Φ is surjective. Moreover, this analysis shows that Φmaps each cell of the CW complex|K| ∗ |L| (see the first definition ofX∗Y, and 4.1.1) homeomorphically onto a cell of|K∗L|. ThusΦ−1is continuous.
WhenK has just one vertexv, the joinK∗Lis called thecone onLwith vertex v andbase L; it is denotedv∗L. Similarly, ifX is the one-point space {v}, the topological joinX∗Y is called thecone onY withvertex vandbase Y; one writesv∗Y for this. A special case of 5.2.5 is
Corollary 5.2.6.Φ:v∗ |L| → |v∗L|is a homeomorphism.
Here are some basic exercises about topological joins of balls and spheres:
Proposition 5.2.7.There are homeomorphisms of pairs as follows:
(Bm∗Bn,(Sm−1∗Bn)∪(Bm∗Sn−1))∼= (Bm+n+1, Sm+n) (Bm∗Sn, Sm−1∗Sn)∼= (Bm+n+1, Sm+n).
If p∈B◦ m, thenp∈∂(Bm∗Bn) andp∈(Bm∗Sn)◦. If σ, τ ∈SK andσ ⊂τ, we say that σis a face ofτ. Consider the set of all simplexes ofK of which σ is a face: the subcomplex generated by these simplexes is called the star of σin Kand is denoted by stKσ. Thelink ofσ inK is the subcomplex, lkKσ, of stKσgenerated by those simplexes of stKσ which contain no vertex ofσ. See Fig. 5.2.
This section, so far, has consisted of two parts, one on simplicial complexes and the other on joins. These are brought together in the next proposition. If σis a simplex ofK, ¯σdenotes the abstract simplicial complex whose vertices are the vertices ofσand whose simplexes are the faces ofσ.
Fig. 5.2.
Proposition 5.2.8.Let σbe a simplex of the abstract simplicial complexK.
Then ¯σ∗lkKσ = stKσ. More precisely, the vertices of σ¯ and of lkKσ form two disjoint subsets ofVK, and the function sending each vertex to itself maps
¯
σ∗lkKσby a simplicial isomorphism onto the subcomplex stKσof K.
The importance of 5.2.8 is that, with 5.2.5, it implies|σ| ∗ |lkKσ|is home- omorphic to|stKσ|. When K is a combinatorialn-manifold (see below) or a CWn-manifold (see Sect. 15.1), each|lkKσ|will be homeomorphic to a sphere or a ball of dimensionn−dimσ−1, so that, by 5.2.7 and 5.2.8,|stKσ| will be ann-ball. To express this better we need some preliminaries.
Let K be a simplicial complex in RN. Asimplicial subdivision ofK is a simplicial complexKinRN such that|K|=|K|and|K|is a subdivision of
|K|(in the sense of Sect. 2.4). LetL be a simplicial complex inRM. A map f : |K| → |L| is piecewise linear (abbreviated to PL) if there is a simplicial subdivisionK ofK such that for each simplexσ ofK, the restriction off to|σ|is affine when regarded as a map intoRM. This property off depends only on the underlying polyhedra|K| and |L|, in the sense that if ¯K and ¯L are simplicial complexes inRN andRM respectively such that|K|=|K¯|and
|L| = |L¯|, then the above map f is also PL when written f : |K¯| → |L¯|. Clearly we have:
Proposition 5.2.9.If f :|K| → |L| is a homeomorphism and if f is also a
PL map, thenf−1 is a PL map.
Piecewise linearity is really a local property, like differentiability. By 5.2.9, the inverse of a PL homeomorphism is a PL homeomorphism.10
Recall that∆n is the (closed) convex hull of the pointsp0,· · ·, pn∈Rn+1 wherepi has (i+ 1)th coordinate 1 and all other coordinates 0. Let nbe the
10Contrast this with the differentiable map f : R → R, f(x) = x3, which is a homeomorphism whose inverse is not differentiable at 0.
134 5 Elementary Geometric Topology
simplicial complex inRn+1whose vertices arep0,· · ·, pnand whose simplexes are the non-empty sets of vertices. Then ∆n = |n|. It is also the case that
∆n =
% n
i=0
tipi|0≤ti≤1 and
i
ti= 1
&
. We denote by ∆•n the subset of ∆n consisting of points for which some ti = 0. Then ∆•n = |n•| for an obvious subcomplexn• of n. It is convenient to also call∆n thestandard PL n-ball and to call ∆• n the standard PL (n−1)-sphere. The pairs (∆n,∆• n) and (Bn, Sn−1) are homeomorphic.
Let K be a simplicial complex in RN. We say K is a combinatorial n- manifoldif for each simplexσofK,|lkKσ|is PL homeomorphic to∆n−dimσ−1 or to∆• n−dimσ: in words, the link of each simplexσis PL homeomorphic to the standard PL ball or PL sphere of dimensionn−dimσ−1. Those simplexes whose links are PL homeomorphic to a standard ball define a subcomplex,
∂K, of K called the combinatorial boundary of K. We say K is a closed combinatorialn-manifold if∂K =∅andK is finite.
We leave it to the reader to formulate and prove a PL version of 5.2.7 (e.g.,
∆• m∗∆n is PL homeomorphic to∆• m+n+1).
Proposition 5.2.10.IfKis a combinatorialn-manifold, then for every sim- plexσ ofK,|stKσ| is PL homeomorphic to∆n. In particular,|K|is a topo-
logical n-manifold, and|∂K|=∂|K|.
Theorem 5.2.11.Let K be a simplicial complex in RN with the property that every point x ∈ |K| has a (closed) neighborhood PL homeomorphic to
∆n. Then K is a combinatorial n-manifold.
IfKis a combinatorial manifold, its geometric realization|K|is apiecewise linear manifold, abbreviated toPL manifold.
Remark 5.2.12.There is an enormous literature behind the last few para- graphs, expounded in part in [90], [136] and [96]. It is difficult to construct a topologicaln-manifold which is not homeomorphic to a piecewise linearn- manifold, though such manifolds exist in all dimensions≥4. Each topological manifold of dimension ≤ 3 is homeomorphic to a piecewise linear manifold which is unique up to PL homeomorphism; see for example [121]. Each closed topological manifold of dimension≥6 admits the structure of a CW complex;
see Essay III of [96].
Exercises
1. If K is a simplicial complex in RN, let |K|1 be its geometric realization and let |K|2 be its underlying polyhedron. Prove that there is a homeomorphism h:|K|1→ |K|2 such that (in the notation of this section)h(v) =vand for each σ∈SK,hmaps|σ|1 onto|σ|2 affinely.
2. Prove 5.2.7, and also a PL version of 5.2.7.
3. Prove that|K|is a CW complex.
4. An abstract simplicial complex K isconnected if for any verticesv,w there is a sequence of vertices v=v0, v1,· · ·, vn =wwhere, for every i,{vi, vi+1}is a simplex ofK. Prove thatKis connected iff|K|is path connected.
5. Prove thatKis a locally finite [resp. finite] abstract simplicial complex iff|K|is a locally finite [resp. finite] CW complex.
6. Prove that everyn-dimensional locally finite abstract simplicial complex is iso- morphic to a simplicial complex inR2n+1.
7. Define thesimplicial boundaryofNK(L) to be the subcomplexN•K(L) ofNK(L) consisting of simplexes which do not have a vertex inL. Give an example where
|N•K(L)| = fr|K|(|NK(L)|).
8. Show thatInis PL homeomorphic to∆n.
9. Show thatRPn,Sn andTn are homeomorphic to PLn-manifolds.
10. Show that the surfaces Tg,d and Uh,d of 5.1.9 are homeomorphic to PL 2- manifolds.
11. Construct a non-triangulable CW complex.
12. Prove 5.2.8.
13. Prove that any conev∗Y is contractible.
14. The Simplicial Approximation Theorem says that ifKandLare simplicial com- plexes in someRN, and if K is finite, then given a mapf :|K| → |L|there is a simplicial subdivisionK of K and a simplicial mapφ:K →Lsuch that f is homotopic to |φ|. This is proved in many books on algebraic topology, e.g., [146]. Using this, prove that every CW complex X has the homotopy type of some |J| where J is an abstract simplicial complex. (Hint: First assume X is finite-dimensional and work by induction on dimension, using 4.1.8.)