Elementary Geometric Topology
5.3 Regular CW complexes
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.)
136 5 Elementary Geometric Topology
We derive some special properties of regular CW complexes (5.3.2 and 5.3.5).
Proposition 5.3.2.Letebe a cell of the regular CW complexX, and letC(e) be the carrier ofe. Then, as spaces,C(e) =e. In other words, each cell ofX is a subcomplex ofX.
For the proof of 5.3.2 we need two lemmas.
Lemma 5.3.3.There is no embedding of Sn inRn.
Proof. If there were such an embedding h : Sn → Rn then, by 5.1.1 and 1.3.7, hwould map every proper open subset of Sn onto an open subset of Rn, henceh(Sn) would be open inRn. But h(Sn) is compact, hence closed inRn. Andh(Sn)=Rn since the latter is not compact. Thus we would have a proper non-empty closed-and-open subset ofRn contradicting the fact that Rn is connected (being obviously path connected).
Lemma 5.3.4.Let enα−1 and enβ be cells of the regular CW complex X. If
◦eα∩•eβ=∅ theneα⊂e•β.
Proof. We have•enβ ⊂Xn−1, and◦enα−1is open inXn−1, soe◦nα−1∩e•nβis open in •enβ. Now •enβ is homeomorphic to Sn−1, so, by 5.1.1, e◦nα−1 ∩•enβ is open in ◦enα−1. But, since •enβ is compact,e◦nα−1∩•enβ is also closed ine◦nα−1, and is non-empty. As in the proof of 5.3.3, this implies ◦enα−1⊂•enβ, from which the
Lemma follows.
Proof (of 5.3.2). We work by induction on the dimension,n, of e. Ifn= 0, the Proposition is trivial. Consider a cell enα−1 in C(e). Then ◦enα−1∩•e=∅, so, by 5.3.4,enα−1⊂e. Since• C(e) consists of etogether with the carriers of suchenα−1, the induction hypothesis completes the proof.
It follows that if hα : (Bn, Sn−1) →(enα,•enα) is a characteristic map for ann-cell in a regular CW complexX,h−α1carries the CW complex structure C(enα) onenαback to a CW complex structure onBn; this structure has exactly onen-cell, andh−α1(C(e•nα)) is a CW complex structure onSn−1. This implies that there is an (n−1)-cellenβ−1 ofX withenβ−1⊂e•nα. In fact, by induction one deduces that for eachn-cellenαof the regular CW complexX and for each k < nthere is ak-cellek withek⊂enα.
Recall that a celle1is aface of a celle2ife1⊂e2. We have just seen that in a regular CW complex an n-cell has faces of all lower dimensions. Easy examples show that non-regular CW complexes need not have this property.
Proposition 5.3.5.Let X be a regular CW complex, and let ekα−2 be a face ofekβ. Then ekα−2 is a face of exactly two (k−1)-dimensional faces of ekβ.
Proof. Since •ekβ is homeomorphic to Sk−1, this follows from the following
lemma.
Lemma 5.3.6.Let Y be a regular CW complex structure on an n-manifold.
Every cell of Y is a face of ann-cell of Y. Every(n−1)-cell of Y is a face of at most two n-cells ofY. An (n−1)-cell, e, of Y is a face of exactly one n-cell ofY iffe⊂∂Y. If eis a face of twon-cells of Y, thene◦⊂Y◦.
Proof. We saw in Sect. 5.1 that every cell ofY has dimension ≤n. If some cell were not a face of an n-cell, there would be k < n and ak-cell ˜e of Y which is not a face of any higher-dimensional cell of Y, implying e◦˜open in Y, contradicting 5.1.6(a). If the (n−1)-celleis a face of exactly one n-cell, then, sinceY is regular, eachx∈e◦has a neighborhood in Y homeomorphic to Rn+. Thus e◦ ⊂ ∂Y, and, since ∂Y is closed in Y, e ⊂ ∂Y. On the other hand, ifeis a face of twon-cells, then everyx∈◦eclearly has a neighborhood homeomorphic toRn, so, by 5.1.6(b),e◦⊂Y◦. The proof thateis not a face of
more than twon-cells is left as an exercise.
The abstract first derived (or barycentric subdivision) of a regular CW complexX is the abstract simplicial complex sdXwhose vertices are the cells ofX and whose simplexes are those finite sets of cells{e0,· · ·, en}which can be ordered so that, for eachi < n,ei is a proper face ofei+1 (i.e.,ei⊂•ei+1).
Convention 5.3.7.We will always list the vertices of a simplex of sd X in order of increasing dimension (of the cells).
Proposition 5.3.8.When X is a regular CW complex, there is a homeo- morphism h:|sdX| →X such that for every simplex {e0,· · · , ek} of sd X, h(|{e0,· · · , ek}|◦)⊂◦ek. Moreover,h−1 is cellular.
Proof. Observe that for any cell e of X, sd C(e) is the cone e∗[sdC(•e)]
whereC(·) denotes “carrier”; note the different meanings ofe! By induction on n we define homeomorphisms hn : |sdXn| → Xn, each extending its predecessor: the requiredh agrees with hn on |sdXn| ⊂ |sdX|. In an ob- vious sense |sd X0| = X0; let h0 = id. Assume that h0 has been extended to hn−1 : |sd Xn−1| → Xn−1. Let e be an n-cell of X. Then hn−1 maps
|sdC(•e)| homeomorphically onto C(•e). By 5.3.2, this is a homeomorphism between (n−1)-spheres. By the above remark about cones, it extends to a homeomorphism|sdC(e)| →C(e) betweenn-balls. Ourhnis defined to agree
with this homeomorphism on|sdC(e)|.
Corollary 5.3.9.Every regular CW complex is triangulable.11
11In Exercise 11 of Sect. 5.2 the reader was asked for a CW complex (non-regular in view of 5.3.9) which is not triangulable.
138 5 Elementary Geometric Topology
The homeomorphismhof 5.3.8 defines a subdivision ofX in the sense of Sect. 2.4 whose cells are {h(|{e0,· · ·, ek}|)}. Call such a subdivision a first derived (or barycentric subdivision) ofX. Onceh is chosen, it is convenient to denote by ˆek the point ofe◦k which is the image underhof the vertexek of
|sdX|and to call ˆekabarycenterofek. In fact, one often identifies|sdX|with X via h, thinking of the barycenters ˆek as being the vertices of |sdX|; the cell (simplex)|{e0,· · ·, ek}|of|sdX|is then identified with a certain subset ofek. See Fig. 5.3.
|sdX| X
h
Fig. 5.3.
We compute incidence numbers in regular CW complexes.
Proposition 5.3.10.Let X be an oriented regular CW complex. For cells enβ−1andenα ofX, the incidence number[enα:enβ−1] is±1 ifenβ−1 is a face of enα, and is 0 otherwise.
Proof. Whenn= 1, this is true by definition; see Sect. 2.5. Let n >1. When enβ−1not a face ofenα, the incidence number is 0 by 2.5.8. LetY be the subcom- plexC(•enα) ofX, and letZ be the CW complex structure onSn−1consisting of one 0-cell ˜e0and one (n−1)-cell ˜en−1. ThenY is homeomorphic toSn−1by 5.3.2. There is a cellular mapf :Y →Z takinge◦nβ−1 homeomorphically onto
◦˜
en−1 and taking the rest ofY to ˜e0. Thenf#(enβ−1) = [enβ−1: ˜en−1:f]˜en−1, and f# takes all other generators of Cn−1(Y;Z) to 0 ∈ Cn−1(Z;Z). In- spection of the diagram in Sect. 2.5 defining mapping degrees reveals that [enβ−1: ˜en−1:f] is the degree of a homeomorphismSn−1→Sn−1and is there- fore±1, by 2.4.20. Since Hn−1(Y;Z)∼=Z, there is a cellular (n−1)-cycle in Y which generatesHn−1(Y;Z) and whose image generatesHn−1(Z;Z)∼=Z. Thus, if we use homeomorphisms to identifyY andZ with Sn−1,f has de- gree±1. Now the diagram in Sect. 2.5 defining incidence numbers shows that
[enα:enβ−1] is this degree, namely±1.
Exercises
1. Find regular CW complex structures which are not triangulations (see 5.3.9) for:
Sn and closed surfaces.
2. Show that the property of being a regular CW complex is preserved under the following constructions: disjoint union, finite product, subcomplex, and covering space.
3. Give an example where the universal cover of a non-regular CW complex is regular.
4. Prove that in ann-manifold no (n−1)-cell is a face of more than twon-cells.
5. LetXbe a presentation complex for the presentationW |R, ρofG=π1(X, v).
Prove that the universal cover ˜X is a regular CW complex iff no element ofW represents 1∈ G, and no proper subword of a relation inρ(R) is conjugate to (i.e., cyclically equivalent to) a relation in ρ(R). Prove that if ˜X is not regular, then there is a sequence of Tietze transformations leading to another presentation W|R, ρofπ1(X, v) so that ifX is the associated presentation complex ˜X is regular. Prove that ifW andRare finite, then this sequence is finite, and each presentation in the sequence is finite.
6. IfY is a regularG-CW complex show that|sdY|is a rigidG-CW complex.
7. Show that ifY is a subcomplex ofX then sdY is a full subcomplex of sdX.