Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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64 II <strong>Simplicial</strong> Complexes<br />
1-simplex “<strong>in</strong>ternal” to the square of Fig. II.10, then they must have equal coefficients<br />
hi = h j. This implies that there exists h ∈ Z such that h j = h for each j,<br />
that is to say, ∑ j h jσ j<br />
2 = hz2 where z2 is the 2-cha<strong>in</strong> ∑ j σ j<br />
2 . It is easy to see that<br />
∂z2 = ∑ j ∂σ j<br />
2 = 0andsoB1 ∩ S = 0, imply<strong>in</strong>g that H1(T 2 ;Z) ∼ = S ∼ = Z2 , with free<br />
generators z1 1 and z21 .<br />
F<strong>in</strong>ally, similar arguments show that any 2-cycle of C2 is a multiple of the 2-cha<strong>in</strong><br />
z2 def<strong>in</strong>ed above (given by the sum of all oriented 2-simplexes of T 2 ) and therefore,<br />
H2(T 2 ;Z) ∼ = Z.<br />
Exercises<br />
1. Let U = {Ux|x ∈ X} be a f<strong>in</strong>ite open cover<strong>in</strong>g of a topological space B,andtake<br />
the set<br />
Φ = {σ ⊂ X| �<br />
Ux �= /0}.<br />
Prove that N(U )=(X,Φ) is a simplicial complex. This is the so-called nerve of U .<br />
2. Let K =(X,Φ) be a simplicial complex. For a given x ∈ X, letSt(x) be the<br />
complement <strong>in</strong> |K| of the union of all |σ| such that x �∈ σ, σ ∈ Φ. St(x) is called<br />
star of x <strong>in</strong> |K|. Prove that S = {St(x) | x ∈ X} is an open cover<strong>in</strong>g of |K|, and<br />
N(S )=K.<br />
3. Let X be a compact metric space and let ε be a positive real number. Take the set<br />
Φ of all f<strong>in</strong>ite subsets of X with diameter less than ε. Prove that K =(X,Φ) is a<br />
simplicial complex (<strong>in</strong>f<strong>in</strong>ite).<br />
4. Exhibit a triangulation of the follow<strong>in</strong>g spaces:<br />
a) Cyl<strong>in</strong>der C – recall that the cyl<strong>in</strong>der C is obta<strong>in</strong>ed from a rectangle by identification<br />
of two opposite sides;<br />
b) Möbius band M obta<strong>in</strong>ed from a rectangle by identification of the “<strong>in</strong>verse”<br />
po<strong>in</strong>ts of two opposite sides; more precisely, let S be the rectangle with vertices<br />
(0,0), (0,1), (2,0),and(2,1) of R2 ;then<br />
x∈σ<br />
M = S/{(0,t) ≡ (2,1 −t)} , 0 ≤ t ≤ 1;<br />
c) Kle<strong>in</strong> bottle K obta<strong>in</strong>ed by identify<strong>in</strong>g the “<strong>in</strong>verse” po<strong>in</strong>ts of the boundary of the<br />
cyl<strong>in</strong>der C;<br />
d) real projective plane RP2 obta<strong>in</strong>ed by the identification of the antipodal po<strong>in</strong>ts<br />
of the boundary ∂D2 ∼ = S1 of the unit disk D2 ⊂ R2 ;<br />
e) G2 obta<strong>in</strong>ed by attach<strong>in</strong>g two handles to the sphere S2 ;provethatG2ishomeo morphic to the space obta<strong>in</strong>ed from an octagon with the suitable identifications<br />
of the edges of its border a1b1a −1<br />
−1<br />
a2b2a<br />
1 b−1<br />
1<br />
2 b−1<br />
2 .