Simplicial Structures in Topology

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