HOMEWORK 3 – 2019
Example 1. Compute the simplicial homology groups of the Klein bottle. Use the
folowing model of the Klein bottle as a ∆-complex:
S
T
a a
b
b
V
VV
V
c
Figure 1. Model of the Klein bottle
Example 2. Compute the homology groups of the projective plane. Use the following
model:
S
T
a a
b
b
V
VW
W
c
Figure 2. Model of the real projective plane
Example 3. We have defined the unreduced suspension of a space X as
SX = X × I/ ∼,
where (x1, 0) ∼ (x2, 0), (x1, 1) ∼ (x2, 2), and the reduced suspension as
ΣX = SX/({x0} × I).
Prove that ΣX = (X, x0) ∧ (S1
, s0).
1