HOMEWORK 3
Example 1. Prove
δ ◦ δ = 0
for the boundary operator
δ(σ) =
n
i=0
(−1)j
σ ◦ εj
n
in a singular chain complex C∗(X).
Example 2. Compute the homology groups of the torus. Use the model of the torus
as a ∆-complex from Hatcher, page 102.
Example 3. Compute the homology groups of the projective plane. Use the following
model:
S
T
a a
b
b
V
VW
W
c
Figure 1. Model of Real projective plane
1