HOMEWORK 5
Exercise 1. Prove that on Sn
there is a nonzero (continuous) vector field if and only
if n is odd.
Exercise 2. Using CW-structure of the Klein bottle and the projective plane compute
their homology groups.
Exercise 3. Let X = Dn+1
∪f Sn
, where f : ∂Dn+1
= Sn
→ Sn
has degree k.
Compute homology groups of X and also the homomorphism
p∗ : Hi(X) → Hi(X/Sn
)
induced by the projection p : X → X/Sn
.
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