Homework number 3
Do two from the following three exercises.
Exercise 7. Consider a map f : Sn
→ X. Prove that the cone Cf of the map f is
homeomorphic to the space Dn+1
∪f X.
Exercise 8. Prove that Hk(Dn
, Sn−1
) ∼= ¯Hk−1(Sn−1
).
Exercise 9. From the long exact sequence for the triple (∆n
, δ∆n
, δ∆n
− one face)
and the excision theorem derive that
Hk(∆n
, δ∆n
) ∼= Hk−1(∆n−1
, δ∆n−1
).
Show by induction that the singular simplex id : ∆n
→ ∆n
is a cycle in Cn(δn
, δ∆n
)
the homology class of which determines a generator of Hn(∆n
, δ∆n
) ∼= Z.
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