HOMEWORK 8
(1) Prove that the cap product in the following case is defined correctly:
∩ : Hn(X, A; R) ⊗ Hk
(X; R) → Hn−k(X, A; R)
(2) Prove that H∗
(S1
∨ S1
∨ S2
) is not isomorphic to H∗
(S1
× S1
).
(3) Show that for a ∈ Hk
(X; R), b ∈ Hl
(X; R), c ∈ Hn(X; R), we have
(a ∩ c) ∩ b = a ∩ (c ∪ d).
Deduce that H∗(X; R) is a right H∗
(X; R) module.
(4) Prove that closed orientable manifolds of odd dimension have Euler characteristics
zero.
Date: April 12, 2013.
1