HOMEWORK 4
(1) Compute the homology groups of S1
∨ S1
.
(2) Using the Mayer–Vietoris exact sequence, compute the homology groups of the
n–hole torus (we cut horizontally)
(3) Prove the exactness of the long exact sequence for the triple A ⊂ B ⊂ X in
the map ∂∗ given the two long exact sequences
· · · // Hn(B) // Hn(B, A) // Hn−1(A) // Hn−1(B)
∼=

// Hn−1(B, A) // · · ·
· · · // Hn(B) // Hn(X) // Hn(X, B)
∂∗
// Hn−1(B) // Hn−1(X) // · · ·
where ∂∗ : Hn(X, B) → Hn−1(B, A) (right, up, right)
(4) Let A ⊂ X Compute the homology groups of CX/A i.e. make a cone and then
collapse part of the base (Hint: make cone CA in the other direction).
Date: March 15, 2013.
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