HOMEWORK 4
Exercise 1. Derive the long exact sequence for the triple A ⊆ B ⊆ X.
Exercise 2. Using the long exact sequence of a triple, prove that
H1([−1, 1], {−1, 1}) ∼= H0({−1, 1}, {−1}).
Then show
H0({−1, 1}, {−1}) ∼= Z
and find a cycle c ∈ C0(({−1, 1}, {−1}) which represents a generator of H0({−1, 1},
{−1}). Using this show that the singular simplex
id : ∆1
→ ∆1
represents a generator of H1(∆1
, ∂∆1
) ∼= Z.
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