HOMEWORK 1
Exercise 1. For the short exact sequence od chain complexes
0 // A∗
f
// B∗
g
// C∗
// 0
there is a long exact sequence of homology groups
. . . // Hn+1(C∗)
∂∗
// Hn(A∗)
f∗
// Hn(B∗)
g∗
// Hn(C∗)
∂∗
// Hn−1(A∗) // . . .
with the connecting homomorphism ∂∗ defined by the prescription
∂∗([c]) = [a], where ∂c = 0, f(a) = ∂b, g(b) = c.
(1) Prove the exactness in Hn(A∗).
(2) Prove the exactness in Hn(B∗).
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