HOMEWORK 2 – 2019
Exercise 1. Given the short exact sequence of Abelian groups
0 // A
f
// B
g
// C // 0
the following conditions are equivalent:
(1) There exists p: B → A such that p ◦ f = idA.
(2) There exists q: C → B such that g ◦ q = idC.
(3) There are p: B → A and q: C → B such that f ◦ q + q ◦ g = idB.
Prove that (3) ⇒ (1) and (2).
Exercise 2. 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 that the definition is independent of the choice of c in the homology class
in Hn(C∗).
(2) Prove the exactness in Hn(C∗).
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