Homework number 1
Do the exercise 1 and one of the exercises 2 and 3.
We have a short exact sequence of chain complexes
0 → A∗
i
−→ B∗
j
−→ C∗ → 0.
In the lecture the boundary homomorphism δ : Hn(C) → Hn−1(A) has been defined.
Exercise 1. Prove that its definition is correct.
Exercise 2. Prove that the long exact sequence of homology groups
· · · → Hn(A)
i∗
−→ Hn(B)
j∗
−→ Hn(C)
δ
−→ Hn−1(A) → . . .
is exact in the term Hn(B).
Exercise 3. Prove exactness in the term Hn(C).
1