Homework 2
March 1, 2013
RULES: To pass the homework you either need to make 2 exercises and hand me the result by
the next friday or you have to make all three exercises – in that case there is no deadline.
1. Finish the proof of the 5–lemma i.e prove that in the diagram of Abelian groups
// A //
a

B //
b

C //
c

D //
d

E //
e

// A // B // C // D // E //
where alll squares commute, the rows are exact sequences and a, b, d, e are isos, the morphism c
is mono.
2. Let us have the following short exact sequence
0 // A
i // B
j
// C // 0
then the following are equivalent:
(a) There exists p : B → A such that p ◦ i = idA.
(b) There exists q : C → B such that j ◦ q = idC.
(c) There are p, q as above such that (i ◦ p + q ◦ j) = idB.
We have shown that the first two conditions are equivalent. Finish the proof.
3. Decide and prove whether the hawaian earring space
H =
n∈N
{x ∈ R2
| x − ( 1
n , 0) =
1
n
}
is a CW complex
1