Math 527/427
Homework 2
9 October 2015
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1. Prove the following fact left unproved in class: if P : C •1 → C •+1
is a chain homotopy between
f : C •1 → C •2
and
g : C •1 → C •2 ,
then f and g induce the same map on homology.
2. Let C • denote the chain complex
0 → C 2 = Z/2Z → C 1 = Z/4Z → C 0 = Z/2Z → 0
where C 1 → C 0 is the surjective reduction map and C 2 → C 1 is the inclusion given by 1 7→ 2.
Calculate the homology of C • . Prove that there is no chain homotopy between the 0-map C • → C •
and the identity id : C • → C • .
3. Suppose
0
0
f
/A
/D
g
/B
s
/F
i
t
j
/C
/0
u
/G
/0
(without the broken arrow C → D) is a commutative diagram of abelian groups, and the rows are
exact. In class, it was asserted that the dashed arrow u : C → D exists making the diagram commutative. Show that this homomorphism u : C → D exists and is unique with the property that the
diagram commutes.
4. Prove the Snake Lemma. [This question will not be graded, so it is not necessary to submit a written
answer.]
5. Do the following problems from Hatcher: 2.1: 14, 18, 22, 23.
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