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Math 527/427 Homework 2 9 October 2015 2 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. 1