Hatcher Exercise 2.1.31


We give an example of the situation in the five-lemma where the middle isomorphism is non-zero but all others are zero. Consider the following commutative diagram:

Certainly, the rows are exact (because is exact and since the endpoints are ’s, we can augment with more ’s). It’s also worth ensuring that the diagram is actually commutative. Only the second and third squares are in question. In the second square, we see that , so the second square is commutative. The third square is verified by a similar observation (note the diagram would not be commutative if we switched the rows!).

So, this gives an example where all vertical maps are zero except the middle one, which is the (non-trivial) identity .