Given exact sequences Ai and Bi where fi: Ai -> Bi as follows:
A1 A2 A3 A4 A5
f1 f2 f3 f4 f5
B1 B2 B3 B4 B5
The five lemma states that if f1 is an epimorphism and f2 and f4 are
monomorphisms, then f3 is a monomorphism.  Similarly, if f5 is a
monomorphism, and f2 and f4 are epimorphisms, then f3 is an
epimorphism.

I need to find examples that all of these conditions are necessary.
Can anyone help?

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Request for Question Clarification by mathtalk-ga on 09 Jul 2004 04:58 PDT

Hi, topologystudent-ga:

I interpret the request as one involving examples for which exact
sequences Ai,Bi support "arrows" fi that make the diagram commute.

So each of the two claims needs a minimum of three examples (or
perhaps "counterexamples"), in which just one of the assumed
epimorphism/monomorphism conditions being violated allows the
conclusion (f3 is a monomorphism or f3 an epimorphism, resp.) to fail.

It might be helpful to know if these examples are to be framed in the
category of groups or another specific category (topological spaces
perhaps?), or if they should be presented as "abstractly" as possible.

Note that if the sequences are "short exact" (i.e. A1 = A5 = B1 = B5 =
0), then necessarily f1 is an epimorphism and f5 is a monomorphism. 
So not all of the required examples can be phrased in terms of short
exact sequences.

regards, mathtalk-ga

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Clarification of Question by topologystudent-ga on 09 Jul 2004 06:27 PDT

Your interpretation was correct.  I am taking a class in Algebraic
Topology, so examples using groups or topological spaces would be
fine.
Thank you for your quick response.