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From: bacle on 2 Aug 2010 14:07
Hi:
I have been going over relative homology of a pair (X,A), and I think I understand well the formalities:
relative cycles (2 cycles being equivalent if they have the same coefficients in X-A), relative boundaries (induced by the "usual" boundary), how we get a LES
from a SES (Snake Lemma and/or Five Lemma), but I cannot find a good geometric example/motivation for it.

Any suggestions/refs., please.?
From: W. Dale Hall on 3 Aug 2010 06:29
bacle wrote:
> Hi: I have been going over relative homology of a pair (X,A), and I
> think I understand well the formalities: relative cycles (2 cycles
> being equivalent if they have the same coefficients in X-A), relative
> boundaries (induced by the "usual" boundary), how we get a LES from a
> SES (Snake Lemma and/or Five Lemma), but I cannot find a good
> geometric example/motivation for it.
>
> Any suggestions/refs., please.?

The relative groups represent what you can see in X if you can ignore
what's going on in A. Given an inclusion (actually, given any mapping)
one can define a geometric sequence of spaces & inclusions


A --> X --> X/A --> SA --> SX --> S(X/A) --> S^2 A --> ...

where S represents the "suspension" of a space.

The "any mapping" version is this: let f: X --> Y be given

X --> Y --> Tf --> SX --> SY --> STf --> S^2 X --> ...

where Tf is the mapping cone of f (i.e., the cone on X glued
to Y via the mapping f), and S still represents suspension.

Google "Puppe sequence". BTW, it's fun for 5-year-olds,
since a crude (umm, in both senses of the word)
English approximation to its pronunciation is
"poopy sequence".

At any rate, the Puppe sequence (together with excision) gives
the long exact sequence you're wondering about.

Dalepuppe sequence
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