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I am refreshing my understanding of homology theory (well, recreating from scratch really!) after a thirty year break and there's something that bugs me in how the texts I've seen write about relative homology.

The relative homology module $H_q(X,A)$ is defined as $ker\ \partial_q^A/Im\ \partial_{q+1}^A$ where $\partial_q^A:S_q(X,A)\to S_{q-1}(X,A)\ $ is given by $\partial_q^A(z+S_q(A))=\partial_q z+S_{q-1}(A)$ and $S_q(X,A)\equiv S_q(X)/S_q(A)$. Note that this is a double quotient (quotient of quotients).

One uses the isomorphism theorem to show that $H_q(X,A)\cong Z_q(X,A)/B_q(X,A)$ where $Z_q(X,A),B_q(X,A)\subseteq S_q(X)$ satisfy certain conditions. Let us call the RHS of this equivalence $H_q^*(X,A)$. Note that the RHS is only a single quotient.

However, when the texts go on to discuss exact sequences, all the proofs I have seen (such as proofs of the Excision Theorem and proofs that the Connecting Map gives an Exact Sequence) take the generic element of $H_q(X,A)$ to be its isomorphic image $z+B_q(X,A)\in H_q^*(X,A)$ rather than what it actually is, which is $(z+S_q(A))+Im\ \partial_{q+1}^A\in H_q(X,A)$. This is presumably done because $H_q^*(X,A)$ is easier to work with, being only a single quotient.

I have yet to see this abuse of notation even acknowledged, let alone justified. Of course, one understands that algebraic properties are preserved by isomorphism. But the proofs tend to involve algebraic, topological, category-theoretic and set-theoretic arguments, and if the things being talked about are not the exact same items as are identified in the premises and conclusion of the theorem, one cannot have any confidence in the non-algebraic parts of the arguments. Hence one cannot trust the proof as a whole. I find this particularly troubling because many of the maps considered, such as the inclusion map, are trivial except for set-theoretic considerations. So set-theoretic considerations, not just algebraic considerations, really matter in this discipline.

I think a rigorous version of these proofs would do something like

  • use premises that refer only to $H_q^*(X,A)$ and then, once an algebraic result is obtained, use the above isomorphism to obtain a corresponding result for $H_q(X,A)$; or
  • in any part of the proof that is not purely algebraic, take special care to avoid abuse of notation. This may require invoking the above isomorphism at the beginning and end of such sections, to allow one to return to the simpler mode of dealing only with $H_q^*(X,A)$.

Has anybody else noticed this?

Is anybody else bothered by this?

Does anybody have any suggestions for how to deal with proofs that cavalierly abuse notation in this way, without even acknowledging that the abuse is occurring?

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The isomorphism between the two definitions is natural in everything in sight (the pair $(X,A)$ and the coefficients —which in you case are just $\mathbb Z$) so for every diagram of pairs of spaces involving one of the the two definitions there is a corresponding diagram involving the other one, and a big commutative diagram comparing the two with isomorphisms.

You will never run into any difficulties, provided you are careful not to do anything that is not given to you by the naturality of the situation.

The attempt to «fix» this or somehow mitigate it is understandable but hopeless. There is an immense number of such natural isomorphisms at play all the time, and taking all of them into account is just impossible. You have to develop a sixth sense which will allow you do not worry at all except when you have to worry.

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    $\begingroup$ Otherwise, you'll some day run into a spectral sequence and die of panic... $\endgroup$ – Mariano Suárez-Álvarez Oct 27 '14 at 4:00
  • $\begingroup$ On advancing further through the material that uses relative homology, so far all of it has implicitly used the single-quotient ($H^*_q(X,A)$) definition of the relative homology module. So I have a follow-up question: Is there any reason not to simply regard the above RHS as the definition of $H_q(X,A)$ rather than an isomorph? That is, is there any commonly-encountered context of which you are aware, in which the original definition of the relative homology module as a double quotient (LHS of isomorphism) needs to be used. $\endgroup$ – Andrew Kirk Oct 29 '14 at 21:16
  • $\begingroup$ I ask this because, not having studied category theory formally, I am not confident in being able to reliably judge whether the 'naturality' of a context is sufficient to justify substituting an isomorph. $\endgroup$ – Andrew Kirk Oct 29 '14 at 21:21
  • $\begingroup$ Well, you are going to have to develop your ability to judge. Essentially nothing is done with the actual single-quotient definition (nor with the double-quotient one): pretty much everything you do with homology is mediated by chains of isomorphisms. In particular, every single homology group (except those of one-point spaces, say...) is determined by sequences of isomorphisms. $\endgroup$ – Mariano Suárez-Álvarez Oct 29 '14 at 22:01
  • $\begingroup$ In other words, changing one of these two definitions for the other would accomplish absolutely nothing. $\endgroup$ – Mariano Suárez-Álvarez Oct 29 '14 at 22:02

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