Abuse of notation in relative homology theory

[andrewkirk]

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?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[mathwonk]

Unless you find an actual statement that is rendered incorrect by this abuse, i.e. that is not true modulo the isomorphism you noted, I think you are worrying unnecessarily. E.g. everybody with different handwriting has a different (but isomorphic) set of symbols for the natural numbers: 1,2,3,4,... but that seems not to bother you. Of course you have noticed something that it helps to aware of. I.e. the inclusion map is isomorphic to just an injection, but that seems not a big deal. And replacing a set theoretic statement like "A = B" by "A equals the image of B under the previously given canonical isomorphism" seems ok too. I do recall being bothered by this sort of thing back when starting out, and you have made it clear you are very observant, a good quality for a mathematician. Or maybe you are really a logician under the skin, since this sort of thing concerns you. It is also quite possible that I just don't understand your point well, but this is my opinion.

 

[andrewkirk]

My complaint is not that I think any of the theorems presented are wrong. I don't, and I'd be a crank if I did.

It's that it makes this stuff unnecessarily hard to learn if the presenter pretends that something is something it isn't. It seems to me a classic case of somebody having been working on a topic for so long that they've forgotten what it's like to learn it. That's why great mathematicians (or experts at anything else) are often very bad teachers of their subject. They've forgotten how to learn this stuff, and what it's like learning it, so they write textbooks as if they were talking to a fellow seasoned expert in the topic, rather than - what it it obviously will be, given it's a textbook - a learner. They use in-jargon, use multiple different synonyms for a single concept without explaining that they are synonyms (because they've been using them as synonyms for so long that they have become blind to the fact that they are actually different words), and assume all sorts of knowledge that nobody but an expert in the field would have.

I posted this same question on Math Stackexchange and got the (in my view) peculiar response that 'you have to develop a sixth sense for these things' (ie when it's OK to treat isomorphs as identical and when it isn't). Under that view, developing a sixth sense for Homology Theory is a prerequisite for commencing the study of Homology Theory. How one is supposed to develop that sixth sense for a subject one has not yet encountered, I have no idea.

The way I dealt with this stuff at uni was to stop the lecturer when they said something that didn't make sense and ask them to justify the questionable step, or explain what something was. But you can't do that with a textbook.

The way I deal with it now is to write out the proofs for myself properly, avoiding invalid equivalences and unjustified steps. It takes a long time but at least, when I'm finished, I feel that I really understand it properly. Plus (an extra bonus) it makes it really easy to explain it to somebody else in a way that they'll understand.

 

[homeomorphic]

I didn't really notice that in algebraic topology so much. I did notice it a bit when I was reading about Hopf algebras. I had to ask my adviser about some omitted isomorphisms in a particularly bad case. I think you just get used to it. It's not so much a matter of not being initiated to a particular subject as it is a matter of being initiated to the way people tend to write. After that first time I had to ask my adviser, I became more aware that people actually did such things, and then it was easier to read more of the material because I expected it.

The bigger problem that made it hard to learn about Hopf algebras was the lack of motivation, not the nit-picky details, which I only obtained by reading some of the material in John Baez's seminar notes, where he shows that the concept can be arrived at very naturally using some diagrams.

Also, I think I agree with Mathwonk that maybe you're secretly a logician. I think most mathematicians tend to adopt a more fluid attitude and not be concerned with the most minute details if they know what is meant and can quickly fill in the gaps, mentally. I didn't find that it made it hard for me to learn at first because I always thought in that way, at least after a while of doing proofs. Let me quote Terence Tao here:

http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/

"It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that “fuzzier” or “intuitive” thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as “non-rigorous”. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education. (Among other things, this can impact one’s ability to read mathematical papers; an overly literal mindset can lead to “compilation errors” when one encounters even a single typo or ambiguity in such a paper.)"

 

[homeomorphic]

I should point out, though, that my adviser and I both agreed that in the particular case we were discussing, the authors really ought to have at least said they were abusing notation and omitting some of the maps.

 

[andrewkirk]

It's interesting that that's the second reference to Terry Tao's essay that I've come across in response to this issue. I have enormous admiration (or is it envy?) for Terry Tao but someone as brilliant as a mathematical practitioner as him is not generally who I would turn to for wisdom about the best way to teach and learn mathematics, and especially about things like intuition. What is intuitively obvious to somebody with his abilities may be a week's work for most of the rest of us (or maybe even an impossible dream). I am reminded of a scene from 'Good Will Hunting' where the uber-genius played by Matt Damon got impatient when the (Fields-medal winning) other mathematician couldn't immediately follow the leaps in his crumpled notes and said, 'don't worry, It's all correct'. Then he set fire to the page and stormed off while the other guy desperately tried to save the page. I loved that movie!

It seems to me that teaching and doing are very different things and there is very little intersection between the sets of best practitioners and best teachers.

I certainly agree with Terry Tao about the importance of intuition. I see it as a capability that always has a key role in one's practice of mathematics, and is there from mid-high-school onwards, rather than as something one attains only as a senior mathematician. The way I see it one uses intuition to find answers to questions. One plays around with numbers, drawings and ideas, looking for patterns and not worrying at all at that stage about whether anything one does is rigorous. It is only when one finds a pattern that looks like it may be useful that one then starts to 'crank up the rigour' to see if the pattern one has found, or maybe only sensed, can stand up to scrutiny. So IMHO intuition and rigour are complementary, and need to always be part of one's armory, rather than one graduating from one to the other.

I get frustrated with many textbooks because of the big gaps in logic they often make, that I can't follow and sometimes need to work for a few days to validate. This is one of the problems of studying this stuff outside a university environment. There is nobody to explain a hard-to-follow step as there is when one has tutorials and lectures. I wish text-books were written more clearly. But I can understand why they're not. There's no money in writing textbooks, unless perhaps you manage to write the gold standard that every educational institution around the world decides they have to set as a core text (John Hull's 'Futures, Options and Other Derivatives' and Spivak's 'Calculus' come to mind). So one just can't justify the time it would take to review one's own writing line by line, trying to put oneself in the shoes of a neophyte.

 

[homeomorphic]

My main point there was in Tao's parenthetical remark that an overly literal mindset can lead to compilation errors.

The question of who is a good teacher is so complicated. Who is a good teacher for whom? I was just trying to watch a lecture of Bill Thurston and couldn't really follow it very well--maybe I will revisit it and go through it really slowly (I'm extremely casual about any pure math these days, since I more or less quit after my PhD, about a year ago). But if I could talk to Thurston one on one, he probably would have been one of the best teachers for me because I would get to talk back and tell him what I don't understand. So, there you see some of my objections to the lecture method of teaching, which are stronger with lower level students who have limited interest, but are still present even at higher levels. Similar comments apply to books, which are similar to lectures in the sense that the student doesn't get to talk back. So, I think the best practitioners are potentially great teachers, but the standard methods of teaching don't allow them to be. Of course, Thurston may or may not have been a great teacher when it comes to undergrads. I don't know. We had a number of professors in my department who were known to be bad lecturers, yet "good in office hours".