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I am taking a course in Algebraic Topology. We are using Hatcher as a textbook. One of the main problems I am facing with the textbook is its level of rigour. Example: On Pg 10, Hatcher mentions in passing that $X^n/X^{n-1}$ is the wedge sum of n-spheres (here $X^n$ is the $n^{th}$ filtration of a CW-complex). While this is intuitively clear, it requires some work to prove. Another example is the proof(?) of the Cellular Boundary Formula on Pg 141. While I can follow what he says (and reproduce it in different contexts), it strikes me as a reason to believe the formula rather than a proof of that fact according to the idea of proof that I have become familiar with from earlier courses in Analysis and Algebra (also I do not think I'll be able to prove this fact at that level of rigour).

My question is: Is this level of rigour acceptable? I feel uncomfortable with the proofs Hatcher gives. But, should I be feeling uncomfortable? Looking back, I was never uncomfortable with the kind of justifications we used to give in high school calculus and this current discomfiture stems from the fact that I have taken a few courses in Analysis in between.

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@Dignaga: I might be misinterpreting you, but I think that you are saying that you do not like my suggestion that cell-complex quotients are immediately viewed as a wedge sum of n-spheres. So allow me to clarify. When when one makes $X^n$, one defines an attaching map of the boundaries of n-spheres to n-1-spheres of $X^{n-1}$. When one takes the quotient, one collapses all of $X^{n-1}$ to a point - so now all the n-spheres are attached to each other by exactly one point. Does that make sense? –  mixedmath Oct 21 '11 at 21:45
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@mixedmath: Of course your right. But, is this the right level of rigour? Also, Lagrange's theorem is also immediate (in your sense) because quotienting a finite group by a subgroup is precisely the collapsing of its cosets to a point, so clearly the number of elements in the quotient is exactly the number of cosets. Or, closer to topology, I could say that collapsing the boundary of a closed disk to a point 'clearly' makes a sphere. Or a simple closed curve in a plane 'clearly' partitions it into two disjoint parts. –  Dignaga Oct 22 '11 at 10:27
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@Dignaga: One point is that the argument mixedmath is giving is something that is directly verifiable at the point-set level. Namely, $X^n$ is formed through a quotient topology on a union of discs and $X^{n-1}$, and if you then impose a further quotient (collapsing $X^{n-1}$ to a point) this is the same as imposing a quotient topology on a union of discs and $*$. Often algebraic topology texts assume that the reader is well acquainted with arguments of a previous course in point-set topology like this (in order not to get trapped on details). –  Tyler Lawson Oct 24 '11 at 2:20
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Some relevant remarks by Terry Tao. –  Steven Taschuk Mar 14 '12 at 4:30
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As for the larger question of how much rigor is necessary, I think a proof should contain enough rigor to provide a high-level sketch of how one would create a mechanically checkable proof, given enough time and energy. –  ShyPerson Jun 11 '12 at 19:31

2 Answers 2

The German mathematician Klaus Janich has a wonderful response to this question in his book on topology, which is intentionally very non-rigorous and intuitive:

It is often said against intuitive, spatial argumentation that it is not really argumentation,but just so much gesticulation-just 'handwaving'.Shall we then abandon all intuitive arguments?Certainly not.As long as it is backed by the gold standard of rigorous proofs,the paper money of gestures is an invaluable aid for quick communication and fast circulation of ideas.Long live handwaving! (Janich, Topology,page 49,translation by Silvio Levy)

It was later said by Levy that Janich told him that this particular passage was inspired by Janich's concerns that German mathematical academia (and textbooks in particular) were beginning to become far too axiomatic and anti-visual and that this was hurting the clarity of presentations to students.

There is indeed much that is wise in this quote and it really gives what I think is an excellent "rule of thumb" for determining when a "proof" in mathematics has crossed the line and really become non-rigorously vague by 21st century mathematical standards to the point is really proves nothing: If the intuitive argument cannot be rewritten in a completely axiomatic and pedantic manner where there is a completely logical progression from premises to a conclusion, then we should seriously reconsider whether or not our intuitive argument is a valuable one. I think you'll notice most of Hatcher's arguments would pass this test,even if it would probably take a considerable amount of spade work to make them completely rigorous in the same sense as a real analysis or algebra proof.But since algebraic topology is so closely related to classical geometry, completely abstract reasoning would probably strip away much understanding of the sources of most of the central concepts,which I believe was Hatcher's reason for writing the text in this manner.

(Sadly, there are too many algebraic topology texts that take the uncompromisingly rigorous and non-visual,categorical/functorial perspective-such as the old classics by Spanier and Dold and more recently, the beautiful texts by May and tom Dieck. The down side of this approach is that it completely disconnects the subject from it's geometric roots and it becomes simply another branch of algebra whose roots are utterly mysterious.No one quite seems to have figured out yet how to effectively interpolate between the 2 approaches in a textbook. The closest anyone's ever come to pulling it off to me is Rotman. )

Of course, as it's stated, this isn't an exact science. The question of what constitutes a proof has constantly been questioned and revised since the beginnings of mathematics in the Ancient World. I have no doubt it will continue to undergo scrutiny in future ages. But I think Janich has given some quite good advice to the novice here.

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Well said, although the thing I really don't like about intuitive handwaving is that intuition differs from person to person. I almost always just tune out when people offer me their intuitions, and develop my own, for this reason. For the same reason, intuitive arguments have I would even say crippled the speed at which I could otherwise read texts, which I understand is the opposite of what most people would say. In fact, people communicating in this "paper currency" is one of the primary reasons I have an account on this site; to resolve the questions that arise from imprecise talk. –  Jeff Jun 27 '12 at 5:46

The level of rigor that is needed depends on your own taste. Of course every mathematician should verify a claim until he feels comfortable that if necessary, he could produce the real argument down to the atomic details. (Unless you are taking something on faith on purpose, but I get the feeling that is not your intent here, but rather you want to understand the material.) For me, this level of "rigor" required lies somewhere between explicitly writing out everything in bare bones set theoretic terms, and the level of detail presented in a graduate analysis text such as Rudin. I imagine that I'll be more slick and fast as I gain more experience, since I'll know the patterns of arguments.

Anyway, I read Hatcher only a few months ago to study for a qualifying exam. The details of the questions you asked can be diligently unwound. You just have to remember what the topology is defined as for quotient spaces, etc., and then it's just a set theoretic exercise. Ask if you need help.

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