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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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    While I think that Hatcher is a bit opaque, I think that both of your examples are fine. The claim that cell-complex quotients is a wedge sum of n-spheres follows immediately from the definition of cell complexes. So he gives it as an example, not a proposition. And the second uses just commutative diagrams.2011-10-21
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    @mixedmath: Looking back now, Lagrange's theorem for finite groups follows immediately from the definition of a coset.2011-10-21
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    It seems like analysis is closer to classical, get-dirty-with-chalk mathmatics, than the more modern algebraic topology, where a lot of things happen under the hood of commutative diagrams.2011-10-21
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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?2011-10-21
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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.2011-10-22
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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).2011-10-24
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    Some [relevant remarks by Terry Tao](http://terrytao.wordpress.com/career-advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/).2012-03-14
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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.2012-06-11
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    @Steven A VERY good commentary by one of the masters at his blog and great suggested reading on the question.2012-06-27

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