I’m planning on asking a big-list CW-question asking for examples of useful homeomorphisms between cones, cylinders, suspensions, spheres and balls etc. together with a proof for them.

When writing the question, I noticed that a lot of such questions are contained in the list of similar questions (such as this one) together with answers and now I feel (even more) uncertain about asking the question.

So bascially, I want to ask a big-list question aiming at creating a large collection of answers to similar-type questions most of which have (probably) already been asked. Is that okay?

For three reasons I still feel like asking the question:

  • It’s a lot of work to dig through all these already-answered questions and it would certainly be nicer to have an array of the more useful ones together at one place.
  • It gives the chance to share an answer to a useful question that hasn’t been asked yet. (I can imagine someoone going “Oh, yeah: I use this homeomorphism all the time which noone else seems to use – it makes my life way easier.”)
  • Over time, the answers will probably be sorted less by the elegance of their arguments, but rather by the usefulness of their statements (which is nice for a change).

So should I go on asking the question or not?

To address the concern of this specific question being too broad raised by GrigoryM, here’s the question I was going to ask in its raw form:

This is about collecting a big list of useful homeomorphisms between

  • cones, mapping cylinders, suspensions and quotients
  • of spheres $S^n$, disks $D^n$ and euclidian spaces $E^n$ (where $E^n \cong \mathbf R^n$)
  • using (obvious) inclusions, projections, embeddings etc.

backed up by a proof. For example, why is

  • $S^n \cong ΣS^{n-1}$,
  • $\operatorname{cone}S^n \cong D^n$, or
  • $D^n/S^{n-1} \cong S^n$?

In Bredon’s Topology and Geometry, examples 13.9 and 13.10 demonstrate techniques to prove such homeomorphisms, and I would like to see more of those.

  • $\begingroup$ A simple question: Does a downvote on this mean “bad meta question” or “No, it’s not okay.”? $\endgroup$ – k.stm Dec 31 '14 at 11:50
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    $\begingroup$ I would guess the latter. $\endgroup$ – mrf Dec 31 '14 at 11:52

As you propose it, I think it is too broad. But if the point is to get that content on MSE, you can ask (and answer) a separate question for each problem and link all of them. This gets the job done and circumvents its large broadness.

I would like it if someone did this.

  • $\begingroup$ I strongly suspect each of 3 questions mentioned in the update has already been asked (e.g. suspension of sphere...). $\endgroup$ – Grigory M Dec 31 '14 at 12:58
  • $\begingroup$ Okay, as a side project, I will now collect all the questions and answers of this kind, and work through them. Then I will see what I do with them. Maybe I will put them on my blog as mrf suggested. $\endgroup$ – k.stm Dec 31 '14 at 13:11

To me it sounds like aiming at creating «a large collection of identities involving addition, multiplication and division» — i.e. hopelessly unfocused (as in 'too broad').

  • $\begingroup$ Okay, that’s a good point. I kind of was hoping that there are 10 or so identities used all the time like the binomial theorems, the geometric series, triangle numbers, …. But that might be just a bottomless pit. $\endgroup$ – k.stm Dec 31 '14 at 10:18
  • $\begingroup$ I mean, maybe you have something more focused in mind — but it's hard to guess from the OP and 1 example... $\endgroup$ – Grigory M Dec 31 '14 at 10:27
  • $\begingroup$ I added the question as I was going to state it – does this sound too broad to you? $\endgroup$ – k.stm Dec 31 '14 at 10:39

Big-list "threads" tend to suffer from limited visibility of answers. The layout of pages here is meant to emphasize 1-2 top answers, not 20 or more. So invariably, the bottom-dwelling answer get little attention: nobody wants to dig that far to vote, edit, flag or even read. Just look at the tail end of any big-list post with many answers.

I suggest to create an thematic index of such posts instead. I did this for certain kinds of calculus exercises. Compiling such an index is a fair bit of work, but it is also a way to tidy up the site: duplicates become more apparent, and so do the topics that are yet to be addressed.

By the way: if you are interested in adding a page "Examples of homeomorphisms" (or other topics in topology) to Math.SE Index site, drop me a message either here or in chat.


This doesn't sound like a good fit for the StackExchange model. I would suggest that you start a blog or a wiki of your own instead.


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