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.