Talk:Exotic R4

Attribution
Simon Donaldson needs to be in the article; see this, for example. Orthografer 05:14, 8 September 2006 (UTC)

I did more investigation, and as far as I understand it, Andrew Casson first had the construction (the Casson handle, which when attached to a standard 0-handle was the first exotic R4), but didn't know whether it was homeomorphic to R4, or whether it was exotic. Then Freedman's and Donaldson's work answered both questions with a "yes." I'm not sure what Taubes had to do with this. Orthografer 19:19, 8 September 2006 (UTC)

Poincaré conjecture
Poincaré conjecture is proven, so someone should modify and explain an exotic 4-sphere cannot exist. —Preceding unsigned comment added by 147.122.3.48 (talk) 09:58, 11 February 2008 (UTC)
 * This is the smooth Poincaré conjecture, which I believe is still open. Algebraist 15:53, 21 May 2008 (UTC)


 * The Poincare conjecture is in three dimensions. I changed this to "generalized Poincare conjecture" to clarify that this is not the standard problem. 86.177.226.10 (talk) 00:27, 20 August 2009 (UTC)

A continuum (quantity)
I've never seen the use of continuum as short for the cardinality. If that isn't standard language in the field, that's quite the misleading abbreviation, since no continuity is implied by having size א1. ᛭ LokiClock (talk) 16:22, 18 March 2012 (UTC)


 * Continuum is standard terminogy. Rybu (talk) 08:11, 21 March 2012 (UTC)


 * And it only refers to the size, without further axioms or structure? ᛭ LokiClock (talk) 07:31, 22 March 2012 (UTC)

Move?

 * The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section. 

The result of the move request was: not moved Armbrust The Homunculus 01:04, 23 December 2013 (UTC)


 * unicode ï¿½ (talk) 20:34, 15 December 2013 (UTC)
 * The special sort of R, when enlarged enough, is shown to be this:  ℝ   . Anthony Appleyard (talk) 09:30, 16 December 2013 (UTC)
 * Oppose WP:TSC This uses two untypable special characters. The article itself doesn't even use either. -- 65.94.78.9 (talk) 15:11, 16 December 2013 (UTC)
 * Oppose - WP:TSC and as the rest of category. In ictu oculi (talk) 00:53, 17 December 2013 (UTC)
 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

A *continuous* continuum of non-diffeomorphic differential structures on R4 ?
The opening paragraph states that

There is a continuum of non-diffeomorphic differentiable structures of R4.

In fact, I've usually heard the statement expressed as "There is an R2 of mutually non-diffeomorphic differentiable structures on R4."

What I'd like to know is whether this continuum of mutually non-diffeomorphic differentiable structures is a continuous continuum or not.

If it were, in the sense I mean, then we could say:

There exists a smooth 6-manifold W that is homeomorphic to R2×R4, say via h: R2×R4 → W, such that

1) For any p ∈ R2, h({p}×R4) is a smooth submanifold of W.

and

2) For any p, q ∈  R2, we have that h({p}×R4) and h({q}×R4) are not diffeomorphic.

It would even be nicer if another condition were also satisfied:

3) The surjective map f: W → R2, defined by f(w) = p where h-1(w) = (p,q) ∈ R2×R4, is smooth.

Can anyone shed light on whether any of these conditions are known to be true or not? If so, I think that should be incorporated into the article.

[Note: Of course, the smooth manifold W must be diffeomorphic to the standard R6, since there is only one differentiable structure on Rn, n ≠ 4, up to diffeomorphism.]Daqu (talk) 04:34, 3 May 2014 (UTC)

Needs to be more explicit
There should be a section explaining how to construct exotic R^4's and explicitly use invariants to show this is true. — Preceding unsigned comment added by 97.122.179.164 (talk) 04:57, 27 April 2017 (UTC)

Exotic hyperbolic spaces?
Do exotic versions of H4 (or indeed, hyperbolic spaces in other dimensions) exist? Wikipedia currently has articles about exotic spheres and exotic Euclidean 4D space, but not anything on exotic hyperbolic spaces, if indeed any exist. 2600:1014:B072:E984:9C45:23C0:15C0:2AAB (talk) 22:47, 3 November 2023 (UTC)