Re: Homotopic maps
- From: ptk <ptk@xxxxxxxxxxxxx>
- Date: 17 Nov 2005 07:45:03 -0500
<x-flowed>On 2005-11-15 12:05:55 -0600, lrudolph@xxxxxxxxx (Lee Rudolph) said:
> Maury asked:
>
>>> Let A be an open subset of R^n, f:S^(n-1)->A a
>>> continuous injective map and D the "interior Jordan
>>> region" bounded by f(S^(n-1)) (for the Jordan-Brouwer
>>> Separation Theorem). Are the following statements
>>> equivelent?
>>>
>>> (I) D is contained in A.
>>> (II) f is homotopic to a constant map g:S^(n-1)->A.
>
> Andrew D. Hwang replied:
>
> ...
>> In short, you can arrange that
>> *neither* component of A minus f(S^{n-1}) (the complement of the image
>> of f) is simply connected, much less homeomorphic to a ball.
>
> but, alas, the reply is to a different question. (In fact, it's to a
> question that Maury asked in sci.math but not sci.math.research; a
> question that was answered there, though later posts of Maury's suggest
> that he didn't read the answer.) The existence of spheres with
> horns inside and out doesn't, at least not on the face of it,
> exclude the following situation: S = f(S^{n-1}) is an embedded sphere
> in R^n, A is an open set containing both S and its "interior
> Jordan region D", and there is a null-homotopy of f to a constant
> the trace of which is contained in A but not contained in D.
>
> Nor, by the way, is it clear (to me) that if S a "sphere with
> internal horns" in R^3 (e.g., the inversion of Alexander's original
> horned sphere) with non-simply-connected "interior
> Jordan region D", then the identity map of S to the closure
> S\cup D of D is not null-homotopic. Actually, if I had to
> bet I'd say that at least some such examples (maybe not Alexander's)
> are aspherical; I have a vague memory that someone (Verjovsky, maybe?)
> has put a hyperbolic structure
> on D in some such cases, and then presumably an application
> of the Side Approximation Theorem would show that id_S could,
> indeed, be homotoped to a point in S\cup D. ... But I swear
> to none of that.
>
> Lee Rudolph
Let S be an Alexander horned sphere in R^3 whose bounded complemenatry
domain U is not simply connected. Observe that X = S union U is a
homology ball in R^3. If X were contactible, then it would be a
homotopy ball in R^3, which is a real ball. Hence the inclusion of S
into X is essential.
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