From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: What is topology
Date: 22 May 1999 06:16:56 -0400
Newsgroups: sci.math
Keywords: What is knot theory
>> => The standard definition of topology is as the study of properties that
>> => are preserved by homeomorphisms.
>>
>> So much for Knot Theory.
>>
>
>Well, you could take the point of view that in knot theory we're
>studying the homeomorphism type of the complement of the knot, rather
>than the homotopy type of the knot.
You could take either point of view, yes. Neither would address
what I imagine to be Gerry's concerns, and neither is actually
taken by any practitioner of my acquaintance. (1) It's only
comparatively recently that it's been proved that (interpreting
"knot" to mean "simple closed curve in 3-space") "the homeomorphism
type of the complement of the knot" determines the knot (up to...hey,
it's your example, *you* tell *us*); and of course it's easily seen
to be false that "the homeomorphism type of the complement" of (for
example) a pair of disjoint simple closed curves in 3-space determines
that two-component link up to anything very useful at all. (And most
practicioners do include the study of links in "Knot Theory".)
(2) It's unclear what you mean by "the homotopy type of the knot";
if you mean the homotopy type of the complement of the knot, then
that this need not determine the knot has been known for a long
time (e.g., the granny knot and the square knot have complements
of the same homotopy type). If you mean something else, I don't
know what it might be but I bet it also doesn't determine the knot.
>But, I agree, definitions like this
>can't be regarded as set in stone, they are always provisional and may
>have to change as the subject grows.
Bring back _Analysis Situs_! Biding that, "What is topology? it's
what topologists do."
Lee Rudolph
==============================================================================
From: lrudolph@panix.com (Lee Rudolph)
Subject: Re: What is topology
Date: 24 May 1999 21:23:37 -0400
Newsgroups: sci.math
Nick Halloway writes:
>In article <7iarsb$hn2$1@gannett.math.niu.edu> rusin@vesuvius.math.niu.edu
>writes:
>
>>Arguably knot theory is precisely the study of topological pairs (X,Y)
>>with Y a subspace of X. Depending on your tolerance for generalization,
>>(X,Y) might be limited to pairs with X and Y spheres, or spheres of
>>dimensions 3 and 1 respectively.
>
>I thought knot theory is about embeddings of subspaces of codimension 2.
That (to use Dave's phrase) depends on your tolerance for generalization,
and (to deviate from his choice of the adverb "precisely") your choice
of categories. In differential topology, you might well want to study
pairs (X,Y) of differentiable manifolds, where Y is smoothly embedded
in X, and your equivalence relation is (maybe) diffeomorphism of pairs.
In that context, Haefliger showed (over 30 years ago) that the 3-sphere
can be knotted in the 6-sphere. On the other hand, his examples unknot
again if you forget the differentiable structure and consider them merely
as pairs of piecewise-linear manifolds; and indeed, in the PL category
codimension 2 is the only codimension where spheres knot in spheres
(starting with 1-spheres in 3-spheres)--but that's not (or didn't used
to be...as Robert Gunning said, when asked "isn't that deep?", "it was,
but mathematics rises to the surface with time") obvious, it's a theorem.
On the other hand, in the topological category, even if we only consider
manifold pairs, it's arguable that knotting already happens in codimension
1 (in dimension 3 and up): would you say that an Alexander horned 2-sphere
in 3-space is knotted, or not?
Lee Rudolph