Newsgroups: sci.math
From: tycchow@math.mit.edu (Timothy Y. Chow)
Subject: Re: Simple homology question
Date: Thu, 27 Oct 94 03:47:03 GMT
Thanks to all those who responded to my queries.
In article , Lee Rudolph wrote:
>tycchow@math.mit.edu (Timothy Y. Chow) writes:
>>Does there exist a subset of R^2 with a nontrivial H_2? Here H_2
>>is the second homology group.
>
>I believe the answer is "no", even for the most pathological
>combinations of subsets and homology theories.
I haven't received a definitive answer for this question yet. There
are opinions on both sides. I didn't expect the answer to be interesting,
but now that I know that it is I'll just trouble one of the local experts
for the answer.
>On the other hand, Milnor (and a co-author whose name I am ashamed
>to have forgotten) published an example of a fairly reasonable
>subset of R^3 (the analogue of the "Hawaiian earring", namely,
>the union of a sequence of spheres of radius 1/n, all tangent
>at a single point) which has non-trivial singular homology in
>infinitely many dimensions, starting (I _think_) with H_3.
Tibor Beke informs me that the other author is Barratt, although I
haven't actually chased down this paper to check the details.
As for my other question, regarding the nonembeddability of various
familiar 2-manifolds, a couple of people pointed out that they all
can be proved pretty easily using invariance of domain, which is a
nice approach that hadn't occurred to me.
--
Tim Chow tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons. ---Popular Mechanics, March 1949