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