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I am going through Hatcher's Algebraic Topology. But I'm stuck with the Exercise $7$ of chapter $0$ in page $18$.

Fill in the details in the following construction from [Edwards 1999] of a compact space $Y \subset \mathbb{R}^3$ with the same properties as the space $Y$ in Exercise 6, that is, $Y$ is contractible but does not deformation retract to any point. To begin, let $X$ be the union of an infinite sequence of cones on the Cantor set arranged end-to-end, as in the figure. Next, form the one-point compactification of $X \times \mathbb{R}$. This embeds in $\mathbb{R}^3$ as a closed disk with curved ‘fins’ attached along circular arcs, and with the one-point compactification of $X$ as a cross-sectional slice. The desired space $Y$ is then obtained from this subspace of $\mathbb{R}^3$ by wrapping one more cone on the Cantor set around the boundary of the disk.

figure

I cannot understand the construction of the space $Y$. After the one-point compactification of $X \times{} \mathbb{R}$, how is this new space embedded in $\mathbb{R}^3$?

Also what does the last line mean?

...by wrapping one more cone on the Cantor set around the boundary of the disk.

Any help about this is appreciated. Can anyone at least provide me a link to the paper [Edwards 1999] mentioned in the question? Thanks!

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  • $\begingroup$ Can you include the problem statement in your question? $\endgroup$ Sep 24, 2014 at 1:57
  • $\begingroup$ I've put the problem in. $\endgroup$ Sep 24, 2014 at 2:07
  • $\begingroup$ By the way, there's is a bibliography at the end of the book. It says that [Edwards 1999] is "R. Edwards, A contractible, nowhere locally connected compactum, Abstracts A.M.S. 20 1999), 494." Unfortunately that paper doesn't seem to be on mathscinet, and I don't know where to find "Abstracts A.M.S.". $\endgroup$ Oct 4, 2014 at 9:22
  • $\begingroup$ @NajibIdrissi, thanks for the info. But I couldn't find a single reference to the paper cited in the bibliography (except maybe for its mention in the bibliography itself)! $\endgroup$
    – ChesterX
    Oct 4, 2014 at 11:16
  • $\begingroup$ Here is a link to Abstracts of the AMS, but it seems that on-line versions only go back to 2009. $\endgroup$
    – Lee Mosher
    Jan 6, 2020 at 20:44

1 Answer 1

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My instructor informed me that the paper in question is by Prof. Robert Edwards of UCLA. I contacted him over e-mail about his paper and he was kind enough to correspond. He informed me that : ..there is no published paper, only his abstract in the AMS Abstracts of the May 1999 AMS meeting in Denton TX.

He also mentioned that there is a clarification needed in the question : ..all of the cones-on-Cantor-sets of the space $X$ are of the same size. So its "baseline" is meant to be $[0,\infty)$, not $[0,1]$.

So now, I can visualize the space $X\times{}\mathbb{R}$ as the upper half plane with 'fins' attached. The one-point compactification, say $C$, is then embedded as a closed disc in $R^3$ with fins attached. This disc has the one-point compactification of the $X$-axis as its boundary. As for its cross-section, we have the one-point compactification of $X$, as shown in the picture. Another cone-on-Cantor-set is wrapped around the boundary to obtain the space $Y$.

The space $Y$ is contractible, since we can retract each of the fins and obtain the closed disc $D^2$, which is contractible.

But $Y$ does not deformation retract to a point. Any neighborhood of a point in $Y$ has infinitely many disjoint fins and thus it is not path connected. For points on the boundary of $C$, the extra cone-on-Cantor-set provides the disjoint fins in its neighborhood. Then by problem 0.5 of Hatcher's book, $Y$ does not deformation retract to a point.

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