I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function assigning to each compact set K a conected component e(K) of the complement of K. I am trying to understand Gompf's article "three exotic $R^{4}$ 's and other anomalies" and he quotes a theorem of Freedman (Corollary 1.2 in "The topology of 4-dimensional manifolds) saying "Any open 4-manifold M with $\pi_{1}(M)$, $H_{1}(M)$ and end collared (topologically) by $S^{3}\times R$ is homeomorphic to $R^{4}$". How can a function be homeomorphic to something? Im interpreting this as end meaning the "hypothetical boundary" of the manifold. Thanks in advance.
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3$\begingroup$ I imagine he means that the complement of a compact set is homeomorphic to $S^3\times\mathbb R$. $\endgroup$– Mariano Suárez-ÁlvarezCommented Jan 20, 2010 at 20:57
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8$\begingroup$ en.wikipedia.org/wiki/End_(topology) $\endgroup$– algoriCommented Jan 20, 2010 at 21:10
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4$\begingroup$ One way to define ends of $M$ is as the direct limit $\lim_KC(M\setminus K)$, where $K$ runs through the compact subsets of $M$, $C(M\setminus K)$ is the set of components of $M\setminus K$, the arrows $C(M\setminus K)\to C(M\setminus K')$, for $K\supseteq K'$, is induced by the inclusion $M\setminus K\to M\setminus K'$. The definition using $e(K)$ is just unraveling the usual construction of this direct limit. $\endgroup$– Mariano Suárez-ÁlvarezCommented Jan 20, 2010 at 21:51
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$\begingroup$ Thanks to everyone who answered me, now everything is much clearer! $\endgroup$– jsosCommented Jan 21, 2010 at 7:19
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$\begingroup$ Note that it is much easier to define an isolated end than an end. Here "collared end" is a special type of isolated end. $\endgroup$– YCorCommented Sep 23, 2021 at 11:26
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1 Answer
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As algori points out in the comments, the definition of end may be found here. Also there you will find the definition of the neighborhood of an end.
When he says "end collared (topologically) by $S^3 \times R$" he means that the end has a neighborhood homeomorphic to $S^3 \times R$. Since he's assuming that $M$ has only one end, this simply means that there is a compact set whose complement is homeomorphic to $S^3 \times R$ (as Mariano said in the comments).
For future reference, it's pretty common in the literature to blur the distinction between an end and a neighborhood of an end. Especially if when there are neighborhoods of the end that are products, as is the case here.
answered Jan 21, 2010 at 0:45
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1$\begingroup$ I wonder for what spaces "ends" are good concepts? Do we need to assume that the space is compactly generated, or even locally compact Hausdorff? $\endgroup$– Z. MCommented Sep 23, 2021 at 20:37