# definition of the end of a manifold?

Hey everybody, I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function asigning 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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I imagine he means that the complement of a compact set is homeomorphic to $S^3\times\mathbb R$. –  Mariano Suárez-Alvarez Jan 20 '10 at 20:57
en.wikipedia.org/wiki/End_(topology) –  algori Jan 20 '10 at 21:10
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. –  Mariano Suárez-Alvarez Jan 20 '10 at 21:51
Thanks to everyone who answered me, now everything is much clearer! –  jsos Jan 21 '10 at 7:19

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).