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Suppose $X$ is a "good enough" Hausdorff topological space; we assume that $X$ is not compact. Now, for a natural number $k$ and an abelian group $G$, consider the group $\varinjlim_{\substack{U\subseteq X\\ \text{$X\setminus U$ is compact}}} H^k(U,G)$.

Could you, please, give me a reference to a text where this object is defined? I would like to learn the standard term and notation for it.

Thank you in advance,

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I think this is called 'cohomology of ends'. It is used by Geoghegan in 'Topological Methods in Group Theory', see 3.8 (numbering of the preprint version). I first learned about this somewhere else, but I don't remember the title of the book, nor the author. – Fernando Muro May 7 '13 at 20:12
I think the following link is relevant:… – J. Martel May 7 '13 at 21:21
up vote 10 down vote accepted

This is the cohomology of $X$ at $\infty$. You'll find a discussion of (singular) homology and cohomology at $\infty$ in Hughes & Ranicki, Ends of Complexes (CUP, 1996).

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@Igor. Oh, so I even guessed the word:))) Thanks a lot for the answer and the reference. – Serge Lvovski May 8 '13 at 5:50

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