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Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or Sullivan's classic textbooks - that last three of which I have read.

But what information about a space is contained in its p-completions? Whilst I have a good handle on the information that may be gleamed from study of the p-localizations, the slightly more abstract nature of the completion leaves me unsure of how to interpret its properties. I am aware of the various arithmetic squares and have no trouble with the concepts or algebra, I just lack any concrete examples of the theory of completions yielding any useful information about the homotopy type of a given space.

Tyrone
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  • If you don't get a good answer here after a day or two or three, this might be a good question for mathoverflow.net – Lee Mosher Sep 04 '15 at 21:26
  • Thanks for the advice. I posted the question over there and it happily garnered some interesting responses.

    http://mathoverflow.net/questions/218207/why-study-the-p-completions-of-a-space

     – Tyrone Sep 21 '15 at 14:35

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