TITLE: Why study the p-completions of a space?
QUESTION [27 upvotes]: 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.

REPLY [21 votes]: I'll give an answer from the point of view of an algebraic topologist,
somebody who cares about examples and computations.  This all goes way back 
and has nothing to do with modern generalities.  First off, calculationally, algebraic 
topologists can compute, like to compute, and need to compute mod p homology.  It is 
far more accessible than p-local homology, and it is the focus of completion, not localization, 
at p.  
An elementary concrete example of 
homotopy information invisible to p-localization is that $K(\mathbb Z/p^{\infty},n)$ 
becomes equivalent to $K(\mathbb Z_p,n+1)$ after $p$-completion.  This jacks up to Quillen's
equivalence between the $p$-completions of the algebraic $K$-theory space $K(\bar{\mathbb F}_q)$ and $BU$ where $p\neq q$.  The homotopy groups of $K(\bar{\mathbb F}_q)$ are 
$\oplus_{p\neq q} \mathbb Z/p^{\infty}$ in odd degrees and $0$ in even degrees.
The homotopy groups of $BU$ are $0$ in odd degrees and $\mathbb {Z}$ in even degrees.
Obviously $p$-localization knows nothing whatsoever about this equivalence.   
Calculationally,
the classical Adams spectral sequence converges to the stable homotopy groups of
the $p$-completion, not the $p$-localization, of the space (or spectrum) one 
starts with; that is also true unstably. The Atiyah-Segal completion theorem says that
the topological $K$-theory of $BG$ is the completion of $R(G)$ at its augmentation
ideal.  If $G$ is a $p$-group, this is just $p$-adic completion (on the reduced level). 
That is a place where use of pro-groups helps calculation, but in most of the 
calculational applications use of pro-objects is not especially helpful.  
There are tons of places where $p$-completion is essential to the proof or statement 
of results: the Sullivan conjecture, the Segal conjecture, the classification of 
$p$-compact groups, the study of atomic spaces, etc.  It is $p$-completion that is 
relevant to all of these and many more.  
A technical reason $p$-completion is so convenient is that there are no phantom maps 
to worry about if one restricts to nilpotent spaces of finite type.  That is also 
true for rationalization but not for localization at $p$.
A more sophisticated example is Mandell's algebraization of $p$-adic homotopy theory, the closest analogue we have of the classical algebraization of rational homotopy theory.
(The last part of Harpaz's answer concerns this.) 
I could go on for many pages.  The more algebraic topology one learns, 
the more frequently $p$-completion appears and the more natural it seems. 
It is worth adding that we do not understand $p$-completion very well for
non-nilpotent spaces, where there are several very different notions,
none well understood calculationally.  This deserves more study.