I think, it is worth to strees, that a classification of open manifolds does not follow from a classification of compact manifolds. Open surfaces were classified, but open 3-folds are not, their classification does not follow from the classification of compact ones at least at the present time. In particular, {\it prime decomposition}, or Kneser's theorem (http://en.wikipedia.org/wiki/Prime_decomposition_(3-manifold)) does not hold for non-compact 3-manfiolds. There is a constructiion  due to Scott (http://plms.oxfordjournals.org/cgi/pdf_extract/s3-34/2/303) of an example of a simply connected 3-fold that can not be a connected sum of finite or infinite number of prime manifolds.

Open 3-manifold are actively studdied now. Let me give two citations that confirm further that our knowlage of compact 3-manfiolds is not sufficient for understending of non-compact ones.

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1) Ricci flow on open 3-manifolds and positive scalar curvature.
Laurent Bessi`eres, G´erard Besson and Sylvain Maillot.
http://arxiv.org/PS_cache/arxiv/pdf/1001/1001.1458v1.pdf

Thanks to G. Perelman’s proof of W. Thurston’s
Geometrisation Conjecture, the topological structure of compact 3-manifolds
is now well understood in terms of the canonical geometric decomposition.
The first step of this decomposition, which goes back to H. Kneser,
consists in splitting such a manifold as a connected sum of prime 3-manifolds,
i.e. 3-manifolds which are not nontrivial connected sums themselves.
It has been known since early work of J. H. C.Whitehead [Whi35] that the
topology of open 3-manifolds is much more complicated. Directly relevant
to the present paper are counterexamples of P. Scott [ST89] and the third
author [Mai08] which show that Kneser’s theorem fails to generalise to open
manifolds, even if one allows infinite connected sums.

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The refference for the article of Maillot is the followning.

2) Some open 3-manifolds and 3-orbifolds
without locally finite canonical decompositions. http://arxiv.org/PS_cache/arxiv/pdf/0802/0802.1438v2.pdf

Here is the citation

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Much of the theory of compact 3-manifolds relies on decompositions into
canonical pieces, in particular the Kneser-Milnor prime decomposition [12,
16], and the Jaco-Shalen-Johannson characteristic splitting [10, 11]. These
have led to important developments in group theory [22, 7, 9, 24], and form
the background of W. Thurston’s geometrization conjecture, which has recently
been proved by G. Perelman [19, 20, 21].

For open 3-manifolds, by contrast, there is not even a conjectural description
of a general 3-manifold in terms of geometric ones. Such a description
would be all the more useful that noncompact hyperbolic 3-manifolds are
now increasingly well-understood, thanks in particular to the recent proofs
of the ending lamination conjecture [17, 4] and the tameness conjecture [5, 1].
The goal of this paper is to present a series of examples which show
that naive generalizations to open 3-manifolds of the canonical decomposition
theorems of compact 3-manifold theory are false.