This answer complement the answers of Henry and Algori. 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.
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.
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
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.
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