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I am wondering whether it is possible to construct any arbitrary connected 4-manifold out of a sequence of surgeries on a simply connected 4-manifold. That is, suppose we are given a simply connected 4-manifold, and a multiply connected 4-manifold. Is it in general possible to construct the latter out of the fomer via a sequence of surgeries?

For example, mathworld states [1] that "Every compact connected 3-manifold comes from Dehn surgery on a link in S^3 (Wallace 1960, Lickorish 1962)." I am looking for a similar statement, but in four dimensions instead of three.

If so, then my next questions:

How many different types of surgeries are there?

Is it possible to construct a set S of generators {g} for the first fundamental group by saying, in effect, that each time we do a surgery, we add a few more generators? In two dimensions, I'm thinking that each surgery results in the addition of two more generators, although I'm not sure about that.

Any help would be appreciated.

David

[1] http://mathworld.wolfram.com/DehnSurgery.html

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# Using surgeries to construct 4-manifolds of arbitrary topology

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