# Using surgeries to construct 4-manifolds of arbitrary topology

#### straycat

Hello all,

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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#### mathwonk

Homework Helper
wallace has a nice little book, called "differential topology, first steps", out of print, but in some libraries. you might like it. i.e. if you learn the techniques, you can think about your own question better.

#### straycat

Thanks - I'll check it out the next time I make it to the math library. (Hopefully in the next week or so!)

David

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