Hello all,(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Using surgeries to construct 4-manifolds of arbitrary topology

Loading...

Similar Threads for Using surgeries construct | Date |
---|---|

I Notations used with vector field and dot product | Jan 22, 2017 |

I How to write the Frenet equations using the vector gradient? | Jun 24, 2016 |

How do I use the geodesic equation for locations on earth | Dec 19, 2015 |

Constructing S^3 from a S^2 and a bunch of S^1's? | Apr 4, 2015 |

**Physics Forums - The Fusion of Science and Community**