Using surgeries to construct 4-manifolds of arbitrary topology

• straycat
In summary, the conversation discusses the possibility of constructing a multiply connected 4-manifold from a simply connected 4-manifold through a sequence of surgeries. This is similar to the statement that every compact connected 3-manifold can be constructed through Dehn surgery on a link in S^3. The conversation also raises questions about the types of surgeries available and the construction of a set of generators for the first fundamental group through surgeries. A suggestion is made to refer to a book by Wallace for further understanding of the techniques involved.
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

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.

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

David

1. What is a 4-manifold?

A 4-manifold is a topological space that has four dimensions. It is a mathematical object used in topology and geometry to study higher-dimensional shapes and spaces.

2. What is the significance of constructing 4-manifolds of arbitrary topology?

Constructing 4-manifolds of arbitrary topology is significant because it allows us to better understand and classify different types of 4-dimensional spaces. This has implications in various fields such as physics, where 4-manifolds are used to model spacetime, and mathematics, where they are studied for their own sake.

3. How are surgeries used in constructing 4-manifolds?

Surgeries are a technique used in topology to modify a space by cutting it and gluing it back together in a different way. In constructing 4-manifolds, surgeries are used to manipulate the topology of a given space, creating a new 4-manifold with desired properties.

4. What are some challenges in using surgeries to construct 4-manifolds?

One of the main challenges in using surgeries to construct 4-manifolds is that the process can be highly technical and require advanced mathematical knowledge. Additionally, it can be difficult to find surgeries that will produce a desired 4-manifold, as there are often multiple ways to perform a surgery on a given space.

5. What are some applications of using surgeries to construct 4-manifolds?

Aside from its use in understanding and classifying 4-dimensional spaces, the construction of 4-manifolds using surgeries has applications in fields such as topology, differential geometry, and algebraic geometry. It also has potential implications in areas such as robotics and computer graphics, where higher-dimensional spaces are used to model complex systems.

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