# Mazur and contractible manifolds

A Mazur manifold is a contractible, compact, smooth $$4$$-manifold with boundary a homology $$3$$-sphere.

It is built from a single $$0$$-handle, a single $$1$$-handle and single $$2$$-handle. It is equivalent that the $$4$$-manifold must be of the form $$\displaystyle S^{1}\times D^{3}$$ union a $$2$$-handle. (Handles are all $$4$$-dimensional.) The following picture is from Akbulut and Durusoy's paper: Here, $$W$$ is a Mazur manifold with the boundary Brieskorn sphere $$\Sigma(2,5,7)$$. The dark blacked dotted circle shows the $$1$$-handle which can be drawn as $$0$$-framing unknot.

Q1. Are we free how we attach $$2$$-handles to $$S^1 \times D^3$$?

Q2. For example, the following picture describes a Mazur manifold? There are contractible $$4$$-manifolds built with a $$0$$-handle, two $$1$$-handles, and two $$2$$-handles. They are the examples of Stern.

Q3. Do we know the classification of contractible $$4$$-manifolds in terms of their handle numbers?

• I would like to point out a few things here...Whenever we talked about "Mazur type" manifolds, there is a natural involution involved associate with the Kirby picture, which is by interchanging the dotted 1 handle with 2-handle. Notice that, for this reason the framing of the 2-handle has to be 0. So the picture in Q2 is not Mazur-type picture. Also I don't think any classification is known as you asked in Q3. If you just consider a contractible 4 manifold then you can attach the 2 handle whatever way you want as long as it is simplyconnected. – Anubhav Mukherjee Sep 27 '20 at 18:24

About terminology: wikipedia defines a Mazur manifold as a contractible compact smooth 4-manifold that is not diffeomorphic to the 4-ball. (It follows from this definition that the boundary of such a manifold is automatically an integral homology sphere.) It also says that frequently the definition is restricted to manifolds constructed with only one handles of each index 0, 1, and 2. I will stick to this latter definition, for consistency with your question(s).

What Anubhav refers to in his comment is a further restriction, but I wouldn't put it in the definition of a Mazur manifold (but rather speak of a Mazur cork if you have such an involution).

Now that we all agree on the objects, let's get to the questions.

Q1. No, we're not free. Suppose you have a presentation of a 4-manifold $$W$$ with one 1-handle and one 2-handle. The 1-handle gives you a generator in the presentation of the fundamental group of the 4-manifold, and the attaching circle of the 2-handle gives you a relation (which is the only relation, since you only have one 2-handle). If you want the fundamental group of $$W$$ to be trivial, you better have that the relation kills the generator, which translates to the attaching circle generating the homology of $$S^1\times S^2$$. Diagrammatically, you're asking for the linking number between the dotted circle and the framed knot to be ±1.

This is almost the only restriction, except that you need to check that the 4-manifold is not the 4-ball, but it follows from the property R that there is only one such knot. This is actually a good segue for question 2.

Q2. No, these are not Mazur manifolds. Even interpreting the 0-framed 2-handle as a 1-handle (otherwise you don't even have the right homology groups), this is just $$B^4$$: the 2-handle geometrically cancels the 1-handle.

Q3. I highly doubt it, and I don't think the question can have a "nice answer". These questions about 4-manifolds are usually incredibly hard.