Understanding Monodromy by examples

What is the intuition behind "monodromy"? Could you explain with some examples?

For instance, what does it mean "monodromy around a singular fiber is a dehn twist"

I don't understand what it means pictorially.

• Have you tried wikipedia or the online encyclopedia of mathematics? Also, maybe this is helpful: math.berkeley.edu/~auroux/277F09/L9.pdf
– Piotr Achinger
Feb 20, 2014 at 3:53
• @PiotrAchinger yes, I read the definitions out there, but still It's not clear to me. Thanks though! Feb 20, 2014 at 3:56
• How can we possibly help you if you don't tell us where you got lost? Feb 20, 2014 at 4:20

I would take a step back and look at the root of monodromy (many math terms can be better understood when looking at their meaning from a Latin / Greek perspective). In this sense, monodromy means running round singly''.

But what does that tell us in a mathematical perspective? In essence, this is nothing more than running around a singularity, or more precisely, a ramification point (that is when we consider the pull-back, we are looking at some branchpoint where a bunch of fibers are being pinched together).

We want to understand the type of action that is occurring when we circle a ramification point. If we consider a homotopy that moves around the ramification point and consider the lifting of this homotopy on our fibers, we see then that monodromy permutes where we land on our fibers as we move around the ramification point.

Consider the following example.

$f(x) = x^2 + t$. For generic values of $t$, we have $2$ solutions, but at $t = 0$, we have a ramification point.

Let's just say $t = 1$. So, now we have the system $f(x) = x^2 + 1$, which has two solutions $\pm i$. Let's consider the positive solution, $i$. Now, let's construct a homotopy that circles around our ramification point of $t = 0$, parametrized by $e^{i\tau}$, for $\tau = 0$ to $2\pi$. After one `loop', we have that our positive solution at $i$ gets permuted to the negative solution at $-i$. Circling around again, we see that the action becomes closed and we return to our original point.

A singular fiber (or how I can best understand what is being meant by this) is a branchpoint, i.e. some point in the pull-back where the generic number of solutions is deficient.

In the sense of a dehn twist, when you look at the lifting of the homotopy along the particular fiber that it is tracking is the dehn twist, i.e. the path that you follow when you swap fibers as you move around the branch point. Note that the projection of the dehn twist is precisely the circle that is parametrized by the homotopy that circles around the ramification point.

I hope this helps.

• Makes sense, totally! Could you explain why "monodromy around a singular fiber is a dehn twist" in the same sense? Feb 20, 2014 at 5:38

A Dehn twist is an operation so we need a before an after. In the picture below imagine the top and bottom of each cylinder are identified so that the end points of the red curve are identified, the end points of the blue curve are identified, and there is no twisting of the red curve on the left, e.g the red and blue curves are unlinked. Here is my guess at what is meant by "monodromy around a singular fiber is a dehn twist". I'll admit I am guessing here a little because I don't have the full context.

The Dehn twist operation will be to take the red curve and replace a neighborhood of it near the green curve with a curve that goes once around the cylinder. The singular fiber is meant to be blue. If we consider the two curves parallel and unlinked on the left, then on the right the linking number of the red and blue curves is 1, i.e the red curve goes once around the singular fiber.

The simplest situation is when your fibration is actually a covering space and the simplest nontrivial covering space is given by $f: S^1\to S^1$ where $f(z)=z^2$. Roughly speaking this map wraps the upstairs (domain) circle around the downstairs (target) circle twice. If you start from any point downstairs and make a full rotation, the corresponding point upstairs makes a half rotation whicn is the same thing as the effect of the antipodal map. So, one says that the monodromy of this fibration equals the antipodal map!

Now note that you can extend $f$ to a singular fibration from the complex plane to itself where the singularity is the origin. This can give you an intuition as why the monodromy around a singular fiber should be a Dehn twist: The Dehn twist around a circle is the antipodal map when restricted to the circle itself.

Warning: If your fibration is not a Lefschetz fibration (locally given around a singularity by $f(z_1,...z_n)=z_1^2+..+.z_n^2$ then its monodromy around a singularity may not be a Dehn twist.

A general result along that by @Reza that may be of interest to some:

If a 4- manifold admits a genus -g fibration ; $$g \geq 2$$, then it admits a Symplectic structure. See theorem in p.5 of https://www.csun.edu/~tf54692/N16-lf.pdf