Monodromy when fiber is connected

Question

When I learned about monodromy, it was in the context of covering spaces. There, if $p$ is the covering map and $l$ is a loop with $\gamma(0)=\gamma(1)=x$, then a $\bar \gamma$ lift of $\gamma$ is uniquely determined by $\bar \gamma (0)$ and the monodromy $m$ map is given by $m(\bar \gamma(0))=\bar \gamma(1)$.
The point is; the fact that the fibers of $p$ are discrete ensures that the lift is unique.

So what happens in the case of a fiber bundle with connected fiber? Isn't it then impossible to talk about monodromy, because if the fiber of $p$ is (path-)connected, then for every $y$ in the fiber, there is a path $\sigma_y$ connecting $\bar \gamma(1)$ with $y$. Hence for every $y$ in the fiber, the concatination $\sigma_y \cdot \bar\gamma$ is a lift of $\gamma$ with starting point $\bar \gamma (0)$ and endpoint $y$.

I wonder because I read about monodromy of mapping tori (see here for example) and I want to make sense of it.

Answer 1

You're close to right. You say that the point is that "the fibers are discrete", which ensures the lift is unique.

That's almost true, but two things are involved: the fibers' discreteness, and the continuity of the bundle-section, i.e., of the lifted curve. Together these make the lift unique.

More generally, you can say "I have a path $\gamma$ in the base space; I have some covering path $\overline{\gamma}$ that has some property that produces a unique covering path, and then I say something about the endpoints of $\overline{\gamma}$.

Here's an example: the base space is the unit sphere, which I'll describe as the earth (or to be a little smaller, a "globe" representing the earth, but small enough for you to pick up). Paths are ... paths. The bundle consists, at each point, of all oriented orthonormal frames on the tangent space (i.e., it's the frame bundle of the tangent bundle). We can make a simple picture of this by drawing, for any frame, only the first of the two vectors...and then you have just the tangent bundle. So a "lift" of a path is simple a tangent-vector-field-along that path. If you allow arbitrary continuous lifts, then as you pointed out in your question, anything is possible -- there's no uniqueness.

Now I'm going to describe a property much loved by geometers. Start out with a large table and imagine that on its surface are drawn tons of little arrows, all pointing the same direction. You might as well imagine that the surface is covered in slightly soft rubber, for reasons you'll see in a moment.

You've got a nice path $\gamma$ -- let's say it's a loop so that $\gamma(0) = \gamma(1)$ drawn on the sphere, and a vector $v$ at $\gamma(0)$, so that the pair $(v, v^\perp)$ (where that denotes $v$ rotated 90 degrees CCW in the tangent plane) is $\overline{gamma}(0)$. You'd like to get a vector at every other point of $\gamma$. Here's how you do it:

Place your sphere so that $\gamma(0)$ is touching the table, and $v$ is aligned with the arrows on the table. Now roll the sphere, without slipping or turning (the rubber surface will help determine what this means), along the table so that at time $t$, $\gamma(t)$ is touching the table surface. Then at each time $t$, lift the arrow from the table surface up to get an arrow $\overline{gamma}(t)$ in the tangent space at $\gamma(t)$. When you're all done, you'll have an arrow $w = \overline{\gamma}(1)$ that's in the same tangent space as $v$ was, but possibly pointing some other direction. The degree of direction shift is the holonomy.

If your curve $\gamma$ traverses the equator (or any great circle), the holonomy will be zero. If it traverses a line of latitude other than the equator, it'll be nonzero. (You can test this with a ping-pong ball rolled along the bottom of a rubberized mouse-pad, for instance.)

In this example, the "lifting criterion" was "parallel translation", and there's only one parallel translate of a vector along a curve in a Riemannian manifold. The holonomy induced by parallel translation is closely tied to the curvature of the area enclosed by the curve, which is generally pretty interesting. If the lifting criterion had been mere continuity, then there'd have been no notion of holonomy, because of the lack of unique-lifting.

Answer 2

You can talk about monodromy or holonomy if the fiber is connected. For this you need to use the notion of connection on a fiber bundle $p:P\rightarrow M$ whose fibre is connected. It is a distribution on $P$, that is for every $y\in P$, a subspace $D_y$ of the tangent space of $y$ such that $dim D_y=dim M$ and the restriction of $dp_y$ to $D_y$ is injective. You can then lift paths on $M$ to paths on $P$ tangent to $D$, if $c:I=[0,1]\rightarrow M$ is path, its lift satisfies $\bar c'(t)\in D_{\bar c(t)}$.

If $p$ is a covering and the fibre is discrete, then the connection is a flat connection, this is equivalent to the fact that it is defined by a foliation transverse to the fibre.