Intuition behind Monodromy Action I am learning a topological concept called monodromy action that I am having difficulty with, may I ask what is the intuition behind this concept and how does it relates to the lifting property? In particular, if I identify the torus with $S^{1} \times S^{1}$, and consider a quotient map from $S^{1} \times S^{1}$ to itself, how should I think about the monodromy action on it?
 A: There are different versions of monodromy. The simplest is that in the situation of covering spaces. So let $\pi:E\rightarrow B$ be a covering space. Given a path $\gamma:[0,1]\rightarrow B$ we know there is a unique lift $\tilde\gamma:[0,1]\rightarrow E$ such that $\gamma=\pi\circ\tilde\gamma$. However, even if $\gamma(0)=\gamma(1)$ there is no reason for $\tilde\gamma(0)=\tilde\gamma(1)$. This defines an action of the fundamental group on the fiber of the point on which you base the fundamental group.
Your example of $S^1\times S^1\rightarrow S^1$ is not a covering space, since it's not a local homeomorphism. A typical example is squaring $sq:S^1\rightarrow S^1$ ($S^1$ are the unit vectors in $\mathbb{C}$, so squaring makes sense). Then the unique lift of $\gamma:[0,1]\rightarrow S^1$ defined by $\gamma(t)=e^{2\pi i t}$ will be $\tilde\gamma(t)=e^{\pi i t}$. So $\gamma$ acts on the fiber, which is $\{e^{\pi i}, 1\}$, by swaping.
Monodromy is always about contexts where you uniquely can lift loops to paths, and then asks what the endpoints are. Apart from covering spaces, this makes sense for principal bundles with connection. In that context the lift is not unique, but there is a preferred one, namely the unique flat lift. So with the structure of a connection, you could make sense of the $S^1\times S^1\rightarrow S^1$ example.
