So I'm reading this book, kind of committed to reading the entire thing, and in the section I'm up to the author starts using some language regarding monodromies and mapping class groups. Having never studied that particular topic, I find myself getting stuck on some really basic stuff.
I looked up monodromies in Ratcliff's Foundations of Hyperbolic Manifolds, and found a nice discussion of moduli spaces, including the definition of a mapping class group. I also found some info about monodromies in Bredon's Topology and Geometry and that part I understood, just don't know how it ties in. Ratcliff defines the mapping class group like so:
Let $M$ be a closed surface of non-positive Euler characteristic. Let $\text{Hom}(M)$ be the set of homeomorphisms from $M$ to itself, given a group structure via composition. Let $\text{Hom}_1(M)$ be the subgroup of $\text{Hom}(M)$ consisting of homeomorphisms homotopic to the identity. The mapping class group is then Map$(M):=\text{Hom}(M)/\text{Hom}_1(M)$.
Okay, first embarrassing question: how can a homeomorphism not be homotopic to the identity map? As far as I know, you can deform an object much more drastically using a homotopy than you can using a homeomorphism. The only example I can think of where a homeomorphism of a surface is not homotopic to the identity is when the surface has more than one connected component and the homeomorphism interchanges some of them. I'm sure there must be less trivial examples or this thing would be kind of useless.
Once I understand the mapping class group, I'm interested in how one would use it to get information about a 3-manifold, knowing that it has a particular fibration with a known surface as the fiber. I expect it should give some information about how the surface had to be deformed to get embedded in the manifold, but right now I can't see how it does.
Thanks in advance for any help!