3

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!

j0equ1nn
  • 3,429
  • 1
    A nontrivial class of examples would be: The antipodal map of the sphere of dimension n is homotopic to the identity iff n is odd. See http://planetmath.org/antipodalmaponsnishomotopictotheidentityifandonlyifnisodd – archipelago Oct 04 '13 at 10:22
  • Ah yes, this is a good point. However, I'm really interested in the case where the manifold is a surface (of course there, the antipodal map on the sphere is trivial). At the moment I'm especially interested in the case where the surface is a punctured torus. If we think of the punctured torus as a twice punctured disk, it seems we would want a nontrivial way of introducing twists (such as the ones that make it a fiber over the figure 8 knot complement), but as far as I can tell the mapping class group does not provide this. – j0equ1nn Oct 04 '13 at 20:52
  • Do you know Dehn Twists? (http://en.m.wikipedia.org/wiki/Dehn_twist) This is the right notion of twisting regarding the mcg. In fact, they more or less generate it. – archipelago Oct 05 '13 at 08:45
  • 1
    If you want to go deeper into mcgs, I recommend http://www.math.ethz.ch/~bgabi/Farb%20Magalit%20January%202011%20version.pdf. – archipelago Oct 05 '13 at 08:47
  • That looks like a good reference, thanks, I will check that out a little later. I do know aobut Dehn twists. After thinking about it more, I can think of other homeomorphisms that aren't homotopic to the identity - like for instance the homeomorphism from two unlinked tori to two linked tori. The role of Dehn twists does seem relevant now. It still doesn't resolve the particular example I'm looking at but I will do some reading. – j0equ1nn Oct 05 '13 at 17:59
  • The first part of my question was resolved rather quickly by the book @archipelago linked. Notice how the definition I gave from Ratcliff specified closed surfaces. Also, in my first comment I mentioned wanting to use this on a punctured torus, which is not closed. Well, the more general definition of the mapping class group of a surface specifies that all the maps, including the ones you quotient by, must restrict to the identity on the boundary. This means that cutting and pasting along an arc is no longer trivial, which was the main problem for me. Thanks @archipelago! – j0equ1nn Oct 08 '13 at 01:38
  • I just came across this going through my old account info, and @archipelago: I recommend posting that reference as an answer, if such a thing is okay on here. It may not be very specific but it's what actually resolved my issue! – j0equ1nn Oct 13 '14 at 18:12

1 Answers1

0

As the OP suggested, I just post the reference, which solved his questions, as an answer.

The book "A primer on mapping class groups" is a good introduction to mapping class groups.

archipelago
  • 4,258