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Let $S$ be a surface with non-empty boundary $\partial S$ and let $f$ be an element of MCG(S), the Mapping Class Group of S, i.e., the group of self-diffeomorphisms of S up to isotopy, but with $f$ restricting to the identity near the boundary. This map $f: S \rightarrow S $ gives rise to a contact structure on a 3-manifold $M^3$ in the following way:

Using some results from Open Books (OB), we can see $f: S \rightarrow S $ as an abstract open book (AOB). With additional results of OB theory, $f: S \rightarrow S$ gives rise to an actual open book by the process of constructing the mapping torus $S_f$, and then filling-in the boundary tori with solid tori, so that the core circles of the solid tori are the components of the binding of the OB. From this process, we have an "actual" (instead of abstract) OB, and, by Giroux's correspondence between contact structures and open books, this OB gives rise to a contact structure, say, $\theta $, on the resulting open book.

So, overall, we have that an automorphism of the surface $S$ gives rise to a contact structure $\theta$ on a three-manifold $M^3$ (constructed from the abstract open book).

Question: Is there a way of determining the properties of the contact structure $\theta$, resulting from $f: S \rightarrow S$, e.g., whether $\theta$ is tight or overtwisted, whether it is Stein-fillable, etc., from the properties of $f$ (up to isotopy)? To simplify, say $S$ is the genus-g surface, so that MCG($S$) has a generating set of $2g+1$ Dehn twists.

Also, if we were to attach a handle and draw a closed curve $C$ about the handle, giving rise to a Dehn twist about a tubular neighborhood of $C$, affecting the monodromy of the OB. Is it known how this change of monodromy would affect the contact structure associated with the open book?

Thanks for any refs, ideas, etc.

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2 Answers

Tightness/overtwistedness is pretty hard. One place to start would be the paper Right-veering diffeomorphisms of compact surfaces with boundary I by Honda-Kazez-Matic, which gives a necessary and sufficient condition in terms of "right-veering" mapping classes. However, their result is not algorithmic since it requires checking something on all open books, not just a particular given one. Andy Wand has recently claimed to have an algorithm that detects tightness. I don't believe that there is a publicly available preprint yet, but there is an online talk by Wand and also some nice blog posts by Peter Lambert-Cole here and here.

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Regarding your last, non-highlighted question: the operation you describe is called Hopf stabilisation, and it correspond to doing a connected sum of $\theta$ with a contact 3-sphere (the binding is connected-summed to a Hopf link -- whence the name). Therefore, on the manifold level this doesn't change anything. Let's see what happens to the contact structure.

If the Dehn twist along $C$ is negative (or left-handed), the Hopf link is negative (i.e. the two components have linking number -1), and the contact structure on the 3-sphere is overtwisted (with $d_3 = +1/2$) and the resulting contact structure $\theta'$ is overtwisted, with the same Chern class as $\theta$ but with a different $d_3$ (namely, $d_3(\theta') = d_3(\theta)+1$).

If the Dehn twist along $C$ is positive (right-handed), the Hopf link is positive (i.e. the two components have linking number +1), then the contact structure on the 3-sphere is the standard one (the only tight one, which has $d_3 = -1/2$), and this doesn't affect $\theta$ at all. I think that this is also called a Giroux stabilisation.

Giroux proved that two contact structures given by two open books $OB_1 = (S_1,f_1)$, $OB_2 = (S_2,f_2)$ are isotopic if and only if $OB_1$ and $OB_2$ are stably equivalent, that is if and only if they have a common (positive) stabilisation.

I could go on an on about this story for quite some time, but I think that I'm better of referring to Etnyre's classical Lectures on open book decompositions and contact structures.

Regarding Stein fillability, a contact structures is Stein fillable if and only if it has a supporting open book whose monodromy can be factorised as the product of positive Dehn twists. If you want to know more about fillability and Stein surfaces, Ozbagci and Stipsicz's Surgery on contact 3-manifolds and Stein surfaces makes for a really pleasant read (in addition to organising a lot of results, proofs and references).

The converse to the positivity statement above is true in some cases. First Wendl proved that if a planar open book supports a Stein fillable contact structure, then the monodromy is a product of right-handed Dehn twists (see here), and more recently Lisca proved the same result for open books supported by the torus with one puncture (see here).

It is not true in general: in the meantime, Wand, and Baker, Etnyre and Van Horn-Morris found example of monodromies that were not isotopic to a product of right-handed Dehn twist, but whose associated open books support Stein fillable contact structures (see here and here). Some of their examples are of genus 2.

The question for genus-1 open books with disconnected boundary is - as far as I know - still open.

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