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.

Let $S$ be a surface with non-empty boundary $\partial S$ and let $f$ be an element of $\mathrm{MCG}(S)$, 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 $\mathrm{MCG}(S)$ 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.

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.

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.