In the paper "Geometry of the complex of curves II: Hierarchical structure" ([Paper](http://arxiv.org/abs/math/9807150)) there is a construction of curve complex for an Annular subdomain (2.4). The construction depends on the domain itself but the definition of domain consist of isotopy classes of subsurfaces. So I have the following doubts.


**1) Why the construction is unique upto isotopy?**


**2) Curves are considered upto isotopy, then what do they mean by 'The' core curve?**


**3) The vertices of curve complex are isotopy classes of curves. Then why the function $\pi\_Y$ defined in the next paragraph is well defined?** 


PS: In para 2.3 they said that "to make discussion clear one *might* fix a complete hyperbolic metric on $S$." So does this metric assumption is necessary for those constructions or we can do them topologically. 



A doubt from "Geometry of the complex of curves II: Hierarchical structure" by Masur and Minsky

> 
> 1) Why the construction is unique upto isotopy?
> 
> 
> 


Recall that $S$ is the surface and $Y$ is the annular subsurface. Masur and Minsy take an annular cover $\tilde{Y}$ and work there. The cover $\tilde{Y}$ depends only on the free homotopy class of any core curve $\alpha \subset Y$. 



> 
> 2) Curves are considered upto isotopy, then what do they mean by 'The'
>  core curve?
> 
> 
> 


They define "curve" in the first sentence of page 12. Also, note that they write $\alpha \in C\_0(S)$, so they mean the isotopy class. 


It is standard practice to use the same notation for curves and their isotopy classes. Tracking the difference between $\alpha$ and $[\alpha]$ is often not necessary. I grant that this abuse of notation can cause confusion the first time you see it. 



> 
> 3) The vertices of curve complex are isotopy classes of curves. Then
>  why the function $\pi\_Y$ defined in the next paragraph is well defined?
> 
> 
> 


The definition of the "curve complex" $C(Y)$ is altered when $Y$ is an annulus. Instead of using isotopy classes of essential curves they use isotopy classes rel endpoints of essential arcs in the Gromov closure of $\tilde{Y}$. The classes downstairs (in $S$) lift to the correct classes upstairs (in $\hat{Y}$, the Gromov closure). 


