r=r$). We only need to remember $(O,E)$ (for the case without corners), and all the related constructions using a choice of $r$ with $O=r(U)$ are independent of such choices. We will omit $E$ in the notation and write retract just as $O$, but this means there is an associated chosen sc-Banach space $E\supset O$[^6] giving rise to a smooth structure on $O$ and all the relevant discussions (e.g. sc-smoothness of a map from $O$ and its tangent) are done via $E$. $O$ is called a **retract**, and in general it cannot be locally modelled by an sc-Banach space.

In fact gluing of (not necessarily $J$-holomorphic) stable maps (see [@PolyfoldGW]) can be described as a retract, and when we turn on the gluing parameter $a\not=0$, the modulus of which is converted via a **gluing profile** into the length of the glued domain cylinder, and two half cylinder maps (arised by localizing near nodes) are twist-glued together into a finite cylinder map, and this process is denoted by $\oplus_a$ which is a linear map from two unglued half cylinders into a twist-glued cylinder (and is the identity when $a=0$). The end regions of maps are not used in gluing, hence the dimension 