ScalingStacks

Definition 3.13 (Collar-gluing). A collar-gluing among $B$-framed $n$-manifolds is a continuous map

$$f\colon M\to[-1,1]$$

to the closed interval for which the restriction $f_{|}\colon M_{|(-1,1)}\to(-1,1)$ is a manifold bundle. We will often denote a collar-gluing $M\xrightarrow{f}[-1,1]$ simply as the open cover

$$M^{\prime}\underset{M_{0}\times\mathbb{R}}{\bigcup}M^{\prime\prime}~\cong~M$$

where $M^{\prime}=f^{-1}[-1,1)$ and $M^{\prime\prime}=f^{-1}(-1,1]$ and $M_{0}=f^{-1}\{0\}$.