103 with all the structure maps above (disjoint union, boundary restriction, etc.). |
103 with all the structure maps above (disjoint union, boundary restriction, etc.). |
104 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
104 Furthermore, if $f: Y\times I \to Y\times I$ is a fiber-preserving homeomorphism |
105 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
105 covering $\bar{f}:Y\to Y$, then $f(c\times I) = \bar{f}(c)\times I$. |
106 \end{enumerate} |
106 \end{enumerate} |
107 |
107 |
108 \nn{need to introduce two notations for glued fields --- $x\bullet y$ and $x\sgl$} |
108 There are two notations we commonly use for gluing. |
109 |
109 One is |
110 \bigskip |
110 \[ |
111 Using the functoriality and $\bullet\times I$ properties above, together |
111 x\sgl \deq \gl(x) \in \cC(X\sgl) , |
|
112 \] |
|
113 for $x\in\cC(X)$. |
|
114 The other is |
|
115 \[ |
|
116 x_1\bullet x_2 \deq \gl(x_1\otimes x_2) \in \cC(X\sgl) , |
|
117 \] |
|
118 in the case that $X = X_1 \du X_2$, with $x_i \in \cC(X_i)$. |
|
119 |
|
120 \medskip |
|
121 |
|
122 Using the functoriality and $\cdot\times I$ properties above, together |
112 with boundary collar homeomorphisms of manifolds, we can define the notion of |
123 with boundary collar homeomorphisms of manifolds, we can define the notion of |
113 {\it extended isotopy}. |
124 {\it extended isotopy}. |
114 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
125 Let $M$ be an $n$-manifold and $Y \subset \bd M$ be a codimension zero submanifold |
115 of $\bd M$. |
126 of $\bd M$. |
116 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |
127 Let $x \in \cC(M)$ be a field on $M$ and such that $\bd x$ is splittable along $\bd Y$. |