93 \end{proof} |
93 \end{proof} |
94 |
94 |
95 For the next proposition we will temporarily restore $n$-manifold boundary |
95 For the next proposition we will temporarily restore $n$-manifold boundary |
96 conditions to the notation. |
96 conditions to the notation. |
97 |
97 |
98 Let $X$ be an $n$-manifold, $\bd X = Y \cup (-Y) \cup Z$. |
98 Let $X$ be an $n$-manifold, $\bd X = Y \cup Y \cup Z$. |
99 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
99 Gluing the two copies of $Y$ together yields an $n$-manifold $X\sgl$ |
100 with boundary $Z\sgl$. |
100 with boundary $Z\sgl$. |
101 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $-Y$ and $Z$, |
101 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $Y$ and $Z$, |
102 we have the blob complex $\bc_*(X; a, b, c)$. |
102 we have the blob complex $\bc_*(X; a, b, c)$. |
103 If $b = -a$ (the orientation reversal of $a$), then we can glue up blob diagrams on |
103 If $b = a$, then we can glue up blob diagrams on |
104 $X$ to get blob diagrams on $X\sgl$. |
104 $X$ to get blob diagrams on $X\sgl$. |
105 This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
105 This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
106 |
106 |
107 \textbf{Property \ref{property:gluing-map}.}\emph{ |
107 \textbf{Property \ref{property:gluing-map}.}\emph{ |
108 There is a natural chain map |
108 There is a natural chain map |
109 \eq{ |
109 \eq{ |
110 \gl: \bigoplus_a \bc_*(X; a, -a, c) \to \bc_*(X\sgl; c\sgl). |
110 \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl). |
111 } |
111 } |
112 The sum is over all fields $a$ on $Y$ compatible at their |
112 The sum is over all fields $a$ on $Y$ compatible at their |
113 ($n{-}2$-dimensional) boundaries with $c$. |
113 ($n{-}2$-dimensional) boundaries with $c$. |
114 `Natural' means natural with respect to the actions of diffeomorphisms. |
114 `Natural' means natural with respect to the actions of diffeomorphisms. |
115 } |
115 } |