97 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $Y$ and $Z$, |
97 Given compatible fields (boundary conditions) $a$, $b$ and $c$ on $Y$, $Y$ and $Z$, |
98 we have the blob complex $\bc_*(X; a, b, c)$. |
98 we have the blob complex $\bc_*(X; a, b, c)$. |
99 If $b = a$, then we can glue up blob diagrams on |
99 If $b = a$, then we can glue up blob diagrams on |
100 $X$ to get blob diagrams on $X\sgl$. |
100 $X$ to get blob diagrams on $X\sgl$. |
101 This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
101 This proves Property \ref{property:gluing-map}, which we restate here in more detail. |
102 \todo{This needs more detail, because this is false without careful attention to non-manifold components, etc.} |
|
103 |
102 |
104 \textbf{Property \ref{property:gluing-map}.}\emph{ |
103 \begin{prop} \label{blob-gluing} |
105 There is a natural chain map |
104 There is a natural chain map |
106 \eq{ |
105 \eq{ |
107 \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl). |
106 \gl: \bigoplus_a \bc_*(X; a, a, c) \to \bc_*(X\sgl; c\sgl). |
108 } |
107 } |
109 The sum is over all fields $a$ on $Y$ compatible at their |
108 The sum is over all fields $a$ on $Y$ compatible at their |
110 ($n{-}2$-dimensional) boundaries with $c$. |
109 ($n{-}2$-dimensional) boundaries with $c$. |
111 ``Natural" means natural with respect to the actions of diffeomorphisms. |
110 ``Natural" means natural with respect to the actions of diffeomorphisms. |
112 } |
111 \end{prop} |
113 |
112 |
114 This map is very far from being an isomorphism, even on homology. |
113 This map is very far from being an isomorphism, even on homology. |
115 We fix this deficit in \S\ref{sec:gluing} below. |
114 We fix this deficit in \S\ref{sec:gluing} below. |