133 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
133 For each $i$ and $j$, we require that either $B_i$ and $B_j$ have disjoint interiors or |
134 $B_i \sub B_j$ or $B_j \sub B_i$. |
134 $B_i \sub B_j$ or $B_j \sub B_i$. |
135 (The case $B_i = B_j$ is allowed. |
135 (The case $B_i = B_j$ is allowed. |
136 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
136 If $B_i \sub B_j$ the boundaries of $B_i$ and $B_j$ are allowed to intersect.) |
137 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
137 If a blob has no other blobs strictly contained in it, we call it a twig blob. |
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138 \nn{need to allow the case where $B\to X$ is not an embedding |
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139 on $\bd B$. this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$ |
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140 and blobs are allowed to meet $\bd X$.} |
138 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
141 \item Fields (boundary conditions) $c_i \in \cC(\bd B_i)$. |
139 (These are implied by the data in the next bullets, so we usually |
142 (These are implied by the data in the next bullets, so we usually |
140 suppress them from the notation.) |
143 suppress them from the notation.) |
141 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
144 $c_i$ and $c_j$ must have identical restrictions to $\bd B_i \cap \bd B_j$ |
142 if the latter space is not empty. |
145 if the latter space is not empty. |
186 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
189 \bd(b) = \sum_{j=0}^{k-1} (-1)^j E_j(b). |
187 } |
190 } |
188 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
191 The $(-1)^j$ factors imply that the terms of $\bd^2(b)$ all cancel. |
189 Thus we have a chain complex. |
192 Thus we have a chain complex. |
190 |
193 |
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194 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
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195 to be the union of the blobs of $b$. |
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196 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
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197 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
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198 |
191 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
199 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
192 but with simplices replaced by a more general class of combinatorial shapes. |
200 but with simplices replaced by a more general class of combinatorial shapes. |
193 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
201 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
194 and cones, and which contains the point. |
202 and cones, and which contains the point. |
195 We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |
203 We can associate an element $p(b)$ of $P$ to each blob diagram $b$ |