equal
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178 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
178 \bd(b) = \sum_{j=1}^{k} (-1)^{j+1} E_j(b). |
179 } |
179 } |
180 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
180 The $(-1)^{j+1}$ factors imply that the terms of $\bd^2(b)$ all cancel. |
181 Thus we have a chain complex. |
181 Thus we have a chain complex. |
182 |
182 |
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183 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, is immediately obvious from the definition. A homeomorphism acts in an obvious on blobs and on fields. |
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184 |
183 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
185 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
184 to be the union of the blobs of $b$. |
186 to be the union of the blobs of $b$. |
185 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
187 For $y \in \bc_*(X)$ with $y = \sum c_i b_i$ ($c_i$ a non-zero number, $b_i$ a blob diagram), |
186 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
188 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
187 |
189 |