31 Moreover some of the pieces |
31 Moreover some of the pieces |
32 into which we cut manifolds below are not themselves manifolds, and it requires special attention |
32 into which we cut manifolds below are not themselves manifolds, and it requires special attention |
33 to define fields on these pieces. |
33 to define fields on these pieces. |
34 |
34 |
35 We of course define $\bc_0(X) = \cF(X)$. |
35 We of course define $\bc_0(X) = \cF(X)$. |
36 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for each $c \in \cF(\bdy X)$. |
36 In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
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37 |
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38 (If $X$ has nonempty boundary, instead define $\bc_0(X; c) = \cF(X; c)$ for $c \in \cF(\bdy X)$. |
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39 The blob complex $\bc_*(X; c)$ will depend on a fixed boundary condition $c\in \cF(\bdy X)$. |
37 We'll omit such boundary conditions from the notation in the rest of this section.) |
40 We'll omit such boundary conditions from the notation in the rest of this section.) |
38 In other words, $\bc_0(X)$ is just the vector space of all fields on $X$. |
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39 |
41 |
40 We want the vector space $\bc_1(X)$ to capture |
42 We want the vector space $\bc_1(X)$ to capture |
41 ``the space of all local relations that can be imposed on $\bc_0(X)$". |
43 ``the space of all local relations that can be imposed on $\bc_0(X)$". |
42 Thus we say a $1$-blob diagram consists of: |
44 Thus we say a $1$-blob diagram consists of: |
43 \begin{itemize} |
45 \begin{itemize} |
146 These configurations are generated by two operations: |
148 These configurations are generated by two operations: |
147 \begin{itemize} |
149 \begin{itemize} |
148 \item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add |
150 \item For any (possibly empty) configuration of blobs on an $n$-ball $D$, we can add |
149 $D$ itself as an outermost blob. |
151 $D$ itself as an outermost blob. |
150 (This is used in the proof of Proposition \ref{bcontract}.) |
152 (This is used in the proof of Proposition \ref{bcontract}.) |
151 \item If $X'$ is obtained from $X$ by gluing, then any permissible configuration of blobs |
153 \item If $X\sgl$ is obtained from $X$ by gluing, then any permissible configuration of blobs |
152 on $X$ gives rise to a permissible configuration on $X'$. |
154 on $X$ gives rise to a permissible configuration on $X\sgl$. |
153 (This is necessary for Proposition \ref{blob-gluing}.) |
155 (This is necessary for Proposition \ref{blob-gluing}.) |
154 \end{itemize} |
156 \end{itemize} |
155 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
157 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
156 a manifold. |
158 a manifold. |
157 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
159 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
164 C & = [-1,0] \times \setc{(y,z)}{z \sin(1/z) \leq y \leq 1, z \in [0,1]} \\ |
166 C & = [-1,0] \times \setc{(y,z)}{z \sin(1/z) \leq y \leq 1, z \in [0,1]} \\ |
165 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z \sin(1/z), z \in [0,1]}. |
167 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z \sin(1/z), z \in [0,1]}. |
166 \end{align*} |
168 \end{align*} |
167 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. |
169 Here $A \cup B = [0,1] \times [-1,1] \times [0,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [0,1]$. |
168 Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, |
170 Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, |
169 and $\{C\}$ is a valid configuration of blobs in $C \cup D$, |
171 and $\{D\}$ is a valid configuration of blobs in $C \cup D$, |
170 so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. |
172 so we must allow $\{A, D\}$ as a configuration of blobs in $[-1,1]^2 \times [0,1]$. |
171 Note however that the complement is not a manifold. |
173 Note however that the complement is not a manifold. |
172 \end{example} |
174 \end{example} |
173 |
175 |
174 \begin{defn} |
176 \begin{defn} |
175 \label{defn:gluing-decomposition} |
177 \label{defn:gluing-decomposition} |
242 and |
244 and |
243 \begin{defn} |
245 \begin{defn} |
244 \label{defn:blobs} |
246 \label{defn:blobs} |
245 The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all |
247 The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all |
246 configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, |
248 configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, |
247 modulo identifying the vector spaces for configurations that only differ by a permutation of the balls |
249 modulo identifying the vector spaces for configurations that only differ by a permutation of the blobs |
248 by the sign of that permutation. |
250 by the sign of that permutation. |
249 The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of |
251 The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of |
250 forgetting one blob from the configuration, preserving the field $r$: |
252 forgetting one blob from the configuration, preserving the field $r$: |
251 \begin{equation*} |
253 \begin{equation*} |
252 \bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r) |
254 \bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r) |
260 having to say anything about gluing together fields on smaller components. |
262 having to say anything about gluing together fields on smaller components. |
261 |
263 |
262 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
264 Note that Property \ref{property:functoriality}, that the blob complex is functorial with respect to homeomorphisms, |
263 is immediately obvious from the definition. |
265 is immediately obvious from the definition. |
264 A homeomorphism acts in an obvious way on blobs and on fields. |
266 A homeomorphism acts in an obvious way on blobs and on fields. |
265 |
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266 We define the {\it support} of a blob diagram $b$, $\supp(b) \sub X$, |
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267 to be the union of the blobs of $b$. |
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268 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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269 we define $\supp(y) \deq \bigcup_i \supp(b_i)$. |
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270 |
267 |
271 \begin{remark} \label{blobsset-remark} \rm |
268 \begin{remark} \label{blobsset-remark} \rm |
272 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
269 We note that blob diagrams in $X$ have a structure similar to that of a simplicial set, |
273 but with simplices replaced by a more general class of combinatorial shapes. |
270 but with simplices replaced by a more general class of combinatorial shapes. |
274 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |
271 Let $P$ be the minimal set of (isomorphisms classes of) polyhedra which is closed under products |