152 \end{itemize} |
152 \end{itemize} |
153 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
153 Combining these two operations can give rise to configurations of blobs whose complement in $X$ is not |
154 a manifold. \todo{example} |
154 a manifold. \todo{example} |
155 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
155 Thus will need to be more careful when speaking of a field $r$ on the complement of the blobs. |
156 |
156 |
157 |
157 \begin{example} |
158 %In order to precisely state the general definition, we'll need a suitable notion of cutting up a manifold into balls. |
158 Consider the four subsets of $\Real^3$, |
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159 \begin{align*} |
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160 A & = [0,1] \times [0,1] \times [-1,1] \\ |
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161 B & = [0,1] \times [-1,0] \times [-1,1] \\ |
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162 C & = [-1,0] \times \setc{(y,z)}{z \sin(1/z) \leq y \leq 1, z \in [-1,1]} \\ |
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163 D & = [-1,0] \times \setc{(y,z)}{-1 \leq y \leq z \sin(1/z), z \in [-1,1]}. |
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164 \end{align*} |
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165 Here $A \cup B = [0,1] \times [-1,1] \times [-1,1]$ and $C \cup D = [-1,0] \times [-1,1] \times [-1,1]$. Now, $\{A\}$ is a valid configuration of blobs in $A \cup B$, and $\{C\}$ is a valid configuration of blobs in $C \cup D$, so we must allow $\{A, C\}$ as a configuration of blobs in $[-1,1]^3$. Note however that the complement is not a manifold. |
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166 \end{example} |
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167 |
159 \begin{defn} |
168 \begin{defn} |
160 \label{defn:gluing-decomposition} |
169 \label{defn:gluing-decomposition} |
161 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
170 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds |
162 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
171 $M_0 \to M_1 \to \cdots \to M_m = X$ such that each $M_k$ is obtained from $M_{k-1}$ |
163 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
172 by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$. |
164 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
173 If, in addition, $M_0$ is a disjoint union of balls, we call it a \emph{ball decomposition}. |
165 \end{defn} |
174 \end{defn} |
166 Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$. |
175 Given a gluing decomposition $M_0 \to M_1 \to \cdots \to M_m = X$, we say that a field is splittable along it if it is the image of a field on $M_0$. |
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176 |
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177 In the example above, note that $$A \sqcup B \sqcup C \sqcup D \to (A \cup B) \sqcup (C \cup D) \to [-1,1]^3$$ is a ball decomposition of $[-1,1]^3$, but other sequences of gluings from $A \sqcup B \sqcup C \sqcup D$ to $[-1,1]^3$ have intermediate steps which are not manifolds. |
167 |
178 |
168 We'll now slightly restrict the possible configurations of blobs. |
179 We'll now slightly restrict the possible configurations of blobs. |
169 \begin{defn} |
180 \begin{defn} |
170 \label{defn:configuration} |
181 \label{defn:configuration} |
171 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. Such a gluing decomposition is \emph{compatible} with the configuration. A blob $B_i$ is a twig blob if no other blob $B_j$ maps into the appropriate $M_r'$. \nn{that's a really clumsy way to say it, but I struggled to say it nicely and still allow boundaries to intersect -S} |
182 A configuration of $k$ blobs in $X$ is an ordered collection of $k$ subsets $\{B_1, \ldots B_k\}$ of $X$ such that there exists a gluing decomposition $M_0 \to \cdots \to M_m = X$ of $X$ and for each subset $B_i$ there is some $0 \leq r \leq m$ and some connected component $M_r'$ of $M_r$ which is a ball, so $B_i$ is the image of $M_r'$ in $X$. Such a gluing decomposition is \emph{compatible} with the configuration. A blob $B_i$ is a twig blob if no other blob $B_j$ maps into the appropriate $M_r'$. \nn{that's a really clumsy way to say it, but I struggled to say it nicely and still allow boundaries to intersect -S} |
176 We describe these as disjoint blobs and nested blobs. |
187 We describe these as disjoint blobs and nested blobs. |
177 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
188 Note that nested blobs may have boundaries that overlap, or indeed coincide. |
178 Blobs may meet the boundary of $X$. |
189 Blobs may meet the boundary of $X$. |
179 Further, note that blobs need not actually be embedded balls in $X$, since parts of the boundary of the ball $M_r'$ may have been glued together. |
190 Further, note that blobs need not actually be embedded balls in $X$, since parts of the boundary of the ball $M_r'$ may have been glued together. |
180 |
191 |
181 \todo{Say something reassuring: that 'most of the time' all the regions are manifolds anyway, and you can take the `trivial' gluing decomposition} |
192 Note that often the gluing decomposition for a configuration of blobs may just be the trivial one: if the boundaries of all the blobs cut $X$ into pieces which are all manifolds, we can just take $M_0$ to be these pieces, and $M_1 = X$. |
182 |
193 |
183 In the informal description above, in the definition of a $k$-blob diagram we asked for any collection of $k$ balls which were pairwise disjoint or nested. We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; this is unsatisfactory because that complement need not be a manifold. Thus, the official definition is |
194 In the informal description above, in the definition of a $k$-blob diagram we asked for any collection of $k$ balls which were pairwise disjoint or nested. We now further insist that the balls are a configuration in the sense of Definition \ref{defn:configuration}. Also, we asked for a local relation on each twig blob, and a field on the complement of the twig blobs; this is unsatisfactory because that complement need not be a manifold. Thus, the official definition is |
184 \begin{defn} |
195 \begin{defn} |
185 \label{defn:blob-diagram} |
196 \label{defn:blob-diagram} |
186 A $k$-blob diagram on $X$ consists of |
197 A $k$-blob diagram on $X$ consists of |
187 \begin{itemize} |
198 \begin{itemize} |
188 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
199 \item a configuration $\{B_1, \ldots B_k\}$ of $k$ blobs in $X$, |
189 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
200 \item and a field $r \in \cC(X)$ which is splittable along some gluing decomposition compatible with that configuration, |
190 \end{itemize} |
201 \end{itemize} |
191 such that |
202 such that |
192 the restriction of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. |
203 the restriction $u_i$ of $r$ to each twig blob $B_i$ lies in the subspace $U(B_i) \subset \cC(B_i)$. |
193 \end{defn} |
204 \end{defn} |
194 \todo{Careful here: twig blobs aren't necessarily balls?} |
205 \todo{Careful here: twig blobs aren't necessarily balls?} |
195 (See Figure \ref{blobkdiagram}. \todo{update diagram}) |
206 (See Figure \ref{blobkdiagram}.) |
196 \begin{figure}[t]\begin{equation*} |
207 \begin{figure}[t]\begin{equation*} |
197 \mathfig{.7}{definition/k-blobs} |
208 \mathfig{.7}{definition/k-blobs} |
198 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
209 \end{equation*}\caption{A $k$-blob diagram.}\label{blobkdiagram}\end{figure} |
199 and |
210 and |
200 \begin{defn} |
211 \begin{defn} |
201 \label{defn:blobs} |
212 \label{defn:blobs} |
202 The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum of all configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, modulo identifying the vector spaces for configurations that only differ by a permutation of the balls by the sign of that permutation. The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of forgetting one blob from the configuration, preserving the field $r$: |
213 The $k$-th vector space $\bc_k(X)$ of the \emph{blob complex} of $X$ is the direct sum over all configurations of $k$ blobs in $X$ of the vector space of $k$-blob diagrams with that configuration, modulo identifying the vector spaces for configurations that only differ by a permutation of the balls by the sign of that permutation. The differential $\bc_k(X) \to \bc_{k-1}(X)$ is, as above, the signed sum of ways of forgetting one blob from the configuration, preserving the field $r$: |
203 \begin{equation*} |
214 \begin{equation*} |
204 \bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r) |
215 \bdy(\{B_1, \ldots B_k\}, r) = \sum_{i=1}^{k} (-1)^{i+1} (\{B_1, \ldots, \widehat{B_i}, \ldots, B_k\}, r) |
205 \end{equation*} |
216 \end{equation*} |
206 \end{defn} |
217 \end{defn} |
207 We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams. |
218 We readily see that if a gluing decomposition is compatible with some configuration of blobs, then it is also compatible with any configuration obtained by forgetting some blobs, ensuring that the differential in fact lands in the space of $k{-}1$-blob diagrams. |