132 finite collection of triples $\{(B_i, X_i, Y_i)\}_{i=1,\ldots, k}$ where $B_i$ is an $n$-ball, $X_i$ is some topological space, and $Y_i$ is pair of disjoint homeomorphic $n-1$-manifolds in the boundary of $X_{i-1}$ (for convenience, $X_0 = Y_1 = \eset$), such that $X_{i+1} = X_i \cup_{Y_i} B_i$ and $X = X_k \cup_{Y_k} B_k$. |
132 finite collection of triples $\{(B_i, X_i, Y_i)\}_{i=1,\ldots, k}$ where $B_i$ is an $n$-ball, $X_i$ is some topological space, and $Y_i$ is pair of disjoint homeomorphic $n-1$-manifolds in the boundary of $X_{i-1}$ (for convenience, $X_0 = Y_1 = \eset$), such that $X_{i+1} = X_i \cup_{Y_i} B_i$ and $X = X_k \cup_{Y_k} B_k$. |
133 |
133 |
134 Equivalently, we can define a ball decomposition inductively. A ball decomposition of $X$ is a topological space $X'$ along with a pair of disjoint homeomorphic $n-1$-manifolds $Y \subset \bdy X$, so $X = X' \bigcup_Y \selfarrow$, and $X'$ is either a disjoint union of balls, or a topological space equipped with a ball decomposition. |
134 Equivalently, we can define a ball decomposition inductively. A ball decomposition of $X$ is a topological space $X'$ along with a pair of disjoint homeomorphic $n-1$-manifolds $Y \subset \bdy X$, so $X = X' \bigcup_Y \selfarrow$, and $X'$ is either a disjoint union of balls, or a topological space equipped with a ball decomposition. |
135 \end{defn} |
135 \end{defn} |
136 |
136 |
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137 Even though our definition of a system of fields only associates vector spaces to $n$-manifolds, we can easily extend this to any topological space admitting a ball decomposition. |
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138 \begin{defn} |
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139 Given an $n$-dimensional system of fields $\cF$, its extension to a topological space $X$ admitting an $n$-ball decomposition is \todo{} |
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140 \end{defn} |
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141 \todo{This is well defined} |
137 |
142 |
138 Before describing the general case we should say more precisely what we mean by |
143 Before describing the general case we should say more precisely what we mean by |
139 disjoint and nested blobs. |
144 disjoint and nested blobs. |
140 Disjoint will mean disjoint interiors. |
145 Two blobs are disjoint if they have disjoint interiors. |
141 Nested blobs are allowed to coincide, or to have overlapping boundaries. |
146 Nested blobs are allowed to have overlapping boundaries, or indeed to coincide. |
142 Blob are allowed to intersect $\bd X$. |
147 Blob are allowed to meet $\bd X$. |
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148 |
143 However, we require of any collection of blobs $B_1,\ldots,B_k \subseteq X$ that |
149 However, we require of any collection of blobs $B_1,\ldots,B_k \subseteq X$ that |
144 $X$ is decomposable along the union of the boundaries of the blobs. |
150 $X$ is decomposable along the union of the boundaries of the blobs. |
145 \nn{need to say more here. we want to be able to glue blob diagrams, but avoid pathological |
151 \nn{need to say more here. we want to be able to glue blob diagrams, but avoid pathological |
146 behavior} |
152 behavior} |
147 \nn{need to allow the case where $B\to X$ is not an embedding |
153 \nn{need to allow the case where $B\to X$ is not an embedding |