131 An \emph{$n$-ball decomposition} of a topological space $X$ is 

   132 A \emph{gluing decomposition} of an $n$-manifold $X$ is a sequence of manifolds $M_0 \to M_1 \to \cdots \to M_m = X$ such that $M_0$ is a disjoint union of balls, and each $M_k$ is obtained from $M_{k-1}$ by gluing together some disjoint pair of homeomorphic $n{-}1$-manifolds in the boundary of $M_{k-1}$.

   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 

   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.

   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.

   135 By `a ball in $X$' we don't literally mean a submanifold homeomorphic to a ball, but rather the image of a map from the pair $(B^n, S^{n-1})$ into $X$, which is an embedding on the interior. The boundary of a ball in $X$ is the image of a locally embedded $n{-}1$-sphere. \todo{examples, e.g. balls which actually look like an annulus, but we remember the boundary} 

   136 

   139 Given an $n$-dimensional system of fields $\cF$, its extension to a topological space $X$ admitting an $n$-ball decomposition is \todo{}

   138 A \emph{ball decomposition} of an $n$-manifold $X$ is a collection of balls in $X$, such that there exists some gluing decomposition $M_0  \to \cdots \to M_m = X$ so that the balls are the images of the components of $M_0$ in $X$. 

   141 \todo{This is well defined}

   140 In particular, the union of all the balls in a ball decomposition comprises all of $X$. \todo{example}

   142 

   141 

   143 Before describing the general case we should say more precisely what we mean by 

   142 We'll now slightly restrict the possible configurations of blobs.

   144 disjoint and nested blobs.

   143 \begin{defn}

   145 Two blobs are disjoint if they have disjoint interiors.

   144 A configuration of $k$ blobs in $X$ is a collection of $k$ balls in $X$ such that there is some gluing decomposition $M_0  \to \cdots \to M_m = X$ of $X$ and each of the balls is the image of some connected component of one of the $M_k$. Such a gluing decomposition is \emph{compatible} with the configuration.

   146 Nested blobs are allowed to have overlapping boundaries, or indeed to coincide.

   145 \end{defn}

   147 Blob are allowed to meet $\bd X$.

   146 In particular, this means that for any two blobs in a configuration of blobs in $X$, they either have disjoint interiors, or one blob is strictly contained in the other. We describe these as disjoint blobs and nested blobs. Note that nested blobs may have boundaries that overlap, or indeed coincide. Blobs may meet the boundary of $X$.

   148 

   147 

   149 However, we require of any collection of blobs $B_1,\ldots,B_k \subseteq X$ that

   148 Note that the boundaries of a configuration of $k$-blobs may cut up in manifold $X$ into components which are not themselves manifolds. \todo{example: the components between the boundaries of the balls may be pathological}

   150 $X$ is decomposable along the union of the boundaries of the blobs.

   149 

   151 \nn{need to say more here.  we want to be able to glue blob diagrams, but avoid pathological

   150 \todo{this notion of configuration of blobs is the minimal one that allows gluing and engulfing}

   152 behavior}

   153 \nn{need to allow the case where $B\to X$ is not an embedding

   154 on $\bd B$.  this is because any blob diagram on $X_{cut}$ should give rise to one on $X_{gl}$

   155 and blobs are allowed to meet $\bd X$.

   156 Also, the complement of the blobs (and regions between nested blobs) might not be manifolds.}