220 \begin{itemize} |
220 \begin{itemize} |
221 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
221 \item For any $b\in \BD_k$ the action map $\Homeo(X) \to \BD_k$, $f \mapsto f(b)$ is continuous. |
222 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
222 \item The gluing maps $\BD_k(M)\to \BD_k(M\sgl)$ are continuous. |
223 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
223 \item For balls $B$, the map $U(B) \to \BD_1(B)$, $u\mapsto (B, u, \emptyset)$, is continuous, |
224 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
224 where $U(B) \sub \bc_0(B)$ inherits its topology from $\bc_0(B)$ and the topology on |
225 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. |
225 $\bc_0(B)$ comes from the generating set $\BD_0(B)$. \nn{This topology is implicitly part of the data of a system of fields, but never mentioned. It should be!} |
226 \end{itemize} |
226 \end{itemize} |
227 |
227 |
228 We can summarize the above by saying that in the typical continuous family |
228 We can summarize the above by saying that in the typical continuous family |
229 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
229 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ and $r(p)$ are induced by a map |
230 $P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. |
230 $P\to \Homeo(X)$, with the twig blob labels $u_i(p)$ varying independently. \nn{``varying independently'' means that \emph{after} you pull back via the family of homeomorphisms to the original twig blob, you see a continuous family of labels, right? We should say this. --- Scott} |
231 We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
231 We note that while we've decided not to allow the blobs $B_i(p)$ to vary independently of the field $r(p)$, |
232 if we did allow this it would not affect the truth of the claims we make below. |
232 if we did allow this it would not affect the truth of the claims we make below. |
233 In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. |
233 In particular, such a definition of $\btc_*(X)$ would result in a homotopy equivalent complex. |
234 |
234 |
235 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |
235 Next we define $\btc_*(X)$ to be the total complex of the double complex (denoted $\btc_{**}$) |