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 $\BD_0(B)$. |
226 \nn{don't we need to say more to specify a topology on an $\infty$-dimensional vector space} |
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227 \end{itemize} |
226 \end{itemize} |
228 |
227 |
229 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 |
230 $P\to \BD_k(X)$, $p\mapsto \left(B_i(p), u_i(p), r(p)\right)$, $B_i(p)$ 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 |
231 $P\to \Homeo(X)$, 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. |
275 the same value (namely $r(y(p))$, for any $p\in P$). |
274 the same value (namely $r(y(p))$, for any $p\in P$). |
276 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
275 Let $e(y - r(y)) \in \btc_{1j}$ denote the $j$-parameter family of 1-blob diagrams |
277 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
276 whose value at $p\in P$ is the blob $B^n$ with label $y(p) - r(y(p))$. |
278 Now define, for $y\in \btc_{0j}$, |
277 Now define, for $y\in \btc_{0j}$, |
279 \[ |
278 \[ |
280 h(y) = e(y - r(y)) + c(r(y)) . |
279 h(y) = e(y - r(y)) - c(r(y)) . |
281 \] |
280 \] |
282 |
281 |
283 We must now verify that $h$ does the job it was intended to do. |
282 We must now verify that $h$ does the job it was intended to do. |
284 For $x\in \btc_{ij}$ with $i\ge 2$ we have |
283 For $x\in \btc_{ij}$ with $i\ge 2$ we have |
285 \begin{align*} |
284 \begin{align*} |
288 &= \bd_b(e(x)) + e(\bd_b x) && \text{(since $\bd_t(e(x)) = e(\bd_t x)$)} \\ |
287 &= \bd_b(e(x)) + e(\bd_b x) && \text{(since $\bd_t(e(x)) = e(\bd_t x)$)} \\ |
289 &= x . && |
288 &= x . && |
290 \end{align*} |
289 \end{align*} |
291 For $x\in \btc_{1j}$ we have |
290 For $x\in \btc_{1j}$ we have |
292 \begin{align*} |
291 \begin{align*} |
293 \bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) + c(r(\bd_b x)) - e(\bd_t x) && \\ |
292 \bd h(x) + h(\bd x) &= \bd_b(e(x)) + \bd_t(e(x)) + e(\bd_b x - r(\bd_b x)) - c(r(\bd_b x)) - e(\bd_t x) && \\ |
294 &= \bd_b(e(x)) + e(\bd_b x) && \text{(since $r(\bd_b x) = 0$)} \\ |
293 &= \bd_b(e(x)) + e(\bd_b x) && \text{(since $r(\bd_b x) = 0$)} \\ |
295 &= x . && |
294 &= x . && |
296 \end{align*} |
295 \end{align*} |
297 For $x\in \btc_{0j}$ with $j\ge 1$ we have |
296 For $x\in \btc_{0j}$ with $j\ge 1$ we have |
298 \begin{align*} |
297 \begin{align*} |
299 \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) - \bd_t(e(x - r(x))) - \bd_t(c(r(x))) + |
298 \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) - \bd_t(e(x - r(x))) + \bd_t(c(r(x))) + |
300 e(\bd_t x - r(\bd_t x)) + c(r(\bd_t x)) \\ |
299 e(\bd_t x - r(\bd_t x)) - c(r(\bd_t x)) \\ |
301 &= x - r(x) - \bd_t(c(r(x))) + c(r(\bd_t x)) \\ |
300 &= x - r(x) + \bd_t(c(r(x))) - c(r(\bd_t x)) \\ |
302 &= x - r(x) + r(x) \\ |
301 &= x - r(x) + r(x) \\ |
303 &= x. |
302 &= x. |
304 \end{align*} |
303 \end{align*} |
305 Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, |
304 Here we have used the fact that $\bd_b(c(r(x))) = 0$ since $c(r(x))$ is a $0$-blob diagram, |
306 as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ |
305 as well as that $\bd_t(e(r(x))) = e(r(\bd_t x))$ |
307 \nn{explain why this is true?} |
306 and $\bd_t(c(r(x))) - c(r(\bd_t x)) = r(x)$. |
308 and $c(r(\bd_t x)) - \bd_t(c(r(x))) = r(x)$ \nn{explain?}. |
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309 |
307 |
310 For $x\in \btc_{00}$ we have |
308 For $x\in \btc_{00}$ we have |
311 \begin{align*} |
309 \begin{align*} |
312 \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ |
310 \bd h(x) + h(\bd x) &= \bd_b(e(x - r(x))) + \bd_t(c(r(x))) \\ |
313 &= x - r(x) + r(x) - r(x)\\ |
311 &= x - r(x) + r(x) - r(x)\\ |