263 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
263 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
264 \node[coordinate] (a1) at (-2,1.2) {}; |
264 \node[coordinate] (a1) at (-2,1.2) {}; |
265 \node[coordinate] (a2) at (-2,0) {}; |
265 \node[coordinate] (a2) at (-2,0) {}; |
266 \node[coordinate] (b1) at (2,1.2) {}; |
266 \node[coordinate] (b1) at (2,1.2) {}; |
267 \node[coordinate] (b2) at (2,0) {}; |
267 \node[coordinate] (b2) at (2,0) {}; |
268 \draw (0.5,1.2) -- (a1) arc (270:90:1) -- +(4,0) arc (90:-90:1); |
268 \draw (a1) arc (270:90:1) -- +(4,0) arc (90:-90:1); |
269 \draw (0.5,0) -- (a2) arc (270:90:2.5) -- +(4,0) arc (90:-90:2.5); |
269 \draw (a2) arc (270:90:2.5) -- +(4,0) arc (90:-90:2.5); |
270 |
270 |
271 % end caps |
271 % end caps |
272 \draw (0.5,1.2) arc (90:450:0.3 and 0.6); |
272 \draw (a1) arc (90:450:0.3 and 0.6); |
273 \draw (b1) arc (90:270:0.3 and 0.6); |
273 \draw (b1) arc (90:270:0.3 and 0.6); |
274 \draw[dashed] (b1) arc (90:-90:0.3 and 0.6); |
274 \draw[dashed] (b1) arc (90:-90:0.3 and 0.6); |
275 |
275 |
276 % the donut hole |
276 % the donut hole |
277 \draw (-2.5,4.2) arc (-135:-45:2); |
277 \draw (-2.5,4.2) arc (-135:-45:2); |
278 \draw (-2,3.9) arc (135:45:1.3); |
278 \draw (-2,3.9) arc (135:45:1.3); |
279 |
279 |
280 % dots |
280 % dots |
281 \draw[dotted] (-2,0.6) ellipse (0.3 and 0.6); |
281 \draw[dotted] (-3.7,2.4) ellipse (0.7 and 0.4); |
282 |
282 |
283 % labels |
283 % labels |
284 \node at (1.8,4) {\Large $ev$}; |
284 \node at (1.8,4) {\Large $v$}; |
|
285 \node at (-3.5,1.4) {\Large $e$}; |
285 \end{tikzpicture} |
286 \end{tikzpicture} |
286 }; |
287 }; |
287 \node (ve) at (1,1) { |
288 \node (ve) at (1,1) { |
288 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
289 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
289 \node[coordinate] (a1) at (-2,1.2) {}; |
290 \node[coordinate] (a1) at (-2,1.2) {}; |
290 \node[coordinate] (a2) at (-2,0) {}; |
291 \node[coordinate] (a2) at (-2,0) {}; |
291 \node[coordinate] (b1) at (2,1.2) {}; |
292 \node[coordinate] (b1) at (2,1.2) {}; |
292 \node[coordinate] (b2) at (2,0) {}; |
293 \node[coordinate] (b2) at (2,0) {}; |
293 \draw (a1) arc (270:90:1) -- +(4,0) arc (90:-90:1) -- (-0.5,1.2); |
294 \draw (a1) arc (270:90:1) -- +(4,0) arc (90:-90:1); |
294 \draw (a2) arc (270:90:2.5) -- +(4,0) arc (90:-90:2.5) -- (-0.5,0); |
295 \draw (a2) arc (270:90:2.5) -- +(4,0) arc (90:-90:2.5); |
295 |
296 |
296 % end caps |
297 % end caps |
297 \draw (a1) arc (90:450:0.3 and 0.6); |
298 \draw (a1) arc (90:450:0.3 and 0.6); |
298 \draw (-0.5,1.2) arc (90:270:0.3 and 0.6); |
299 \draw (b1) arc (90:270:0.3 and 0.6); |
299 \draw[dashed] (-0.5,1.2) arc (90:-90:0.3 and 0.6); |
300 \draw[dashed] (b1) arc (90:-90:0.3 and 0.6); |
300 |
301 |
301 % dots |
302 % dots |
302 \draw[dotted] (2,0.6) ellipse (0.3 and 0.6); |
303 \draw[dotted] (3.7,2.4) ellipse (0.7 and 0.4); |
303 |
304 |
304 % the donut hole |
305 % the donut hole |
305 \draw (-2.5,4.2) arc (-135:-45:2); |
306 \draw (-2.5,4.2) arc (-135:-45:2); |
306 \draw (-2,3.9) arc (135:45:1.3); |
307 \draw (-2,3.9) arc (135:45:1.3); |
307 |
308 |
308 % labels |
309 % labels |
309 \node at (1.8,4) {\Large $ve$}; |
310 \node at (1.8,4) {\Large $v$}; |
|
311 \node at (3.5,1.4) {\Large $e$}; |
310 \end{tikzpicture} |
312 \end{tikzpicture} |
311 }; |
313 }; |
312 \node (b) at (0,0) { |
314 \node (b) at (0,0) { |
313 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
315 \begin{tikzpicture}[x=0.5cm,y=0.5cm] |
314 \node[coordinate] (a1) at (-2,1.2) {}; |
316 \node[coordinate] (a1) at (-2,1.2) {}; |
321 % the donut hole |
323 % the donut hole |
322 \draw (-2.5,4.2) arc (-135:-45:2); |
324 \draw (-2.5,4.2) arc (-135:-45:2); |
323 \draw (-2,3.9) arc (135:45:1.3); |
325 \draw (-2,3.9) arc (135:45:1.3); |
324 |
326 |
325 % dots |
327 % dots |
326 \draw[dotted] (-2,0.6) ellipse (0.3 and 0.6); |
328 \draw[dotted] (-3.7,2.4) ellipse (0.7 and 0.4); |
327 \draw[dotted] (2,0.6) ellipse (0.3 and 0.6); |
329 \draw[dotted] (3.7,2.4) ellipse (0.7 and 0.4); |
|
330 \draw[dotted] (0,0.6) ellipse (0.3 and 0.6); |
328 |
331 |
329 % labels |
332 % labels |
330 \node at (1.8,4) {$ve = ev$}; |
333 \node at (1.8,4) {$ve \sim ev$}; |
331 \end{tikzpicture} |
334 \end{tikzpicture} |
332 }; |
335 }; |
333 \draw[->] (a) -- (ev); |
336 \draw[->] (a) -- (ev); |
334 \draw[->] (a) -- (ve); |
337 \draw[->] (a) -- (ve); |
335 \draw[->] (ev) -- (b); |
338 \draw[->] (ev) -- (b); |
336 \draw[->] (ve) -- (b); |
339 \draw[->] (ve) -- (b); |
337 \end{tikzpicture} |
340 \end{tikzpicture} |
338 $$ |
341 $$ |
339 \caption{Isotopic fields on the glued manifold} |
342 \caption{$ve$ and $ev$ differ by a collar shift on the glued manifold} |
340 \label{fig:ev-ve} |
343 \label{fig:ev-ve} |
341 \end{figure} |
344 \end{figure} |
342 |
345 |
343 There is a map the other way, too. There isn't quite a map $\cF(X \bigcup_Y \selfarrow) \to \cF(X)$, since a field on $X \bigcup_Y \selfarrow$ need not be splittable along $Y$. Nevertheless, every field is isotopic to one that is splittable along $Y$, and combining this with the lemma above we obtain a map $\cF(X \bigcup_Y \selfarrow) / (\text{isotopy}) \to A(X) \Tensor_{A(Y)} \selfarrow$. We now need to show that this descends to a map from $A(X \bigcup_Y \selfarrow)$. Consider an field of the form $u \bullet f$, for some ball $B$ embedded in $X \bigcup_Y \selfarrow$ and $u \in \cU(B), f \in \cF(X \bigcup_Y \selfarrow \setminus B)$. Now $B$ might cross $Y$, but we can choose an isotopy of $X \bigcup_Y \selfarrow$ so that it doesn't. Thus $u \bullet f$ is sent to a field in $\cU(X)$, and is zero in $A(X) \Tensor_{A(Y)} \selfarrow$. |
346 There is a map the other way, too. There isn't quite a map $\cF(X \bigcup_Y \selfarrow) \to \cF(X)$, since a field on $X \bigcup_Y \selfarrow$ need not be splittable along $Y$. Nevertheless, every field is isotopic to one that is splittable along $Y$, and combining this with the lemma above we obtain a map $\cF(X \bigcup_Y \selfarrow) / (\text{isotopy}) \to A(X) \Tensor_{A(Y)} \selfarrow$. We now need to show that this descends to a map from $A(X \bigcup_Y \selfarrow)$. Consider an field of the form $u \bullet f$, for some ball $B$ embedded in $X \bigcup_Y \selfarrow$ and $u \in \cU(B), f \in \cF(X \bigcup_Y \selfarrow \setminus B)$. Now $B$ might cross $Y$, but we can choose an isotopy of $X \bigcup_Y \selfarrow$ so that it doesn't. Thus $u \bullet f$ is sent to a field in $\cU(X)$, and is zero in $A(X) \Tensor_{A(Y)} \selfarrow$. |
344 |
347 |