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talks/201101-Teichner/notes.tex
changeset 698 f2a4a0788f06
parent 696 b98e16c7a2ba
child 700 172cf5fc2629
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696:b98e16c7a2ba 698:f2a4a0788f06 
   222 
   222 
   223 \begin{thm}
 
   223 \begin{thm}
 
   224 $$A(X \bigcup_Y \selfarrow) \iso A(X) \Tensor_{A(Y)} \selfarrow$$
 
   224 $$A(X \bigcup_Y \selfarrow) \iso A(X) \Tensor_{A(Y)} \selfarrow$$
 
   225 \end{thm}
 
   225 \end{thm}
 
   226 \begin{proof}
 
   226 \begin{proof}
 
   227 Certainly there is a map $A(X) \selfarrow \to A(X \bigcup_Y \selfarrow)$. We send an element of $A(X)$ to the corresponding `glued up' element of $A(X \bigcup_Y \selfarrow)$. This is well-defined since $\cU(X)$ maps into $\cU(X \bigcup_Y \selfarrow)$. This map descends down to a map
 
   227 Certainly there is a map $A(X) \to A(X \bigcup_Y \selfarrow)$. We send an element of $A(X)$ to the corresponding `glued up' element of $A(X \bigcup_Y \selfarrow)$. This is well-defined since $\cU(X)$ maps into $\cU(X \bigcup_Y \selfarrow)$. This map descends down to a map
 
   228 $$A(X) \Tensor_{A(Y)} \selfarrow \to A(X \bigcup_Y \selfarrow)$$
 
   228 $$A(X) \Tensor_{A(Y)} \selfarrow \to A(X \bigcup_Y \selfarrow)$$
 
   229 since the fields $ev$  and $ve$ (here $e \in A(Y), v \in A(X)$) are isotopic on $X \bigcup_Y \selfarrow$ (see Figure \ref{fig:ev-ve}).
 
   229 since the fields $ev$  and $ve$ (here $e \in A(Y), v \in A(X)$) are isotopic on $X \bigcup_Y \selfarrow$ (see Figure \ref{fig:ev-ve}).
 
   230 
   230 
   231 \begin{figure}[!ht]
 
   231 \begin{figure}[!ht]
 
   232 $$
 
   232 $$
 
   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
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