2576 (a) supported away from $E$, or (b) modify $E$ in the simple manner described above. |
2576 (a) supported away from $E$, or (b) modify $E$ in the simple manner described above. |
2577 \end{proof} |
2577 \end{proof} |
2578 |
2578 |
2579 It follows from the lemma that we can construct an isomorphism |
2579 It follows from the lemma that we can construct an isomorphism |
2580 between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
2580 between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
2581 This construction involves on a choice of simple ``moves" (as above) to transform |
2581 This construction involves a choice of simple ``moves" (as above) to transform |
2582 $E$ to $E'$. |
2582 $E$ to $E'$. |
2583 We must now show that the isomorphism does not depend on this choice. |
2583 We must now show that the isomorphism does not depend on this choice. |
2584 We will show below that it suffice to check two ``movie moves". |
2584 We will show below that it suffices to check two ``movie moves". |
2585 |
2585 |
2586 The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
2586 The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
2587 The result is equivalent to doing nothing. |
2587 The result is equivalent to doing nothing. |
2588 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
2588 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
2589 inverse, so we have invariance under this movie move. |
2589 inverse, so we have invariance under this movie move. |
2673 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2673 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2674 |
2674 |
2675 %The third movie move could be called ``locality" or ``disjoint commutativity". |
2675 %The third movie move could be called ``locality" or ``disjoint commutativity". |
2676 %\nn{...} |
2676 %\nn{...} |
2677 |
2677 |
2678 If $n\ge 2$, these two movie move suffice: |
2678 If $n\ge 2$, these two movie moves suffice: |
2679 |
2679 |
2680 \begin{lem} |
2680 \begin{lem} |
2681 Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
2681 Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
2682 Then any two sequences of elementary moves connecting $E$ to $E'$ |
2682 Then any two sequences of elementary moves connecting $E$ to $E'$ |
2683 are related by a sequence of the two movie moves defined above. |
2683 are related by a sequence of the two movie moves defined above. |
2694 (b) have boundaries corresponding to the two movie moves above. |
2694 (b) have boundaries corresponding to the two movie moves above. |
2695 Finally, observe that the space of $E$'s is simply connected. |
2695 Finally, observe that the space of $E$'s is simply connected. |
2696 (This fails for $n=1$.) |
2696 (This fails for $n=1$.) |
2697 \end{proof} |
2697 \end{proof} |
2698 |
2698 |
2699 For $n=1$ we have to check an additional ``global" relations corresponding to |
2699 For $n=1$ we have to check an additional ``global" relation corresponding to |
2700 rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
2700 rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
2701 But if $n=1$, then we are in the case of ordinary algebroids and bimodules, |
2701 But if $n=1$, then we are in the case of ordinary algebroids and bimodules, |
2702 and this is just the well-known ``Frobenius reciprocity" result for bimodules \cite{MR1424954}. |
2702 and this is just the well-known ``Frobenius reciprocity" result for bimodules \cite{MR1424954}. |
2703 |
2703 |
2704 \medskip |
2704 \medskip |