2046 |
2046 |
2047 We will show that if the sphere modules are equipped with a ``compatible family of |
2047 We will show that if the sphere modules are equipped with a ``compatible family of |
2048 non-degenerate inner products", then there is a coherent family of isomorphisms |
2048 non-degenerate inner products", then there is a coherent family of isomorphisms |
2049 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
2049 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$. |
2050 This will allow us to define $\cS(X; c)$ independently of the choice of $E$. |
2050 This will allow us to define $\cS(X; c)$ independently of the choice of $E$. |
2051 \nn{also need to (simultaneously) show compatibility with action of homeos of boundary} |
|
2052 |
2051 |
2053 First we must define ``inner product", ``non-degenerate" and ``compatible". |
2052 First we must define ``inner product", ``non-degenerate" and ``compatible". |
2054 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
2053 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image. |
2055 (We assume we are working in the unoriented category.) |
2054 (We assume we are working in the unoriented category.) |
2056 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
2055 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$ |
2218 It follows from the lemma that we can construct an isomorphism |
2217 It follows from the lemma that we can construct an isomorphism |
2219 between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
2218 between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$. |
2220 This construction involves on a choice of simple ``moves" (as above) to transform |
2219 This construction involves on a choice of simple ``moves" (as above) to transform |
2221 $E$ to $E'$. |
2220 $E$ to $E'$. |
2222 We must now show that the isomorphism does not depend on this choice. |
2221 We must now show that the isomorphism does not depend on this choice. |
2223 We will show below that it suffice to check three ``movie moves". |
2222 We will show below that it suffice to check two ``movie moves". |
2224 |
2223 |
2225 The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
2224 The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back. |
2226 The result is equivalent to doing nothing. |
2225 The result is equivalent to doing nothing. |
2227 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
2226 As we remarked above, the isomorphisms corresponding to these two pushes are mutually |
2228 inverse, so we have invariance under this movie move. |
2227 inverse, so we have invariance under this movie move. |
2309 \label{jun23d} |
2308 \label{jun23d} |
2310 \end{figure} |
2309 \end{figure} |
2311 Invariance under this movie move follows from the compatibility of the inner |
2310 Invariance under this movie move follows from the compatibility of the inner |
2312 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2311 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$. |
2313 |
2312 |
2314 The third movie move could be called ``locality" or ``disjoint commutativity". |
2313 %The third movie move could be called ``locality" or ``disjoint commutativity". |
2315 \nn{...} |
2314 %\nn{...} |
2316 |
2315 |
2317 If $n\ge 2$, these three movie move suffice: |
2316 If $n\ge 2$, these two movie move suffice: |
2318 |
2317 |
2319 \begin{lem} |
2318 \begin{lem} |
2320 Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
2319 Assume $n\ge 2$ and fix $E$ and $E'$ as above. |
2321 The any two sequences of elementary moves connecting $E$ to $E'$ |
2320 The any two sequences of elementary moves connecting $E$ to $E'$ |
2322 are related by a sequence of the three movie moves defined above. |
2321 are related by a sequence of the two movie moves defined above. |
2323 \end{lem} |
2322 \end{lem} |
2324 |
2323 |
2325 \begin{proof} |
2324 \begin{proof} |
2326 (Sketch) |
2325 (Sketch) |
2327 Consider a two parameter family of diffeomorphisms (one parameter family of isotopies) |
2326 Consider a two parameter family of diffeomorphisms (one parameter family of isotopies) |
2328 of $\bd X$. |
2327 of $\bd X$. |
2329 Up to homotopy, |
2328 Up to homotopy, |
2330 such a family is homotopic to a family which can be decomposed |
2329 such a family is homotopic to a family which can be decomposed |
2331 into small families which are either |
2330 into small families which are either |
2332 (a) supported away from $E$, |
2331 (a) supported away from $E$, |
2333 (b) have boundaries corresponding to the three movie moves above. |
2332 (b) have boundaries corresponding to the two movie moves above. |
2334 Finally, observe that the space of $E$'s is simply connected. |
2333 Finally, observe that the space of $E$'s is simply connected. |
2335 (This fails for $n=1$.) |
2334 (This fails for $n=1$.) |
2336 \end{proof} |
2335 \end{proof} |
2337 |
2336 |
2338 For $n=1$ we have to check an additional ``global" relations corresponding to |
2337 For $n=1$ we have to check an additional ``global" relations corresponding to |
2339 rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
2338 rotating the 0-sphere $E$ around the 1-sphere $\bd X$. |
2340 \nn{should check this global move, or maybe cite Frobenius reciprocity result} |
2339 \nn{should check this global move, or maybe cite Frobenius reciprocity result} |
2341 |
2340 |
2342 \nn{...} |
2341 \medskip |
|
2342 |
|
2343 We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$. |
|
2344 We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$. |
|
2345 Choosing an equator $E\sub \bd X$ we have |
|
2346 \[ |
|
2347 \cS(X; c) \cong \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) . |
|
2348 \] |
|
2349 We define $f: \cS(X; c) \to \cS(X', f(c))$ to be the tautological map |
|
2350 \[ |
|
2351 f: \cS(X; c; E) \to \cS(X'; f(c); f(E)) . |
|
2352 \] |
|
2353 It is easy to show that this is independent of the choice of $E$. |
|
2354 Note also that this map depends only on the restriction of $f$ to $\bd X$. |
|
2355 In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by |
|
2356 Axiom \ref{axiom:extended-isotopies} of \S\ref{ss:n-cat-def}. |
|
2357 |
|
2358 |
|
2359 \nn{still to do: gluing, associativity, collar maps} |
2343 |
2360 |
2344 \medskip |
2361 \medskip |
2345 \hrule |
2362 \hrule |
2346 \medskip |
2363 \medskip |
2347 |
|
2348 \nn{to be continued...} |
|
2349 \medskip |
|
2350 |
|
2351 |
|
2352 |
|
2353 |
2364 |
2354 |
2365 |
2355 |
2366 |
2356 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
2367 Stuff that remains to be done (either below or in an appendix or in a separate section or in |
2357 a separate paper): |
2368 a separate paper): |