blob: comparison text/ncat.tex

equal deleted inserted replaced

-:6712876d73e0
+:8ed3aeb78778
 We will show that if the sphere modules are equipped with a ``compatible family of
 non-degenerate inner products", then there is a coherent family of isomorphisms
 $\cS(X; c; E) \cong \cS(X; c; E')$ for all pairs of choices $E$ and $E'$.
 This will allow us to define $\cS(X; c)$ independently of the choice of $E$.
-\nn{also need to (simultaneously) show compatibility with action of homeos of boundary}
 First we must define ``inner product", ``non-degenerate" and ``compatible".
 Let $Y$ be a decorated $n$-ball, and $\ol{Y}$ it's mirror image.
 (We assume we are working in the unoriented category.)
 Let $Y\cup\ol{Y}$ denote the decorated $n$-sphere obtained by gluing $Y$ and $\ol{Y}$
 It follows from the lemma that we can construct an isomorphism
 between $\cS(X; c; E)$ and $\cS(X; c; E')$ for any pair $E$, $E'$.
 This construction involves on a choice of simple ``moves" (as above) to transform
 $E$ to $E'$.
 We must now show that the isomorphism does not depend on this choice.
-We will show below that it suffice to check three ``movie moves".
+We will show below that it suffice to check two ``movie moves".
 The first movie move is to push $E$ across an $n$-ball $B$ as above, then push it back.
 The result is equivalent to doing nothing.
 As we remarked above, the isomorphisms corresponding to these two pushes are mutually
 inverse, so we have invariance under this movie move.
 \label{jun23d}
 \end{figure}
 Invariance under this movie move follows from the compatibility of the inner
 product for $B_1\cup B_2$ with the inner products for $B_1$ and $B_2$.
-The third movie move could be called ``locality" or ``disjoint commutativity".
+%The third movie move could be called ``locality" or ``disjoint commutativity".
-\nn{...}
+%\nn{...}
-If $n\ge 2$, these three movie move suffice:
+If $n\ge 2$, these two movie move suffice:
 \begin{lem}
 Assume $n\ge 2$ and fix $E$ and $E'$ as above.
 The any two sequences of elementary moves connecting $E$ to $E'$
-are related by a sequence of the three movie moves defined above.
+are related by a sequence of the two movie moves defined above.
 \end{lem}
 \begin{proof}
 (Sketch)
 Consider a two parameter family of diffeomorphisms (one parameter family of isotopies)
 of $\bd X$.
 Up to homotopy,
 such a family is homotopic to a family which can be decomposed
 into small families which are either
 (a) supported away from $E$,
-(b) have boundaries corresponding to the three movie moves above.
+(b) have boundaries corresponding to the two movie moves above.
 Finally, observe that the space of $E$'s is simply connected.
 (This fails for $n=1$.)
 \end{proof}
 For $n=1$ we have to check an additional ``global" relations corresponding to
 rotating the 0-sphere $E$ around the 1-sphere $\bd X$.
 \nn{should check this global move, or maybe cite Frobenius reciprocity result}
-\nn{...}
+\medskip
+We have now defined $\cS(X; c)$ for any $n{+}1$-ball $X$ with boundary decoration $c$.
+We must also define, for any homeomorphism $X\to X'$, an action $f: \cS(X; c) \to \cS(X', f(c))$.
+Choosing an equator $E\sub \bd X$ we have
+\[
+	\cS(X; c) \cong \cS(X; c; E) \deq \hom_{\cS(E_c)}(\cS(\bd_-X_c), \cS(\bd_+X_c)) .
+\]
+We define $f: \cS(X; c) \to \cS(X', f(c))$ to be the tautological map
+\[
+	f: \cS(X; c; E) \to \cS(X'; f(c); f(E)) .
+\]
+It is easy to show that this is independent of the choice of $E$.
+Note also that this map depends only on the restriction of $f$ to $\bd X$.
+In particular, if $F: X\to X$ is the identity on $\bd X$ then $f$ acts trivially, as required by
+Axiom \ref{axiom:extended-isotopies} of \S\ref{ss:n-cat-def}.
+\nn{still to do: gluing, associativity, collar maps}
 \medskip
 \hrule
 \medskip
-\nn{to be continued...}
-\medskip
 Stuff that remains to be done (either below or in an appendix or in a separate section or in
 a separate paper):

changeset 505	8ed3aeb78778
parent 497	18b742b1b308
child 506	4a23163843a9